# Patent

## Title
Enhanced precise location

## Abstract

A line locator includes a signal detector to detect signals from an underground line; an error modeler that models a phase error in the signal from neighboring underground lines; and an enhanced electromagnetic field modeler that provides a location of the underground line based on the signal and a result from the error modeler.

## Background
CROSS REFERENCE TO RELATED APPLICATIONS
This is a continuation of U.S. application Ser. No. 12/122,136, filed May 16, 2008, which claims priority to Provisional Application No. 60/930,753, filed May 18, 2007, all of which are incorporated herein by reference, in their entirety, for all purposes.


BACKGROUND
1. Field of the Invention
The present invention relates to detection of electromagnetic (EM) signals from targeted concealed conductors and, in particular, to the precise location of such conductors in the presence of signal distortion.
2. Discussion of Related Art
Utility conduits are often buried underground and not readily accessible. It is often necessary to locate these concealed utility conduits in order to repair and replace them. It is also important to know the location of utility conduits so that excavators can avoid them while excavating an area. If these locations are not accurate for an excavation, harmful results may occur such as property damage, serious physical harm to a person, or even death.
In an attempt to overcome these harmful results, there are various ways to locate concealed conduits, for example, using pipe and cable locators. Underground pipe and cable locators (sometimes termed line locators) have existed for many years and are well known. Line locator systems typically include a mobile receiver and a transmitter. The transmitter can be coupled to a target conductor, either by direct electrical connection or through induction, to provide a current signal on the target conductor. The receiver detects and processes signals resulting from the EM field generated at the target conductor as a result of the current signal. The signal detected at the line locator can be a continuous wave sinusoidal signal provided to the target conductor by the transmitter.
The transmitter is often physically separated from the receiver, with a typical separation distance of several meters or in some cases up to many kilometers. The transmitter couples the current signal, whose frequency can be user selected from a selectable set of frequencies, to the target conductor. The frequency of the current signal applied to the target conductor can be referred to as the active locate frequency. The target conductor generates an EM field at the active locate frequency in response to the current signal.
Different location methodologies and underground environments can call for different active frequencies. The typical range of active locate frequencies can be from several Hertz (for location of the target conductor over separation distances between the transmitter and receiver of many kilometers) to 100 kHz or more. Significant radio frequency interference on the EM field detected by the receiver can be present in the environment over this range. Therefore, receivers of line location systems have often included highly tuned filters to preclude interference from outside sources from affecting the measurement of signals at the desired active locate frequency from the target conductor. These filters can be tuned to receive signals resulting from EM fields at each of the selectable active locate frequencies and reject signals resulting from EM fields at frequencies other than the active locate frequencies.
In line location systems, the signal strength parameter determined from detection of the EM field provides a basis for derived quantities of the current signal (i.e., the line current in the targeted conductor), the position of the line locator receiver relative to the center of the conductor, the depth of the conductor from the line locator receiver, and the input to a peak or null indicator (depending on the orientation of the electromagnetic field to which the detector is sensitive). Line location systems measure signal strength at one or more active frequencies, also referred to as measurement channels.
For detection of cables or pipes laying in a continuous path (e.g., buried in a trench or occupying an underground conduit extending over some distance), an assumption is often made that the induced electromagnetic field is concentric around the cable and that signal strength depends only on the local ground conductivity, the depth and horizontal position of the target cable, and the magnitude of AC current flowing in the cable. When this is the case, the EM field at the detector of the line locator, on which the signal strength depends, is said to be free of distortion.
Nearly all locating systems present a “peak” indication that results from a horizontally-oriented coil, with axis orthogonal to the direction of the cable, that has a maximum deflection over the presumed centerline of the cable, assuming the ideal undistorted field. Some locating systems also present a “null” output from a vertically-oriented coil, which has a minimum at the same position, again making the same ideal assumption about the electromagnetic field. Moreover, some locating systems also provide a top-oriented coil to further assist in estimating a conductor lateral position and depth using a non-linear optimization algorithm.
Often in a crowded underground utility environment of metallic pipes and cables, coupling of signals at the active locating frequency from the target conductor to other adjacent underground conductors can occur. These conductors (lines) are not intended to be tracked by the line location system, but coupling of currents from the target conductor to those neighboring conductors through various means (resistive, inductive, or capacitive), termed “bleedover,” can lead a line locator astray such that the operator of the line location system ceases tracking the targeted conductor (e.g., pipe or cable of interest) and instead begins following an adjacent line. In some cases, there may be bias in the targeted conductor's estimated centerline as a result of distortion due to bleedover.
In conventional receivers, it is very difficult to determine whether the receiver is tracking the targeted conductor or whether the receiver is erroneously tracking a neighboring conductor. Therefore, there is a need for refinement of walkover line location systems to allow for more precise locating of an underground conductor in the presence of distortion due to significant bleedover to other conductors. As stated above, the more precise a locating can be, the less likely that property damage, physical injury, or even death will occur due to an erroneous locate.
SUMMARY
Consistent with some embodiments of the present invention, a line locator system used in walkover locate mode is presented that can more accurately provide a precise centerline and depth of the targeted conductor even in the presence of high field distortion due to bleedover.
Some embodiments of the invention provide a line locator including a plurality of coil detectors, each of the plurality of coil detectors oriented to measure a component of an electromagnetic field; circuitry coupled to receive signals from the plurality of coil detectors and provide quadrature signals indicating a complex electromagnetic field strength; a position locator for indicating a position of the line locator; a processor coupled to receive the complex electromagnetic field strength and the position and calculate values related to a target line; a memory configured to stare instructions; and a display coupled to the processor, the display indicating to a user the values related to the target line. Further, the processor executes the instructions for performing the following: modeling an expected complex field strength corresponding to complex field strengths and a phase error measured at the plurality of coil detectors at each of a plurality of positions determined by the position locator with the target line and a hypothesized number of bleedover cables to form a set of hypothesized values corresponding to a set of individual models for the target line for each of the hypothesized number of bleedover cables; determining which of the set of individual models is a best model; and determining parameters related to the target line from the best model.
Some embodiments of the present invention provide a method of locating an underground line. The method includes measuring an electromagnetic field; modeling a phase error based on the presence of lines neighboring the underground target line; and utilizing the phase error in an enhanced electromagnetic field model to locate the target line.
Some embodiments of the present invention also provide a line locator. The line locator includes a signal detector to detect signals from an underground line; an error modeler that models a phase error in the signal from neighboring underground lines; and an enhanced electromagnetic field modeler that provides a location of the underground line based on the signal and a result from the error modeler.
These and other embodiments are further discussed below with reference to the following figures.



BRIEF DESCRIPTION OF THE DRAWINGS
FIG. 1 illustrates the aboveground use of a locate receiver consistent with some embodiments of the present invention to locate the position of an underground targeted conductor in the presence of bleedover to an adjacent conductor.
FIG. 2 illustrates a block diagram of a locator receiver consistent with some embodiments of the present invention.
FIG. 3 illustrates the processing flow for a locator consistent with some embodiments of the present invention.
FIGS. 4A & 4B are diagrammatic representations of a walkover locate process where the induced electromagnetic field on a target conductor is distorted by a field on a bleedover conductor.
FIGS. 5A & 5B illustrate a component of a display for a line locator consistent with the present invention.
FIG. 6 illustrates an exemplary “grid and search” method for estimating a cable location.
In the figures, elements having the same designation have the same or similar functions. Elements in the figures are not drawn to scale.



DETAILED DESCRIPTION
The following exemplary embodiments describe using several parameters used by a line locator to more accurately locate a concealed line. These parameters can include the following:

a=[x,z,I,φ,dI,dφ]

    
    
        where, x is the horizontal position of each each line (whether it be the target line and/or any bleedover lines;
        z is the vertical position (depth) of each line;
        I is the current of each line;
        φ is the phase for each line;
        dI is the current difference between two nearest side tones; and
        dφ is the phase difference between the two nearest side tones.
    
    


FIG. 1 illustrates a line location environment with a line locator 11 (also referred to as “receiver”). A target line 15 energized by an electric current from transmitter 10 emits an electromagnetic (EM) field 17. EM field 17 induces a current in nearby bleedover conductors such as lines 16a and 16b (collectively lines 16) because of resistive, inductive, or capacitive bleedover. The bleedover currents in lines 16 also emit an EM field 18. There may be any number of lines 16.
The sum of fields generated from lines 16 and line 15 can be detected at the surface by EM detector coils such as EM detector coils 14, 19, and 20 of line locator 11. If locator 11 includes horizontal coil 14, vertical coil 19, and top coil 20, three orthogonal sets of measurements of the strength of the EM field as a function of position over the target line can be obtained. Further, a coil orthogonal to the horizontal and vertical (not illustrated in this figure; e.g., illustrated as orthogonal coil 24 of FIG. 2) can be used to measure the electromagnetic field lying in the plane defined by horizontal coil 14, vertical coil 19, and/or top coil 20. In general, line locator 11 may include any number of detector coils.
Using a walkover locate method, field measurements can be detected at multiple positions of line locator 11 as an operator 13 walks transversely across target line 15. With the enhanced model-based method described below, the centerline, depth, and current in target line 15 can be computed and presented to operator 13 on a display 12.
Although line locator 11 illustrated in FIG. 1 is a hand-held locator, embodiments of a line locator consistent with the present invention can be mounted on vehicles, pullcarts, or included in other devices that can be moved relative to target line 15. Line locator 11 is movable for sampling the EM field generated by target line 15 and bleedover lines 16 that significantly contribute to the EM field sampled by line locator 11.
It is often difficult to distinguish signals resulting from target line 15 and from neighboring conductors (sometimes referred to as bleedover lines) 16a and 16b where bleedover has occurred, even if the receiver of line locator 11 provides an indication of the signal direction as well as signal strength. A system that provides an indication of the signal direction as well as signal strength is described in U.S. patent application Ser. No. 10/622,376, now U.S. Pat. No. 7,062,414, by James W. Waite and Johan D. Överby (the '376 application), which is assigned to Metrotech Corporation and herein incorporated by reference in its entirety.
U.S. patent application Ser. No. 10/842,239, now U.S. Pat. No. 7,057,383, by Hubert Schlapp and Johan D. Överby (the '239 application), which is assigned to Metrotech Corporation and herein incorporated by reference in its entirety, discloses a signal processing structure called “bleedover decoupling” for allowing a line locator system to distinguish between signals received from a targeted line 15 and signals received as a result of bleedover from neighboring lines 16. The '239 application also describes an example of a “walkover” technique.
In the '239 application, a “walkover locate” algorithm is described that utilizes a bleedover decoupling processing system to create more accurate estimates of centerline, depth, and current. The walkover locate process includes repeated quadrature measurement of the electromagnetic field as a function of a transect distance, i.e., as the receiver moves transversely above a target cable such as target line 15. The data are fit to a model using a numerical optimization approach, and from the model the cable centerline, depth, and current estimates are obtained.
Systems disclosed in the '239 application represent an important advance in the detection and reporting of signal distortion due to bleedover. However, in the particular case of long haul cables resident in ducts with many other cables, the effect of the phase transfer function that occurs between the locating system transmitter 10 and locator 11, the estimation of confidence bounds for centerline and depth estimates, and the rejection of false positive locates due to the presence of noise and interference were not discussed. The disclosure in the '239 application focused on the demodulation method used to determine the phase of the bleedover signal, but did not address the case where the induced electromagnetic field has nulls where the measurable field disappears. At the null points, the digital phase locked loop (“DPLL”) described in the '376 application can fail to hold a lock to the transmitter signal and the measurement results can become imprecise for some distance segment around the location where there is a null in the EM field. Therefore, further corrections can be made to the bleedover calculation so that more accurate prediction of true centerline and depth of the conductor results.
U.S. patent application Ser. No. 11/100,696, now U.S. Pat. No. 7,356,421, by Thorkell Gudmundsson, Johan D. Överby, Stevan Polak, James W. Waite, and Niklas Lindstrom (the '696 application), which is assigned to Metrotech Corporation and herein incorporated by reference in its entirety, discloses a signal processing structure for refining models to account for the phase transfer function occurring between system transmitter 10 and locator 11 along targeted line 15, for the present confidence bounds for both the centerline and depth estimates. These models also reject false positive locates due to the presence of noise and interference in the environment. Furthermore, to account for the presence of an arbitrarily complex electromagnetic field (again, due to distortion), the refinements help ensure that the estimates of electromagnetic field strength are more accurate over the entire walkover to avoid systematic errors in the field modeling and resulting biases in the estimates of centerline and depth.
The embodiments disclosed in the '696 application represent. an important advance in the detection and reporting of signal distortion due to bleedover. However, those embodiments did not address utilizing an enhanced electromagnetic field model including a phase. measurement error term to precisely locate an underground object.
The refinement of the models disclosed herein can account for a phase measurement error term used to more precisely locate an underground object. In some embodiments, the phase error term can be modeled as a function of location, current, and phase of each cable in addition to a current amplitude and phase variation.
As stated above, transmitter 10 applies a current signal, whose frequency can be user selected from a selectable set of frequencies, to target line 15. The frequency of the current signal applied to the target conductor can be referred to as the active locate frequency. Target line 15 generates EM field 17 at the active locate frequency in response to the current signal. The phase of a tone (e.g., current signal of a determined frequency) placed on a line 15 by transmitter 10 carries information that may be used by locator 11 to determine the degree of signal distortion, as well as the direction of the signal. Locator 11 measures the signal phase relative to a reference signal detectable at locator 11. This reference signal may be transmitted to locator 11 separately from a signal source (such as via an aboveground cable or wireless radio link between transmitter 10 and locator 11), may be determined from an independent synchronized time source (such as a global positioning system (GPS)), or may be embedded in the signal measured from line 15 itself. The latter approach requires no external equipment or wires to implement.
Any transmission medium can affect the phase of a tone traveling along it. This effect is a function of the electrical characteristics of the medium, the distance the signal travels, and the frequency of the tone. For long-haul fiber cables with metallic sheaths and a tone signal frequency above a few Hz, this distortion can become significant. Both measurement and transmission-line models indicate that the phase can easily change by about 1° to about 3° for each mile traveled and depends on local soil conditions, making the phase at a locate site somewhere on a 50 mile cable run difficult to determine. Soil dampness and the cable termination impedance can particularly affect phase distortion. In particular, non-uniform soil conditions, such as conductivity, lead to non-uniform phase distortion along a cable, and different termination impedances lead to further distortion near the termination.
Bleedover distortion at a locate site generally results from an interplay of several complex phenomena. An approach for separating this distortion from the main signal from target line 15 when the phases of the two differ by 90° is described in the '239 application. In some cases (as for long-haul fiber optic cables collocated in duct structures), the distortion is not necessarily orthogonal to the signal from target line 15, so the distortion cannot be fully removed without further correction. In particular, unless the bleedover signal is almost 90° out of phase with the signal from target line 15, a significant distortion component can be present in the in-phase part of the measured field. This bleedover component may appear as a positive or negative current in the bleedover cable, depending on the exact phase offset of the bleedover signal. Thus, a measurement of the current due to bleedover by locator 11 as described in the '239 application may appear similar to a measurement of resistively-coupled “return current” (i.e., current that results from a galvanic connection to the target cable, likely because of a common ground at a splice).
To accurately identify the depth and centerline of target line 15, the presence of non-orthogonal bleedover can be incorporated into the optimization method disclosed in the '239 application. As disclosed herein, the model presented in the '239 application is extended using a generalized formulation that vectors an aggregate measurement of the electromagnetic field to one target line 15 by decoupling the influence of additional bleedover cables (such as lines 16, for example) that are hypothesized to exist and by taking into consideration the phase error measurement term. This bleedover decoupling model can be solved via a non-linear numerical optimization method, such as the Levenberg-Marquardt algorithm and other model-based methods (R. Fletcher, “Practical Methods of Optimization”, Wiley, 2003.).
The EM field measurements at line locator 11 are complex values and represent the aggregate phase of the electromagnetic field measurement at the particular aboveground position of the measurement. To show this, the constituent parts of the EM field can be modeled as a sum of the constituent individual electromagnetic fields. The field measured at locator 11 is distorted from that generated by the primary cable (target line 15) by fields generated by nearby cables, such as lines 16, that carry bleedover current induced by target line 15 and/or return current through a common ground connection. Therefore, the measured EM field at line locator 11 is a summation of three terms,

hmeasured=hprimary+hreturn+hbleedover,  (1)

where hprimary is the electromagnetic field generated by target line 15, hreturn is the electromagnetic field generated by return current to transmitter 10, and hbleedover is the electromagnetic field generated by bleedover cables such as lines 16a and 16b shown in FIG. 1.

To correctly model Equation (1), an operator should estimate the phase of the EM field generated by each cable. This estimation can be done using the signal select modulation scheme (as described in U.S. Pat. No. 6,411,073, which is herein incorporated by reference in its entirety) and the bleedover decoupling signal processing scheme described in the '239 application. Using these approaches, the phase of the EM field measured at line locator 11 can be determined, relative to a reference imparted at transmitter 10. A resistive model best describes the galvanic (direct) coupling related to the terms hprimary and hreturn. In this case, the electromagnetic field can be added directly using only a sign change for whether the current in the cable is flowing in a forward or reverse direction (phase 0° or 180°).
Bleedover models for hbleedover vary depending on the coupling mechanism, Inductive or capacitive coupling (or a combination of both) can occur to other co-located cables as well. The phase of the signal from a cable carrying bleedover current is not as constrained as for resistive coupling. Furthermore, strong return currents can be the source of bleedover, resulting in an ambiguity in the sign of the bleedover phase. Therefore, for the purpose of this discussion, hreturn and hbleedover are treated as indistinguishable, since each can be composed of both in-phase and quadrature components.
An unknown and not insignificant phase offset exists for long cables, for example, cables over several kilometers in length. This offset phase θ can be modeled by a uniform rotation of all the individual cable phases at the measurement location by the same amount. In a system identification framework, the offset phase θ can be viewed as the phase transfer function between transmitter 10 and line locator receiver 11, which can be separated by significant distances (tens of kilometers) in some cases.
Therefore, Equation (1) can be rewritten as:

hmeasured=eiθ(hprimary+hinphase+hquadrature)  (2)

Generally hprimary is real (with zero phase), as is hinphase by definition. An exception to this is when the conductivity of the ground in which the cables are buried is high. In that case, the electromagnetic field induced by the primary current may not be completely in-phase and the bleedover field can be further distorted.
To estimate the location of target line 15, several of the electromagnetic field measurements can be taken as locator 11 is traversed over a cross-section of target line 15. These measurements, along with the horizontal coordinates at which they are taken, are then matched with a model of the electromagnetic field generated by the system of underground lines (target line 15 and surrounding bleedover lines 16) as a function of position to estimate the depth of line 15 and the horizontal centerline coordinate. In addition, an estimate of the current in line 15 and the soil conductivity may be extracted from the results.
The model consistent with some embodiments of the present invention assumes that target line 15 is locally long enough that the induced electromagnetic field is cylindrical. A further assumption is that target line 15 and bleedover lines 16 are substantially parallel. If this latter assumption is incorrect, the model may not accurately estimate the locations of bleedover lines 16. However, identifying the locations of bleedover lines 16 is not the goal. The accurate location of target line 15 can still be obtained even if the locations of the bleedover cables are inaccurate.
Each field source generates a radially decaying field according to the equation:




  
    
      
        
          
            h
            ⁡
            
              (
              r
              )
            
          
          =
          
            
              I
              
                2
                ⁢
                π
              
            
            ⁢
            
              
                ∫
                0
                ∞
              
              ⁢
              
                
                  
                    2
                    ⁢
                    
                      ξe
                      
                        
                          -
                          r
                        
                        ⁢
                        
                          
                            
                              ξ
                              2
                            
                            +
                            
                              2
                              ⁢
                              j
                              ⁢
                              
                                
                              
                              ⁢
                              
                                δ
                                
                                  -
                                  2
                                
                              
                            
                          
                        
                      
                    
                  
                  
                    ξ
                    +
                    
                      
                        
                          ξ
                          2
                        
                        +
                        
                          2
                          ⁢
                          
                            jδ
                            
                              -
                              2
                            
                          
                        
                      
                    
                  
                
                ⁢
                
                  d
                  ξ
                
              
            
          
        
        ,
      
    
    
      
        (
        3
        )
      
    
  




where r is the distance from the source, j=√{square root over (−1)}, and δ is the skin depth of the ground at the signal frequency, where




  
    
      
        
          δ
          =
          
            1
            
              
                πμσ
                ⁢
                
                  
                
                ⁢
                f
              
            
          
        
        ,
      
    
    
      
        (
        4
        )
      
    
  




where,

    
    
        μ=permeability of free space (4π10−7),
        σ=conductivity of the ground, and
        ƒ=signal frequency in Hz.
    
    


If the source of an electromagnetic field from a long conductor is at depth z and centerline x, and this field is measured by a horizontal coil 14 at x-coordinate xn, the coil is oriented at an angle γ with respect to the full field, where




  
    
      
        
          
            cos
            ⁢
            
              
            
            ⁢
            
              γ
              n
            
          
          =
          
            z
            
              r
              n
            
          
        
        ⁢
        
          

        
        ⁢
        
          
            r
            n
            2
          
          =
          
            
              
                (
                
                  
                    x
                    n
                  
                  -
                  x
                
                )
              
              2
            
            +
            
              
                z
                2
              
              .
            
          
        
      
    
    
      
        (
        5
        )
      
    
  




So the measurement at the nth position would be





  
    
      
        
          h
          n
        
        =
        
          
            h
            ⁡
            
              (
              
                r
                n
              
              )
            
          
          ⁢
          
            
              z
              
                r
                n
              
            
            .
          
        
      
    
    
      
        (
        6
        )
      
    
  




In general, the measurement is a complex number, representing the non-zero phase of the signal.

In the limit when σ goes to 0, Equation (6) reduces to:




  
    
      
        
          h
          n
        
        =
        
          Iz
          
            2
            ⁢
            π
            ⁢
            
              
            
            ⁢
            
              r
              n
              2
            
          
        
      
    
    
      
        (
        7
        )
      
    
  




In an ideal world, the electromagnetic field generated by a buried cable carrying a current I0 measured by a horizontal coil at position (xj, z=0) can be modeled as follows:




  
    
      
        
          
            h
            n
          
          ⁡
          
            (
            
              x
              j
            
            )
          
        
        =
        
          
            
              z
              0
            
            ⁢
            
              I
              0
            
            ⁢
            
              cos
              ⁡
              
                (
                
                  
                    ω
                    ⁢
                    
                      
                    
                    ⁢
                    t
                  
                  +
                  
                    φ
                    0
                  
                
                )
              
            
          
          
            
              
                (
                
                  
                    x
                    j
                  
                  -
                  
                    x
                    0
                  
                
                )
              
              2
            
            +
            
              z
              0
              2
            
          
        
      
    
    
      
        (
        8
        )
      
    
  




where,

    
    
        z0 is the depth of the primary cable;
        x0 is the centerline of the primary cable;
        I0 cos(ωt+φ0) is the primary current;
        ω is the active locate frequency; and
        φ0 is the channel phase distortion at ω.
    
    


If a vertically oriented coil is used to measure the field strength rather than the horizontal peak coil, the optimization takes a slightly different form:




  
    
      
        
          
            h
            v
          
          ⁡
          
            (
            
              x
              j
            
            )
          
        
        =
        
          
            
              (
              
                
                  x
                  j
                
                -
                
                  x
                  0
                
              
              )
            
            ⁢
            
              I
              0
            
            ⁢
            
              cos
              ⁡
              
                (
                
                  
                    ω
                    ⁢
                    
                      
                    
                    ⁢
                    t
                  
                  +
                  
                    φ
                    0
                  
                
                )
              
            
          
          
            
              
                (
                
                  
                    x
                    j
                  
                  -
                  
                    x
                    0
                  
                
                )
              
              2
            
            +
            
              z
              0
              2
            
          
        
      
    
    
      
        (
        9
        )
      
    
  




Moreover, the electromagnetic field measured by top coil can be modeled as follows:





  
    
      
        
          
            h
            i
          
          ⁡
          
            (
            
              x
              j
            
            )
          
        
        =
        
          
            
              (
              
                
                  z
                  0
                
                +
                h
              
              )
            
            ⁢
            
              I
              0
            
            ⁢
            
              cos
              ⁡
              
                (
                
                  
                    ω
                    ⁢
                    
                      
                    
                    ⁢
                    t
                  
                  +
                  
                    φ
                    0
                  
                
                )
              
            
          
          
            
              
                (
                
                  
                    x
                    j
                  
                  -
                  
                    x
                    0
                  
                
                )
              
              2
            
            +
            
              
                (
                
                  
                    z
                    0
                  
                  +
                  h
                
                )
              
              2
            
          
        
      
    
    
      
        (
        10
        )
      
    
  




where h is the vertical distance between the horizontal and top coils. Likewise, expressions for the field strength can be derived if the sensing coils are mounted at other arbitrary geometric angles.

FIG. 2 illustrates an embodiment of locator 11 consistent with embodiments of the invention. The embodiment of locator 11 shown in FIG. 2 includes sensing coils 14, 19, 20, and 24, although any number of sensing coils oriented in any orientations can be included. Sensing coils (or detectors) 14, 19, and 20 are coupled to analog front-end (AFE) electronics 21. AFE electronics 21 provides signal conditioning, including amplification, alias protection, and interference filtering prior to analog-to-digital conversion. A digital signal processor implements the phase and field strength amplitude detection algorithm described in the '239 application. These results, as well as distance measurements from a position info 22, are provided to a model optimization 23. The output signal from AFE electronics 21 is also input to model optimization 23. Model optimization 23 optimizes the model utilized to estimate unknown parameters and provides the estimated parameters to display 12 for presentation.
FIG. 3 illustrates the processing flow for a locator consistent with some embodiments of the present invention. Because in a highly distorted field, sharp nulls can exist at various points in the walkover, a dual, nested, digital phase locked loop (DPLL, as described in the '376 application) can be separately connected to each of detector coils 14 and 19, which in turn is used to demodulate both sets of complex measurements of hn, hv from the horizontal and vertical coils. While detector coils 14 and 19 are illustrated in this embodiment for simplicity purposes, one of ordinary skill in the art would appreciate implementing one or more additional coils, such as top coil 20, along with its corresponding DPLLs into this embodiment as well. Nested PLL pairs formed by DPLL 603 and DPLL 605 and by DPLL 604 and DPLL 606 are used to demodulate the phase signal from coils 14 and 19, respectively. To ensure that a solid phase lock is obtained for every x-position at which measurements are taken, duplicate field strength signals can be computed for each of coils 14 and 19. In this way, each walkover generates two pairs of complex signals, [hn, hv]hpll (611, 613, 615, and 617) and [hn, hv]vpll (612, 614, 616, and 618), where the hpll and vpll subscripts denote which digital phase locked loop was used in processing.
The field optimization model utilized in model optimization 23 can use the complex field strength vector (with independent variable xn) for the signal strength measured at each of coils 14 and 19. Thus, it is possible to define a selection method (per each xn) of which pair of [hn, hv] signals to use. In some embodiments the field strength signal magnitudes can be compared (element by element), progressively building the output field strength vector from either the hpll and vpll generated values based upon which is larger than the other by a defined margin. In the end, given a reasonably good amplitude and phase calibration of the PLLs downstream from detector coils 14 and 19, the complex vector of field strength used as input to optimization 23 can be arbitrarily constructed of measurement segments that embody the most accurate field strengths. This method minimizes the places in the walkover where sharp nulls in the EM field degrade the quality of the measurement.  Zone Name: a4,AMD,M
Position info 22 of FIG. 2 provides the current position information xn to model optimization 23. The measured values of xn can be obtained in a variety of ways. In some embodiments, the transect distance increments are computed in situ (within the locate receiver) at the same time intervals that drive the measurement of EM field strength from coils 19 and 14. Inertial methods such as, for example, from accelerometers, gyroscopes, and digital compasses are the best example of this class of distance measurement devices. However, on-board rangefinders, stereo imaging systems, and measuring wheels also can be utilized because they do not require a separate device to be managed by the locate technician. A line locator that includes an inertial position-tracking device is described in U.S. patent application Ser. No. 10/407,705, now U.S. Pat. No. 7,120,546, entitled “Buried Line Locator with Integral Position Sensing”, herein incorporated by reference, by Gordon Pacey and assigned to Metrotech Corporation, which is herein incorporated by reference in its entirety. U.S. patent application Ser. No. 10/997,729, now U.S. Pat. No. 7,113,124, entitled “Centerline and Depth Locating Method for Non-Metallic Buried Utility Lines,” by James W. Waite and assigned to Metrotech Corporation, discloses utilization of inertial sensors to result in transverse (to the line) position estimates xn.
Clearly, simple methods like electromagnetic or optical detection of tick marks on a tape, or simply pressing a button on every desired measurement interval (as determined by a tape measure) can also be used to collect the xn array during the walkover transect. Yet other methods utilized in position info 22 may make assumptions about the walkover speed being constant, and thus use elapsed time to derive position estimates xn. These methods may be appropriate, especially where line locator 11 is mounted and towed at constant velocity.
Onboard inertial measurement units can also be useful in being able to detect pitch and yaw angles from the ideal walkover transect direction, as might occur during a walkover using a handheld receiver. As such, position info 22 can provide pitch and yaw information to model optimization 23. Pitch and yaw angles can be used to correct the field measurements to represent a fixed direction or, alternatively, the field model can be extended to incorporate the pitch and yaw information.
In the presence of multiple bleedover cables, the optimization algorithm matches the measured horizontal coil data to the following model, derived from Equation (8), representing the target cable and a number of secondary cables that contribute to the EM field through bleedover:




  
    
      
        
          
            h
            n
          
          ⁡
          
            (
            
              x
              j
            
            )
          
        
        =
        
          
            
              
                z
                0
              
              ⁢
              
                I
                0
              
              ⁢
              
                cos
                ⁡
                
                  (
                  
                    
                      ω
                      ⁢
                      
                        
                      
                      ⁢
                      t
                    
                    +
                    
                      φ
                      0
                    
                  
                  )
                
              
            
            
              
                
                  (
                  
                    
                      x
                      j
                    
                    -
                    
                      x
                      0
                    
                  
                  )
                
                2
              
              +
              
                z
                0
                2
              
            
          
          +
          
            
              ∑
              
                k
                =
                1
              
              M
            
            ⁢
            
              
            
            ⁢
            
              
                
                  z
                  k
                
                ⁢
                
                  I
                  k
                
                ⁢
                
                  cos
                  ⁡
                  
                    (
                    
                      
                        ω
                        ⁢
                        
                          
                        
                        ⁢
                        t
                      
                      +
                      
                        φ
                        k
                      
                    
                    )
                  
                
              
              
                
                  
                    (
                    
                      
                        x
                        j
                      
                      -
                      
                        x
                        k
                      
                    
                    )
                  
                  2
                
                +
                
                  z
                  k
                  2
                
              
            
          
        
      
    
    
      
        (
        11
        )
      
    
  




where,

    
    
        M is the number of secondary cables;
        zk is the depth of the kth secondary cable;
        xk is the centerline of the kth secondary cable; and
        Ik cos(ωt+φk) is the current in the kth secondary cable.
    
    


Similarly, from Equation (9), the EM field sensed by the vertical coil can be modeled as:




  
    
      
        
          
            h
            n
          
          ⁡
          
            (
            
              x
              j
            
            )
          
        
        =
        
          
            
              
                (
                
                  
                    x
                    j
                  
                  -
                  
                    x
                    0
                  
                
                )
              
              ⁢
              
                I
                0
              
              ⁢
              
                cos
                ⁡
                
                  (
                  
                    
                      ω
                      ⁢
                      
                        
                      
                      ⁢
                      t
                    
                    +
                    
                      φ
                      0
                    
                  
                  )
                
              
            
            
              
                
                  (
                  
                    
                      x
                      j
                    
                    -
                    
                      x
                      0
                    
                  
                  )
                
                2
              
              +
              
                z
                0
                2
              
            
          
          +
          
            
              ∑
              
                k
                =
                1
              
              M
            
            ⁢
            
              
            
            ⁢
            
              
                
                  (
                  
                    
                      x
                      j
                    
                    -
                    
                      x
                      k
                    
                  
                  )
                
                ⁢
                
                  I
                  k
                
                ⁢
                
                  cos
                  ⁡
                  
                    (
                    
                      
                        ω
                        ⁢
                        
                          
                        
                        ⁢
                        t
                      
                      +
                      
                        φ
                        k
                      
                    
                    )
                  
                
              
              
                
                  
                    (
                    
                      
                        x
                        j
                      
                      -
                      
                        x
                        k
                      
                    
                    )
                  
                  2
                
                +
                
                  z
                  k
                  2
                
              
            
          
        
      
    
    
      
        (
        12
        )
      
    
  




and, from Equation (10), the electromagnetic filed sensed by the top coil can be modeled as:




  
    
      
        
          
            h
            n
          
          ⁡
          
            (
            
              x
              j
            
            )
          
        
        =
        
          
            
              
                (
                
                  
                    z
                    0
                  
                  +
                  h
                
                )
              
              ⁢
              
                I
                0
              
              ⁢
              
                cos
                ⁡
                
                  (
                  
                    
                      ω
                      ⁢
                      
                        
                      
                      ⁢
                      t
                    
                    +
                    
                      φ
                      0
                    
                  
                  )
                
              
            
            
              
                
                  (
                  
                    
                      x
                      j
                    
                    -
                    
                      x
                      0
                    
                  
                  )
                
                2
              
              +
              
                
                  (
                  
                    
                      z
                      0
                    
                    +
                    h
                  
                  )
                
                2
              
            
          
          +
          
            
              ∑
              
                k
                =
                1
              
              M
            
            ⁢
            
              
            
            ⁢
            
              
                
                  
                    (
                    
                      
                        z
                        k
                      
                      +
                      h
                    
                    )
                  
                  ⁢
                  
                    I
                    k
                  
                  ⁢
                  
                    cos
                    ⁡
                    
                      (
                      
                        
                          ω
                          ⁢
                          
                            
                          
                          ⁢
                          t
                        
                        +
                        
                          φ
                          k
                        
                      
                      )
                    
                  
                
                
                  
                    
                      (
                      
                        
                          x
                          j
                        
                        -
                        
                          x
                          k
                        
                      
                      )
                    
                    2
                  
                  +
                  
                    
                      (
                      
                        
                          z
                          k
                        
                        +
                        h
                      
                      )
                    
                    2
                  
                
              
              .
            
          
        
      
    
    
      
        (
        13
        )
      
    
  




To obtain the in-phase and quadratic components of hn(xj) Equation (11) can be rewritten as:




  
    
      
        
          
            
              
                
                  h
                  n
                
                ⁡
                
                  (
                  
                    x
                    j
                  
                  )
                
              
              =
              
              ⁢
              
                
                  
                    [
                    
                      
                        ∑
                        
                          k
                          =
                          0
                        
                        M
                      
                      ⁢
                      
                        
                      
                      ⁢
                      
                        
                          
                            z
                            k
                          
                          ⁢
                          
                            I
                            k
                          
                          ⁢
                          
                            cos
                            ⁡
                            
                              (
                              
                                φ
                                k
                              
                              )
                            
                          
                        
                        
                          
                            
                              (
                              
                                
                                  x
                                  j
                                
                                -
                                
                                  x
                                  k
                                
                              
                              )
                            
                            2
                          
                          +
                          
                            z
                            k
                            2
                          
                        
                      
                    
                    ]
                  
                  ·
                  
                    cos
                    ⁡
                    
                      (
                      
                        ω
                        ⁢
                        
                          
                        
                        ⁢
                        t
                      
                      )
                    
                  
                
                -
              
            
          
        
        
          
            
              
              ⁢
              
                
                  [
                  
                    
                      ∑
                      
                        k
                        =
                        0
                      
                      M
                    
                    ⁢
                    
                      
                    
                    ⁢
                    
                      
                        
                          z
                          k
                        
                        ⁢
                        
                          I
                          k
                        
                        ⁢
                        
                          sin
                          ⁡
                          
                            (
                            
                              φ
                              k
                            
                            )
                          
                        
                      
                      
                        
                          
                            (
                            
                              
                                x
                                j
                              
                              -
                              
                                x
                                k
                              
                            
                            )
                          
                          2
                        
                        +
                        
                          z
                          k
                          2
                        
                      
                    
                  
                  ]
                
                ·
                
                  sin
                  ⁡
                  
                    (
                    
                      ω
                      ⁢
                      
                        
                      
                      ⁢
                      t
                    
                    )
                  
                
              
            
          
        
        
          
            
              =
              
              ⁢
              
                
                  
                    
                      h
                      n
                      i
                    
                    ⁡
                    
                      (
                      
                        x
                        j
                      
                      )
                    
                  
                  ·
                  
                    cos
                    ⁡
                    
                      (
                      
                        ω
                        ⁢
                        
                          
                        
                        ⁢
                        t
                      
                      )
                    
                  
                
                +
                
                  
                    
                      h
                      n
                      Q
                    
                    ⁡
                    
                      (
                      
                        x
                        j
                      
                      )
                    
                  
                  ·
                  
                    sin
                    ⁡
                    
                      (
                      
                        ω
                        ⁢
                        
                          
                        
                        ⁢
                        t
                      
                      )
                    
                  
                
              
            
          
        
      
    
    
      
        (
        14
        )
      
    
  




where, hnI(xj) and hnQ(xj) are defined as the in-phase and quadrature components of hn(xj), respectively, wherein these values equate to:




  
    
      
        
          
            
              h
              n
              I
            
            ⁡
            
              (
              
                x
                j
              
              )
            
          
          =
          
            [
            
              
                ∑
                
                  k
                  =
                  0
                
                M
              
              ⁢
              
                
              
              ⁢
              
                
                  
                    z
                    k
                  
                  ⁢
                  
                    I
                    k
                  
                  ⁢
                  
                    cos
                    ⁡
                    
                      (
                      
                        φ
                        k
                      
                      )
                    
                  
                
                
                  
                    
                      (
                      
                        
                          x
                          j
                        
                        -
                        
                          x
                          k
                        
                      
                      )
                    
                    2
                  
                  +
                  
                    z
                    k
                    2
                  
                
              
            
            ]
          
        
        ⁢
        
          

        
        ⁢
        
          
            
              h
              n
              Q
            
            ⁡
            
              (
              
                x
                j
              
              )
            
          
          =
          
            
              [
              
                
                  ∑
                  
                    k
                    =
                    0
                  
                  M
                
                ⁢
                
                  
                
                ⁢
                
                  
                    
                      z
                      k
                    
                    ⁢
                    
                      I
                      k
                    
                    ⁢
                    
                      sin
                      ⁡
                      
                        (
                        
                          φ
                          k
                        
                        )
                      
                    
                  
                  
                    
                      
                        (
                        
                          
                            x
                            j
                          
                          -
                          
                            x
                            k
                          
                        
                        )
                      
                      2
                    
                    +
                    
                      z
                      k
                      2
                    
                  
                
              
              ]
            
            .
          
        
      
    
    
      
        (
        15
        )
      
    
  




Similarly, the vertical and top coils in-phase and quadrature components are found as:




  
    
      
        
          
            
              h
              v
              I
            
            ⁡
            
              (
              
                x
                j
              
              )
            
          
          =
          
            [
            
              
                ∑
                
                  k
                  =
                  0
                
                M
              
              ⁢
              
                
              
              ⁢
              
                
                  
                    (
                    
                      
                        x
                        j
                      
                      -
                      
                        x
                        k
                      
                    
                    )
                  
                  ⁢
                  
                    I
                    k
                  
                  ⁢
                  
                    cos
                    ⁡
                    
                      (
                      
                        φ
                        k
                      
                      )
                    
                  
                
                
                  
                    
                      (
                      
                        
                          x
                          j
                        
                        -
                        
                          x
                          k
                        
                      
                      )
                    
                    2
                  
                  +
                  
                    z
                    k
                    2
                  
                
              
            
            ]
          
        
        ⁢
        
          

        
        ⁢
        
          
            
              
                h
                v
                Q
              
              ⁡
              
                (
                
                  x
                  j
                
                )
              
            
            =
            
              [
              
                
                  ∑
                  
                    k
                    =
                    0
                  
                  M
                
                ⁢
                
                  
                
                ⁢
                
                  
                    
                      (
                      
                        
                          x
                          j
                        
                        -
                        
                          x
                          k
                        
                      
                      )
                    
                    ⁢
                    
                      I
                      k
                    
                    ⁢
                    
                      sin
                      ⁡
                      
                        (
                        
                          φ
                          k
                        
                        )
                      
                    
                  
                  
                    
                      
                        (
                        
                          
                            x
                            j
                          
                          -
                          
                            x
                            k
                          
                        
                        )
                      
                      2
                    
                    +
                    
                      z
                      k
                      2
                    
                  
                
              
              ]
            
          
          ;
        
      
    
    
      
        (
        16
        )
      
    
  
  
    
      
        
          
            
              h
              t
              I
            
            ⁡
            
              (
              
                x
                j
              
              )
            
          
          =
          
            [
            
              
                ∑
                
                  k
                  =
                  0
                
                M
              
              ⁢
              
                
              
              ⁢
              
                
                  
                    (
                    
                      
                        z
                        k
                      
                      +
                      h
                    
                    )
                  
                  ⁢
                  
                    I
                    k
                  
                  ⁢
                  
                    cos
                    ⁡
                    
                      (
                      
                        φ
                        k
                      
                      )
                    
                  
                
                
                  
                    
                      (
                      
                        
                          x
                          j
                        
                        -
                        
                          x
                          k
                        
                      
                      )
                    
                    2
                  
                  +
                  
                    
                      (
                      
                        
                          z
                          k
                        
                        +
                        h
                      
                      )
                    
                    2
                  
                
              
            
            ]
          
        
        ⁢
        
          

        
        ⁢
        
          
            
              h
              t
              Q
            
            ⁡
            
              (
              
                x
                j
              
              )
            
          
          =
          
            
              [
              
                
                  ∑
                  
                    k
                    =
                    0
                  
                  M
                
                ⁢
                
                  
                
                ⁢
                
                  
                    
                      (
                      
                        
                          z
                          k
                        
                        +
                        h
                      
                      )
                    
                    ⁢
                    
                      I
                      k
                    
                    ⁢
                    
                      sin
                      ⁡
                      
                        (
                        
                          φ
                          k
                        
                        )
                      
                    
                  
                  
                    
                      
                        (
                        
                          
                            x
                            j
                          
                          -
                          
                            x
                            k
                          
                        
                        )
                      
                      2
                    
                    +
                    
                      
                        (
                        
                          
                            z
                            k
                          
                          +
                          h
                        
                        )
                      
                      2
                    
                  
                
              
              ]
            
            .
          
        
      
    
    
      
        (
        17
        )
      
    
  




In general, the EM field 18 generated by primary line 15 is distorted by EM field 18 generated by nearby lines 16 that carry current induced by primary line 15 and/or return current through a ground connection. As shown in the above equations, the measured field is a function of line locations and the electrical currents. To estimate the line location, several field measurements are taken over a cross-section of the line. The measurements, along with the horizontal coordinates at which they are taken, are then matched with Equation (14) to estimate the depth of the line and the horizontal centerline coordinate, along with an estimate of the current in the line, as described in U.S. patent application Ser. No. 11/100,696, “Precise Location of buried metallic pipes and cables in the presence of signal distortion” by Thorkell Gudmundsson, James W. Waite, Johan D. Överby, Stevan Polak, and Niklas Lindstrom, which is assigned to Metrotech Corporation.
The locating problem is formulated to minimize the following cost function:




  
    
      
        
          ∑
          j
        
        ⁢
        
          
        
        ⁢
        
          {
          
            
              
                
                  
                    
                      [
                      
                        
                          
                            
                              h
                              n
                              I
                            
                            ⁡
                            
                              (
                              
                                x
                                j
                              
                              )
                            
                          
                          meas
                        
                        -
                        
                          
                            h
                            n
                            I
                          
                          ⁡
                          
                            (
                            
                              x
                              j
                            
                            )
                          
                        
                      
                      ]
                    
                    2
                  
                  +
                  
                    
                      [
                      
                        
                          
                            
                              h
                              n
                              Q
                            
                            ⁡
                            
                              (
                              
                                x
                                j
                              
                              )
                            
                          
                          meas
                        
                        -
                        
                          
                            h
                            n
                            Q
                          
                          ⁡
                          
                            (
                            
                              x
                              j
                            
                            )
                          
                        
                      
                      ]
                    
                    2
                  
                  +
                
              
            
            
              
                
                  
                    
                      [
                      
                        
                          
                            
                              h
                              v
                              I
                            
                            ⁡
                            
                              (
                              
                                x
                                j
                              
                              )
                            
                          
                          meas
                        
                        -
                        
                          
                            h
                            v
                            I
                          
                          ⁡
                          
                            (
                            
                              x
                              j
                            
                            )
                          
                        
                      
                      ]
                    
                    2
                  
                  +
                  
                    
                      [
                      
                        
                          
                            
                              h
                              v
                              Q
                            
                            ⁡
                            
                              (
                              
                                x
                                j
                              
                              )
                            
                          
                          meas
                        
                        -
                        
                          
                            h
                            v
                            Q
                          
                          ⁡
                          
                            (
                            
                              x
                              j
                            
                            )
                          
                        
                      
                      ]
                    
                    2
                  
                  +
                
              
            
            
              
                
                  
                    
                      [
                      
                        
                          
                            
                              h
                              t
                              I
                            
                            ⁡
                            
                              (
                              
                                x
                                j
                              
                              )
                            
                          
                          meas
                        
                        -
                        
                          
                            h
                            t
                            I
                          
                          ⁡
                          
                            (
                            
                              x
                              j
                            
                            )
                          
                        
                      
                      ]
                    
                    2
                  
                  +
                  
                    
                      [
                      
                        
                          
                            
                              h
                              t
                              Q
                            
                            ⁡
                            
                              (
                              
                                x
                                j
                              
                              )
                            
                          
                          meas
                        
                        -
                        
                          
                            h
                            t
                            Q
                          
                          ⁡
                          
                            (
                            
                              x
                              j
                            
                            )
                          
                        
                      
                      ]
                    
                    2
                  
                
              
            
          
          }
        
      
    
    
      
        (
        18
        )
      
    
  




where, hnI(xj)meas, hnQ(xj)meas are the in-phase and quadrature field from horizontal coil 14 as measured by locator 11; hvI(xj)meas, hvQ(xj)meas are the in-phase and quadrature field from vertical coil 19 as measured by locator 11; htI(xj)meas, htQ(xj)meas are the in-phase and quadrature field from top coil 20 as measured by locator 11.

In the presence of bleedover lines, the phase reference derived from a signal-select modulation scheme can be affected. In signal-select mode, the transmitter signal is composed of a number of whole cycles at a frequency slightly lower than a nominal carrier, followed by the same number of cycles at a frequency slightly higher. And this pattern is repeated indefinitely.

ω0−=ω0−ωm=(2n−1)ωm 

ω0+=ω0+ωm=(2n+1)ωm  (19)

where,

    
    
        ωc is the nominal carrier frequency;
        ω0 is the actual carrier frequency;
        ωm is the modulation frequency;
        ω0+ and ω0− are the two nearest side tones; and
        n is the number of whole cycles at each frequency, for a total of 2n cycles in the carrier frequency.
    
    


Accordingly, the following relationships occur:




  
    
      
        
          
        
        ⁢
        
          
            
              ω
              1
            
            =
            
              
                ω
                c
              
              ⁡
              
                (
                
                  1
                  +
                  Δ
                
                )
              
            
          
          ⁢
          
            

          
          ⁢
          
            
          
          ⁢
          
            
              
                ω
                1
              
              ·
              
                T
                1
              
            
            =
            
              
                n
                ·
                2
              
              ⁢
              π
            
          
          ⁢
          
            

          
          ⁢
          
            
          
          ⁢
          
            
              T
              1
            
            =
            
              
                2
                ⁢
                
                  
                
                ⁢
                n
                ⁢
                
                  
                
                ⁢
                π
              
              
                ω
                1
              
            
          
          ⁢
          
            

          
          ⁢
          
            
          
          ⁢
          
            
              ω
              2
            
            =
            
              
                ω
                c
              
              ⁡
              
                (
                
                  1
                  -
                  Δ
                
                )
              
            
          
          ⁢
          
            

          
          ⁢
          
            
          
          ⁢
          
            
              
                ω
                2
              
              ·
              
                T
                2
              
            
            =
            
              
                n
                ·
                2
              
              ⁢
              π
            
          
          ⁢
          
            

          
          ⁢
          
            
          
          ⁢
          
            
              T
              2
            
            =
            
              
                2
                ⁢
                
                  
                
                ⁢
                n
                ⁢
                
                  
                
                ⁢
                π
              
              
                ω
                2
              
            
          
          ⁢
          
            

          
          ⁢
          
            
              ω
              m
            
            =
            
              
                
                  2
                  ⁢
                  π
                
                
                  
                    T
                    1
                  
                  +
                  
                    T
                    2
                  
                
              
              =
              
                
                  
                    2
                    ⁢
                    π
                  
                  
                    
                      
                        2
                        ⁢
                        
                          
                        
                        ⁢
                        n
                        ⁢
                        
                          
                        
                        ⁢
                        π
                      
                      
                        ω
                        1
                      
                    
                    +
                    
                      
                        2
                        ⁢
                        
                          
                        
                        ⁢
                        n
                        ⁢
                        
                          
                        
                        ⁢
                        π
                      
                      
                        ω
                        2
                      
                    
                  
                
                =
                
                  
                    
                      2
                      ⁢
                      π
                    
                    
                      
                        
                          2
                          ⁢
                          
                            
                          
                          ⁢
                          n
                          ⁢
                          
                            
                          
                          ⁢
                          π
                        
                        
                          
                            ω
                            c
                          
                          ⁡
                          
                            (
                            
                              1
                              +
                              Δ
                            
                            )
                          
                        
                      
                      +
                      
                        
                          2
                          ⁢
                          
                            
                          
                          ⁢
                          n
                          ⁢
                          
                            
                          
                          ⁢
                          π
                        
                        
                          
                            ω
                            c
                          
                          ⁡
                          
                            (
                            
                              1
                              -
                              Δ
                            
                            )
                          
                        
                      
                    
                  
                  =
                  
                    
                      
                        ω
                        c
                      
                      ⁡
                      
                        (
                        
                          1
                          -
                          
                            Δ
                            2
                          
                        
                        )
                      
                    
                    
                      2
                      ⁢
                      n
                    
                  
                
              
            
          
          ⁢
          
            

          
          ⁢
          
            
          
          ⁢
          
            
              ω
              0
            
            =
            
              
                2
                ⁢
                
                  n
                  ·
                  
                    ω
                    m
                  
                
              
              =
              
                
                  ω
                  c
                
                ⁡
                
                  (
                  
                    1
                    -
                    
                      Δ
                      2
                    
                  
                  )
                
              
            
          
        
      
    
    
      
        (
        20
        )
      
    
  




where,

    
    
        Δ is a ratio of offset frequency to nominal carrier frequency.
    
    


Further, in some embodiments, the output current can be set to a constant at transmitter 10. For example, in some embodiments FSK modulation is utilized. In FSK modulation, two closely spaced frequencies are alternatively active at a rate defined by the modulation frequency. The transmitted FSK signal could be expressed as follows:




  
    
      
        
          I
          ⁡
          
            (
            t
            )
          
        
        =
        
          
            
              
                I
                tx
              
              ⁢
              
                
                  ∑
                  
                    k
                    =
                    0
                  
                  ∞
                
                ⁢
                
                  
                
                ⁢
                
                  (
                  
                    cos
                    ⁢
                    
                      
                        {
                        
                          
                            ω
                            1
                          
                          ⁡
                          
                            [
                            
                              t
                              +
                              
                                k
                                ⁡
                                
                                  (
                                  
                                    
                                      T
                                      1
                                    
                                    +
                                    
                                      T
                                      2
                                    
                                  
                                  )
                                
                              
                            
                            ]
                          
                        
                        }
                      
                      ·
                      
                        {
                        
                          
                            u
                            ⁡
                            
                              (
                              
                                t
                                -
                                
                                  k
                                  ⁡
                                  
                                    (
                                    
                                      
                                        T
                                        1
                                      
                                      +
                                      
                                        T
                                        2
                                      
                                    
                                    )
                                  
                                
                              
                              )
                            
                          
                          -
                          
                            u
                            ⁡
                            
                              (
                              
                                t
                                -
                                
                                  k
                                  ⁡
                                  
                                    (
                                    
                                      
                                        T
                                        1
                                      
                                      +
                                      T
                                    
                                    )
                                  
                                
                                -
                                
                                  T
                                  1
                                
                              
                              )
                            
                          
                        
                        }
                      
                    
                  
                  )
                
              
            
            +
            
              
                I
                tx
              
              ⁢
              
                
                  ∑
                  
                    k
                    =
                    0
                  
                  ∞
                
                ⁢
                
                  
                
                ⁢
                
                  (
                  
                    cos
                    ⁢
                    
                      
                        {
                        
                          
                            ω
                            2
                          
                          ⁡
                          
                            [
                            
                              t
                              +
                              
                                k
                                ⁡
                                
                                  (
                                  
                                    
                                      T
                                      1
                                    
                                    +
                                    
                                      T
                                      2
                                    
                                  
                                  )
                                
                              
                              -
                              
                                T
                                1
                              
                            
                            ]
                          
                        
                        }
                      
                      ·
                      
                        {
                        
                          
                            u
                            ⁡
                            
                              (
                              
                                t
                                -
                                
                                  k
                                  ⁡
                                  
                                    (
                                    
                                      
                                        T
                                        1
                                      
                                      +
                                      
                                        T
                                        2
                                      
                                    
                                    )
                                  
                                
                                -
                                
                                  T
                                  1
                                
                              
                              )
                            
                          
                          -
                          
                            u
                            ⁡
                            
                              (
                              
                                t
                                -
                                
                                  
                                    (
                                    
                                      k
                                      +
                                      1
                                    
                                    )
                                  
                                  ⁢
                                  
                                    (
                                    
                                      
                                        T
                                        1
                                      
                                      +
                                      T
                                    
                                    )
                                  
                                
                                -
                                
                                  T
                                  1
                                
                              
                              )
                            
                          
                        
                        }
                      
                    
                  
                  )
                
              
            
          
          =
          
            
              a
              0
            
            +
            
              
                ∑
                
                  k
                  =
                  1
                
                ∞
              
              ⁢
              
                
              
              ⁢
              
                [
                
                  
                    
                      a
                      k
                    
                    ⁢
                    
                      cos
                      ⁡
                      
                        (
                        
                          
                            k
                            ·
                            
                              ω
                              m
                            
                          
                          ⁢
                          t
                        
                        )
                      
                    
                  
                  +
                  
                    
                      b
                      k
                    
                    ⁢
                    
                      sin
                      ⁡
                      
                        (
                        
                          
                            k
                            ·
                            
                              ω
                              m
                            
                          
                          ⁢
                          t
                        
                        )
                      
                    
                  
                
                ]
              
            
          
        
      
    
    
      
        (
        21
        )
      
    
  




where, u(t) is the standard unit step function, and ak and bk are the Fourier series coefficients computed, as shown in Appendix I. In some embodiments, when in signal-select mode, a primary PLL, such as PLL 603 or 604 provided in FIG. 3, can lock the 2nth harmonics and use the two side tones harmonics at (2n−1)ωm and (2n+1)ωm, to derive the signal phase information with the nested PLL, such as nested PLL 605 or 606 provided in FIG. 3.

By neglecting the other harmonics outside of 2nth harmonics and the two nearest side tones, ω0+ and ω0−, Equation (21) can be rewritten as:




  
    
      
        
          I
          ⁡
          
            (
            t
            )
          
        
        =
        
          
            
              
                
                  I
                  tx
                
                ·
                
                  sin
                  ⁡
                  
                    (
                    
                      n
                      ⁢
                      
                        
                      
                      ⁢
                      π
                      ⁢
                      
                        
                      
                      ⁢
                      Δ
                    
                    )
                  
                
              
              
                n
                ⁢
                
                  
                
                ⁢
                π
                ⁢
                
                  
                
                ⁢
                Δ
              
            
            ·
            
              cos
              ⁡
              
                (
                
                  
                    
                      ω
                      0
                    
                    ⁢
                    t
                  
                  +
                  
                    n
                    ⁢
                    
                      
                    
                    ⁢
                    π
                    ⁢
                    
                      
                    
                    ⁢
                    Δ
                  
                
                )
              
            
          
          +
          
            
              
                
                  
                    I
                    tx
                  
                  ·
                  4
                
                ⁢
                n
                ⁢
                
                  
                
                ⁢
                
                  Δ
                  /
                  π
                
              
              
                1
                -
                
                  
                    (
                    
                      2
                      ⁢
                      n
                      ⁢
                      
                        
                      
                      ⁢
                      Δ
                    
                    )
                  
                  2
                
              
            
            ·
            
              sin
              ⁡
              
                (
                
                  
                    
                      ω
                      0
                      -
                    
                    ⁢
                    
                      T
                      1
                    
                  
                  2
                
                )
              
            
            ·
            
              cos
              ⁡
              
                (
                
                  
                    
                      ω
                      0
                      -
                    
                    ⁢
                    t
                  
                  -
                  
                    
                      
                        ω
                        0
                        -
                      
                      ⁢
                      
                        T
                        1
                      
                    
                    2
                  
                
                )
              
            
          
          +
          
            
              
                
                  
                    I
                    tx
                  
                  ·
                  4
                
                ⁢
                n
                ⁢
                
                  
                
                ⁢
                
                  Δ
                  /
                  π
                
              
              
                1
                -
                
                  
                    (
                    
                      2
                      ⁢
                      n
                      ⁢
                      
                        
                      
                      ⁢
                      Δ
                    
                    )
                  
                  2
                
              
            
            ·
            
              sin
              (
              
                
                  
                    ω
                    0
                    +
                  
                  ⁢
                  
                    T
                    1
                  
                
                2
              
              )
            
            ·
            
              cos
              (
              
                
                  
                    ω
                    0
                    +
                  
                  ⁢
                  t
                
                -
                
                  
                    
                      ω
                      0
                      +
                    
                    ⁢
                    
                      T
                      1
                    
                  
                  2
                
              
              )
            
          
        
      
    
    
      
        (
        22
        )
      
    
  




Accordingly, at locator 11, the target line current can be given by:




  
    
      
        
          
            I
            0
          
          ⁡
          
            (
            t
            )
          
        
        =
        
          
            
              
                
                  I
                  0
                
                ·
                
                  sin
                  ⁡
                  
                    (
                    
                      n
                      ⁢
                      
                        
                      
                      ⁢
                      π
                      ⁢
                      
                        
                      
                      ⁢
                      Δ
                    
                    )
                  
                
              
              
                n
                ⁢
                
                  
                
                ⁢
                π
                ⁢
                
                  
                
                ⁢
                Δ
              
            
            ·
            
              cos
              ⁡
              
                (
                
                  
                    
                      ω
                      0
                    
                    ⁢
                    t
                  
                  +
                  
                    n
                    ⁢
                    
                      
                    
                    ⁢
                    π
                    ⁢
                    
                      
                    
                    ⁢
                    Δ
                  
                  +
                  
                    φ
                    0
                  
                
                )
              
            
          
          +
          
            
              
                
                  
                    I
                    0
                    -
                  
                  ·
                  4
                
                ⁢
                n
                ⁢
                
                  
                
                ⁢
                
                  Δ
                  /
                  π
                
              
              
                1
                -
                
                  
                    (
                    
                      2
                      ⁢
                      n
                      ⁢
                      
                        
                      
                      ⁢
                      Δ
                    
                    )
                  
                  2
                
              
            
            ·
            
              sin
              ⁡
              
                (
                
                  
                    
                      ω
                      0
                      -
                    
                    ⁢
                    
                      T
                      1
                    
                  
                  2
                
                )
              
            
            ·
            
              cos
              ⁡
              
                (
                
                  
                    
                      ω
                      0
                      -
                    
                    ⁢
                    t
                  
                  -
                  
                    
                      
                        ω
                        0
                        -
                      
                      ⁢
                      
                        T
                        1
                      
                    
                    2
                  
                  +
                  
                    φ
                    0
                    -
                  
                
                )
              
            
          
          +
          
            
              
                
                  
                    I
                    0
                    +
                  
                  ·
                  4
                
                ⁢
                n
                ⁢
                
                  
                
                ⁢
                
                  Δ
                  /
                  π
                
              
              
                1
                -
                
                  
                    (
                    
                      2
                      ⁢
                      n
                      ⁢
                      
                        
                      
                      ⁢
                      Δ
                    
                    )
                  
                  2
                
              
            
            ·
            
              sin
              (
              
                
                  
                    ω
                    0
                    +
                  
                  ⁢
                  
                    T
                    1
                  
                
                2
              
              )
            
            ·
            
              cos
              (
              
                
                  
                    ω
                    0
                    +
                  
                  ⁢
                  t
                
                -
                
                  
                    
                      ω
                      0
                      +
                    
                    ⁢
                    
                      T
                      1
                    
                  
                  2
                
                +
                
                  φ
                  0
                  +
                
              
              )
            
          
        
      
    
    
      
        (
        23
        )
      
    
  




where, I0−, I0, I0+ and φ0−, φ0, φ0+ are different due to the channel attenuation and phase characteristics at different frequencies.
Similarly, the current in bleedover line 16 could be expressed as:




  
    
      
        
          
            
              I
              k
            
            ⁡
            
              (
              t
              )
            
          
          =
          
            
              
                
                  
                    I
                    k
                  
                  ·
                  
                    sin
                    ⁡
                    
                      (
                      
                        n
                        ⁢
                        
                          
                        
                        ⁢
                        π
                        ⁢
                        
                          
                        
                        ⁢
                        Δ
                      
                      )
                    
                  
                
                
                  n
                  ⁢
                  
                    
                  
                  ⁢
                  π
                  ⁢
                  
                    
                  
                  ⁢
                  Δ
                
              
              ·
              
                cos
                ⁡
                
                  (
                  
                    
                      
                        ω
                        0
                      
                      ⁢
                      t
                    
                    +
                    
                      n
                      ⁢
                      
                        
                      
                      ⁢
                      π
                      ⁢
                      
                        
                      
                      ⁢
                      Δ
                    
                    +
                    
                      φ
                      k
                    
                  
                  )
                
              
            
            +
            
              
                
                  
                    
                      I
                      k
                      -
                    
                    ·
                    4
                  
                  ⁢
                  n
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Δ
                    /
                    π
                  
                
                
                  1
                  -
                  
                    
                      (
                      
                        2
                        ⁢
                        n
                        ⁢
                        
                          
                        
                        ⁢
                        Δ
                      
                      )
                    
                    2
                  
                
              
              ·
              
                sin
                ⁡
                
                  (
                  
                    
                      
                        ω
                        0
                        -
                      
                      ⁢
                      
                        T
                        1
                      
                    
                    2
                  
                  )
                
              
              ·
              
                cos
                ⁡
                
                  (
                  
                    
                      
                        ω
                        0
                        -
                      
                      ⁢
                      t
                    
                    -
                    
                      
                        
                          ω
                          0
                          -
                        
                        ⁢
                        
                          T
                          1
                        
                      
                      2
                    
                    +
                    
                      φ
                      k
                      -
                    
                  
                  )
                
              
            
            +
            
              
                
                  
                    
                      I
                      k
                      +
                    
                    ·
                    4
                  
                  ⁢
                  n
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Δ
                    /
                    π
                  
                
                
                  1
                  -
                  
                    
                      (
                      
                        2
                        ⁢
                        n
                        ⁢
                        
                          
                        
                        ⁢
                        Δ
                      
                      )
                    
                    2
                  
                
              
              ·
              
                sin
                (
                
                  
                    
                      ω
                      0
                      +
                    
                    ⁢
                    
                      T
                      1
                    
                  
                  2
                
                )
              
              ·
              
                cos
                (
                
                  
                    
                      ω
                      0
                      +
                    
                    ⁢
                    t
                  
                  -
                  
                    
                      
                        ω
                        0
                        +
                      
                      ⁢
                      
                        T
                        1
                      
                    
                    2
                  
                  +
                  
                    φ
                    k
                    +
                  
                
                )
              
            
          
        
        ,
        
          

        
        ⁢
        
          
            
              
                
                  
                
                ⁢
                
                  
                    k
                    =
                    1
                  
                  ,
                  2
                  ,
                  ...
                  ⁢
                  
                    
                  
                  ,
                  M
                
              
            
            
              
                
              
            
          
        
      
    
    
      
        (
        24
        )
      
    
  




Assuming the EM field sensed by horizontal coil of locator 11 at location (x,z) is:




  
    
      
        
          
            h
            n
          
          ⁡
          
            (
            t
            )
          
        
        =
        
          
            
              
                sin
                ⁡
                
                  (
                  
                    n
                    ⁢
                    
                      
                    
                    ⁢
                    π
                    ⁢
                    
                      
                    
                    ⁢
                    Δ
                  
                  )
                
              
              
                n
                ⁢
                
                  
                
                ⁢
                π
                ⁢
                
                  
                
                ⁢
                Δ
              
            
            ·
            
              
                ∑
                
                  k
                  =
                  0
                
                M
              
              ⁢
              
                
                  A
                  k
                
                ·
                
                  cos
                  ⁡
                  
                    (
                    
                      
                        
                          ω
                          0
                        
                        ⁢
                        t
                      
                      +
                      
                        n
                        ⁢
                        
                          
                        
                        ⁢
                        π
                        ⁢
                        
                          
                        
                        ⁢
                        Δ
                      
                      +
                      
                        φ
                        k
                      
                    
                    )
                  
                
              
            
          
          +
          
            
              
                4
                ⁢
                n
                ⁢
                
                  
                
                ⁢
                
                  Δ
                  /
                  π
                
              
              
                1
                -
                
                  
                    (
                    
                      2
                      ⁢
                      n
                      ⁢
                      
                        
                      
                      ⁢
                      Δ
                    
                    )
                  
                  2
                
              
            
            ·
            
              sin
              ⁡
              
                (
                
                  
                    
                      ω
                      0
                      -
                    
                    ⁢
                    
                      T
                      1
                    
                  
                  2
                
                )
              
            
            ·
            
              
                ∑
                
                  k
                  =
                  0
                
                M
              
              ⁢
              
                
                  A
                  k
                  -
                
                ·
                
                  cos
                  ⁡
                  
                    (
                    
                      
                        
                          ω
                          0
                          -
                        
                        ⁢
                        t
                      
                      -
                      
                        
                          
                            ω
                            0
                            -
                          
                          ⁢
                          
                            T
                            1
                          
                        
                        2
                      
                      +
                      
                        φ
                        k
                        -
                      
                    
                    )
                  
                
              
            
          
          +
          
            
              
                4
                ⁢
                n
                ⁢
                
                  
                
                ⁢
                
                  Δ
                  /
                  π
                
              
              
                1
                -
                
                  
                    (
                    
                      2
                      ⁢
                      n
                      ⁢
                      
                        
                      
                      ⁢
                      Δ
                    
                    )
                  
                  2
                
              
            
            ·
            
              sin
              (
              
                
                  
                    ω
                    0
                    +
                  
                  ⁢
                  
                    T
                    1
                  
                
                2
              
              )
            
            ·
            
              
                ∑
                
                  k
                  =
                  0
                
                M
              
              ⁢
              
                
                  A
                  k
                  +
                
                ·
                
                  cos
                  (
                  
                    
                      
                        ω
                        0
                        +
                      
                      ⁢
                      t
                    
                    -
                    
                      
                        
                          ω
                          0
                          +
                        
                        ⁢
                        
                          T
                          1
                        
                      
                      2
                    
                    +
                    
                      φ
                      k
                      +
                    
                  
                  )
                
              
            
          
        
      
    
    
      
        (
        25
        )
      
    
  
  
    
      
        
          
        
        ⁢
        
          where
          ,
        
      
    
    
      
        
      
    
  
  
    
      
        
          
        
        ⁢
        
          
            
              A
              k
            
            =
            
              
                
                  z
                  k
                
                ⁢
                
                  I
                  k
                
              
              
                
                  
                    (
                    
                      x
                      -
                      
                        x
                        k
                      
                    
                    )
                  
                  2
                
                +
                
                  z
                  k
                  2
                
              
            
          
          ,
          
            k
            =
            0
          
          ,
          1
          ,
          ...
          ⁢
          
            
          
          ,
          
            M
            .
          
        
      
    
    
      
        (
        26
        )
      
    
  
  
    
      
        
          
        
        ⁢
        
          
            
              A
              k
              -
            
            =
            
              
                
                  z
                  k
                
                ⁢
                
                  I
                  k
                  -
                
              
              
                
                  
                    (
                    
                      x
                      -
                      
                        x
                        k
                      
                    
                    )
                  
                  2
                
                +
                
                  z
                  k
                  2
                
              
            
          
          ,
          
            k
            =
            0
          
          ,
          1
          ,
          ...
          ⁢
          
            
          
          ,
          M
        
      
    
    
      
        (
        27
        )
      
    
  
  
    
      
        
          
        
        ⁢
        
          
            
              A
              k
              +
            
            =
            
              
                
                  z
                  k
                
                ⁢
                
                  I
                  k
                  +
                
              
              
                
                  
                    (
                    
                      x
                      -
                      
                        x
                        k
                      
                    
                    )
                  
                  2
                
                +
                
                  z
                  k
                  2
                
              
            
          
          ,
          
            k
            =
            0
          
          ,
          1
          ,
          ...
          ⁢
          
            
          
          ,
          M
          ,
        
      
    
    
      
        (
        28
        )
      
    
  




then Equation (25) can be rewritten as:





  
    
      
        
          
            h
            n
          
          ⁡
          
            (
            t
            )
          
        
        =
        
          
            
              
                sin
                ⁡
                
                  (
                  
                    n
                    ⁢
                    
                      
                    
                    ⁢
                    π
                    ⁢
                    
                      
                    
                    ⁢
                    Δ
                  
                  )
                
              
              
                n
                ⁢
                
                  
                
                ⁢
                π
                ⁢
                
                  
                
                ⁢
                Δ
              
            
            ·
            A
            ·
            
              cos
              ⁡
              
                (
                
                  
                    
                      ω
                      0
                    
                    ⁢
                    t
                  
                  +
                  
                    n
                    ⁢
                    
                      
                    
                    ⁢
                    π
                    ⁢
                    
                      
                    
                    ⁢
                    Δ
                  
                  +
                  
                    φ
                    0
                  
                
                )
              
            
          
          +
          
            
              
                4
                ⁢
                n
                ⁢
                
                  
                
                ⁢
                
                  Δ
                  /
                  π
                
              
              
                1
                -
                
                  
                    (
                    
                      2
                      ⁢
                      n
                      ⁢
                      
                        
                      
                      ⁢
                      Δ
                    
                    )
                  
                  2
                
              
            
            ·
            
              sin
              ⁡
              
                (
                
                  
                    
                      ω
                      0
                      -
                    
                    ⁢
                    
                      T
                      1
                    
                  
                  2
                
                )
              
            
            ·
            
              A
              -
            
            ·
            
              cos
              ⁡
              
                (
                
                  
                    
                      ω
                      0
                      -
                    
                    ⁢
                    t
                  
                  -
                  
                    
                      
                        ω
                        0
                        -
                      
                      ⁢
                      
                        T
                        1
                      
                    
                    2
                  
                  +
                  
                    φ
                    0
                    -
                  
                
                )
              
            
          
          +
          
            
              
                4
                ⁢
                n
                ⁢
                
                  
                
                ⁢
                
                  Δ
                  /
                  π
                
              
              
                1
                -
                
                  
                    (
                    
                      2
                      ⁢
                      n
                      ⁢
                      
                        
                      
                      ⁢
                      Δ
                    
                    )
                  
                  2
                
              
            
            ·
            
              sin
              (
              
                
                  
                    ω
                    0
                    +
                  
                  ⁢
                  
                    T
                    1
                  
                
                2
              
              )
            
            ·
            
              A
              +
            
            ·
            
              cos
              (
              
                
                  
                    ω
                    0
                    +
                  
                  ⁢
                  t
                
                -
                
                  
                    
                      ω
                      0
                      +
                    
                    ⁢
                    
                      T
                      1
                    
                  
                  2
                
                +
                
                  φ
                  0
                  +
                
              
              )
            
          
        
      
    
    
      
        (
        29
        )
      
    
  
  
    
      
        
          
        
        ⁢
        
          where
          ,
        
      
    
    
      
        
      
    
  
  
    
      
        
          
            
              
                
              
              ⁢
              
                A
                =
                
                ⁢
                
                  
                    
                      
                        [
                        
                          
                            ∑
                            
                              k
                              =
                              0
                            
                            M
                          
                          ⁢
                          
                            
                              A
                              k
                            
                            ⁢
                            
                              cos
                              ⁡
                              
                                (
                                
                                  φ
                                  k
                                
                                )
                              
                            
                          
                        
                        ]
                      
                      2
                    
                    +
                    
                      
                        [
                        
                          
                            ∑
                            
                              k
                              =
                              0
                            
                            M
                          
                          ⁢
                          
                            
                              A
                              k
                            
                            ⁢
                            
                              sin
                              ⁡
                              
                                (
                                
                                  φ
                                  k
                                
                                )
                              
                            
                          
                        
                        ]
                      
                      2
                    
                  
                
              
            
          
        
        
          
            
              =
              
              ⁢
              
                
                  
                    ∑
                    
                      j
                      =
                      0
                    
                    M
                  
                  ⁢
                  
                    
                      ∑
                      
                        k
                        =
                        0
                      
                      M
                    
                    ⁢
                    
                      
                        A
                        j
                      
                      ⁢
                      
                        A
                        k
                      
                      ⁢
                      
                        cos
                        ⁡
                        
                          (
                          
                            
                              φ
                              j
                            
                            -
                            
                              φ
                              k
                            
                          
                          )
                        
                      
                    
                  
                
              
            
          
        
      
    
    
      
        (
        30
        )
      
    
  
  
    
      
        
          
        
        ⁢
        
          
            φ
            0
          
          =
          
            
              tan
              
                -
                1
              
            
            (
            
              
                
                  ∑
                  
                    k
                    =
                    0
                  
                  M
                
                ⁢
                
                  
                    A
                    k
                  
                  ⁢
                  
                    sin
                    ⁡
                    
                      (
                      
                        φ
                        k
                      
                      )
                    
                  
                
              
              
                
                  ∑
                  
                    k
                    =
                    0
                  
                  M
                
                ⁢
                
                  
                    A
                    k
                  
                  ⁢
                  
                    cos
                    ⁡
                    
                      (
                      
                        φ
                        k
                      
                      )
                    
                  
                
              
            
            )
          
        
      
    
    
      
        (
        31
        )
      
    
  
  
    
      
        
          
        
        ⁢
        
          
            A
            -
          
          =
          
            
              
                ∑
                
                  j
                  =
                  0
                
                M
              
              ⁢
              
                
                  ∑
                  
                    k
                    =
                    0
                  
                  M
                
                ⁢
                
                  
                    A
                    j
                    -
                  
                  ⁢
                  
                    A
                    k
                    -
                  
                  ⁢
                  
                    cos
                    ⁡
                    
                      (
                      
                        
                          φ
                          j
                          -
                        
                        -
                        
                          φ
                          k
                          -
                        
                      
                      )
                    
                  
                
              
            
          
        
      
    
    
      
        (
        32
        )
      
    
  
  
    
      
        
          
        
        ⁢
        
          
            φ
            0
            -
          
          =
          
            
              tan
              
                -
                1
              
            
            (
            
              
                
                  ∑
                  
                    k
                    =
                    0
                  
                  M
                
                ⁢
                
                  
                    A
                    k
                    -
                  
                  ⁢
                  
                    sin
                    ⁡
                    
                      (
                      
                        φ
                        k
                        -
                      
                      )
                    
                  
                
              
              
                
                  ∑
                  
                    k
                    =
                    0
                  
                  M
                
                ⁢
                
                  
                    A
                    k
                    -
                  
                  ⁢
                  
                    cos
                    ⁡
                    
                      (
                      
                        φ
                        k
                        -
                      
                      )
                    
                  
                
              
            
            )
          
        
      
    
    
      
        (
        33
        )
      
    
  
  
    
      
        
          
        
        ⁢
        
          
            A
            +
          
          =
          
            
              
                ∑
                
                  j
                  =
                  0
                
                M
              
              ⁢
              
                
                  ∑
                  
                    k
                    =
                    0
                  
                  M
                
                ⁢
                
                  
                    A
                    j
                    +
                  
                  ⁢
                  
                    A
                    k
                    +
                  
                  ⁢
                  
                    cos
                    ⁡
                    
                      (
                      
                        
                          φ
                          j
                          +
                        
                        -
                        
                          φ
                          k
                          +
                        
                      
                      )
                    
                  
                
              
            
          
        
      
    
    
      
        (
        34
        )
      
    
  
  
    
      
        
          
        
        ⁢
        
          
            φ
            0
            +
          
          =
          
            
              
                tan
                
                  -
                  1
                
              
              (
              
                
                  
                    ∑
                    
                      k
                      =
                      0
                    
                    M
                  
                  ⁢
                  
                    
                      A
                      k
                      +
                    
                    ⁢
                    
                      sin
                      ⁡
                      
                        (
                        
                          φ
                          k
                          +
                        
                        )
                      
                    
                  
                
                
                  
                    ∑
                    
                      k
                      =
                      0
                    
                    M
                  
                  ⁢
                  
                    
                      A
                      k
                      +
                    
                    ⁢
                    
                      cos
                      ⁡
                      
                        (
                        
                          φ
                          k
                          +
                        
                        )
                      
                    
                  
                
              
              )
            
            .
          
        
      
    
    
      
        (
        35
        )
      
    
  




From Equations (20) and (29) above, the carrier frequency is ω0=2nωm, and the phase is ω0t+nπΔ+φ0. The demodulated FM signal provided to a signal direction detector of DPLL 603/604 of FIG. 3, which is further described in the '376 application, would then be:

hn(t)·[−sin(ω0t+φ0+nπΔ)].  (36)

After a band-pass filter, which is centered at ωm and which is further described in the '376 application, the FM signal input to Phase DPLL block 605/606 of FIG. 3 would be (see Appendix II for further detail):




  
    
      
        
          
            
              
                
                  
                    -
                    2
                  
                  ⁢
                  n
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Δ
                    /
                    π
                  
                
                
                  1
                  -
                  
                    
                      (
                      
                        2
                        ⁢
                        n
                        ⁢
                        
                          
                        
                        ⁢
                        Δ
                      
                      )
                    
                    2
                  
                
              
              ·
              
                sin
                ⁡
                
                  (
                  
                    
                      
                        ω
                        0
                        -
                      
                      ⁢
                      
                        T
                        1
                      
                    
                    2
                  
                  )
                
              
              ·
              
                A
                -
              
              ·
              
                sin
                ⁡
                
                  (
                  
                    
                      
                        ω
                        m
                      
                      ⁢
                      t
                    
                    +
                    
                      
                        
                          ω
                          0
                          -
                        
                        ⁢
                        
                          T
                          1
                        
                      
                      2
                    
                    +
                    
                      n
                      ⁢
                      
                        
                      
                      ⁢
                      π
                      ⁢
                      
                        
                      
                      ⁢
                      Δ
                    
                    +
                    
                      φ
                      0
                    
                    -
                    
                      φ
                      0
                      -
                    
                  
                  )
                
              
            
            +
            
              
                
                  
                    -
                    2
                  
                  ⁢
                  n
                  ⁢
                  
                    
                  
                  ⁢
                  
                    Δ
                    /
                    π
                  
                
                
                  1
                  -
                  
                    
                      (
                      
                        2
                        ⁢
                        n
                        ⁢
                        
                          
                        
                        ⁢
                        Δ
                      
                      )
                    
                    2
                  
                
              
              ·
              
                sin
                ⁡
                
                  (
                  
                    
                      
                        ω
                        0
                        +
                      
                      ⁢
                      
                        T
                        1
                      
                    
                    2
                  
                  )
                
              
              ·
              
                A
                +
              
              ·
              
                sin
                (
                
                  
                    
                      -
                      
                        ω
                        m
                      
                    
                    ⁢
                    t
                  
                  +
                  
                    
                      
                        ω
                        0
                        +
                      
                      ⁢
                      
                        T
                        1
                      
                    
                    2
                  
                  +
                  
                    n
                    ⁢
                    
                      
                    
                    ⁢
                    π
                    ⁢
                    
                      
                    
                    ⁢
                    Δ
                  
                  +
                  
                    φ
                    0
                  
                  -
                  
                    φ
                    0
                    +
                  
                
                )
              
            
          
          =
          
            
              
                
                  
                    n
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Δ
                      /
                      π
                    
                  
                  
                    1
                    -
                    
                      
                        (
                        
                          2
                          ⁢
                          n
                          ⁢
                          
                            
                          
                          ⁢
                          Δ
                        
                        )
                      
                      2
                    
                  
                
                ·
                
                  A
                  -
                
                ·
                
                  [
                  
                    
                      cos
                      ⁡
                      
                        (
                        
                          
                            
                              ω
                              m
                            
                            ⁢
                            t
                          
                          +
                          
                            
                              ω
                              0
                              -
                            
                            ⁢
                            
                              T
                              1
                            
                          
                          +
                          
                            n
                            ⁢
                            
                              
                            
                            ⁢
                            π
                            ⁢
                            
                              
                            
                            ⁢
                            Δ
                          
                          +
                          
                            φ
                            0
                          
                          -
                          
                            φ
                            0
                            -
                          
                        
                        )
                      
                    
                    -
                    
                      cos
                      ⁡
                      
                        (
                        
                          
                            
                              ω
                              m
                            
                            ⁢
                            t
                          
                          +
                          
                            n
                            ⁢
                            
                              
                            
                            ⁢
                            π
                            ⁢
                            
                              
                            
                            ⁢
                            Δ
                          
                          +
                          
                            φ
                            0
                          
                          -
                          
                            φ
                            0
                            -
                          
                        
                        )
                      
                    
                  
                  ]
                
              
              +
              
                
                  
                    n
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Δ
                      /
                      π
                    
                  
                  
                    1
                    -
                    
                      
                        (
                        
                          2
                          ⁢
                          n
                          ⁢
                          
                            
                          
                          ⁢
                          Δ
                        
                        )
                      
                      2
                    
                  
                
                ·
                
                  A
                  +
                
                ·
                
                  [
                  
                    
                      cos
                      ⁡
                      
                        (
                        
                          
                            
                              -
                              
                                ω
                                m
                              
                            
                            ⁢
                            t
                          
                          +
                          
                            
                              ω
                              0
                              +
                            
                            ⁢
                            
                              T
                              1
                            
                          
                          +
                          
                            n
                            ⁢
                            
                              
                            
                            ⁢
                            π
                            ⁢
                            
                              
                            
                            ⁢
                            Δ
                          
                          +
                          
                            φ
                            0
                          
                          -
                          
                            φ
                            0
                            -
                          
                        
                        )
                      
                    
                    -
                    
                      cos
                      ⁡
                      
                        (
                        
                          
                            
                              -
                              
                                ω
                                m
                              
                            
                            ⁢
                            t
                          
                          +
                          
                            n
                            ⁢
                            
                              
                            
                            ⁢
                            π
                            ⁢
                            
                              
                            
                            ⁢
                            Δ
                          
                          +
                          
                            φ
                            0
                          
                          -
                          
                            φ
                            0
                            +
                          
                        
                        )
                      
                    
                  
                  ]
                
              
            
            =
            
              
                
                  
                    n
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Δ
                      /
                      π
                    
                  
                  
                    1
                    -
                    
                      
                        (
                        
                          2
                          ⁢
                          n
                          ⁢
                          
                            
                          
                          ⁢
                          Δ
                        
                        )
                      
                      2
                    
                  
                
                ·
                
                  
                    
                      A
                      -
                    
                    +
                    A
                    +
                  
                  2
                
                ·
                
                  {
                  
                    
                      
                        
                          
                            
                              (
                              
                                1
                                +
                                δ
                              
                              )
                            
                            ·
                            
                              [
                              
                                
                                  
                                    
                                      
                                        cos
                                        ⁢
                                        
                                          (
                                          
                                            
                                              
                                                ω
                                                m
                                              
                                              ⁢
                                              t
                                            
                                            +
                                            
                                              
                                                ω
                                                0
                                                -
                                              
                                              ⁢
                                              
                                                T
                                                1
                                              
                                            
                                            +
                                            
                                              n
                                              ⁢
                                              
                                                
                                              
                                              ⁢
                                              π
                                              ⁢
                                              
                                                
                                              
                                              ⁢
                                              Δ
                                            
                                            +
                                            
                                              φ
                                              0
                                            
                                            -
                                            
                                              φ
                                              0
                                              -
                                            
                                          
                                          )
                                        
                                      
                                      -
                                    
                                  
                                
                                
                                  
                                    
                                      cos
                                      ⁡
                                      
                                        (
                                        
                                          
                                            
                                              ω
                                              m
                                            
                                            ⁢
                                            t
                                          
                                          +
                                          
                                            n
                                            ⁢
                                            
                                              
                                            
                                            ⁢
                                            π
                                            ⁢
                                            
                                              
                                            
                                            ⁢
                                            Δ
                                          
                                          +
                                          
                                            φ
                                            0
                                          
                                          -
                                          
                                            φ
                                            0
                                            -
                                          
                                        
                                        )
                                      
                                    
                                  
                                
                              
                              ]
                            
                          
                          +
                        
                      
                    
                    
                      
                        
                          
                            (
                            
                              1
                              -
                              δ
                            
                            )
                          
                          ·
                          
                            [
                            
                              
                                
                                  
                                    
                                      cos
                                      ⁡
                                      
                                        (
                                        
                                          
                                            
                                              -
                                              
                                                ω
                                                m
                                              
                                            
                                            ⁢
                                            t
                                          
                                          +
                                          
                                            
                                              ω
                                              0
                                              +
                                            
                                            ⁢
                                            
                                              T
                                              1
                                            
                                          
                                          +
                                          
                                            n
                                            ⁢
                                            
                                              
                                            
                                            ⁢
                                            π
                                            ⁢
                                            
                                              
                                            
                                            ⁢
                                            Δ
                                          
                                          +
                                          
                                            φ
                                            0
                                          
                                          -
                                          
                                            φ
                                            0
                                            +
                                          
                                        
                                        )
                                      
                                    
                                    -
                                  
                                
                              
                              
                                
                                  
                                    cos
                                    ⁡
                                    
                                      (
                                      
                                        
                                          
                                            -
                                            
                                              ω
                                              m
                                            
                                          
                                          ⁢
                                          t
                                        
                                        +
                                        
                                          n
                                          ⁢
                                          
                                            
                                          
                                          ⁢
                                          π
                                          ⁢
                                          
                                            
                                          
                                          ⁢
                                          Δ
                                        
                                        +
                                        
                                          φ
                                          0
                                        
                                        -
                                        
                                          φ
                                          0
                                          +
                                        
                                      
                                      )
                                    
                                  
                                
                              
                            
                            ]
                          
                        
                      
                    
                  
                  }
                
              
              ≈
              
                
                  -
                  4
                
                ⁢
                
                  
                    cos
                    ⁡
                    
                      (
                      
                        
                          φ
                          0
                        
                        -
                        
                          
                            
                              φ
                              0
                              -
                            
                            +
                            
                              φ
                              0
                              +
                            
                          
                          2
                        
                      
                      )
                    
                  
                  ·
                  
                    cos
                    ⁡
                    
                      (
                      
                        Δ
                        ⁢
                        
                          
                        
                        ⁢
                        n
                        ⁢
                        
                          
                        
                        ⁢
                        π
                      
                      )
                    
                  
                  ·
                  
                    cos
                    ⁡
                    
                      (
                      
                        
                          Δ
                          ⁢
                          
                            
                          
                          ⁢
                          π
                        
                        2
                      
                      )
                    
                  
                  ·
                  
                    cos
                    ⁡
                    
                      (
                      
                        
                          
                            ω
                            m
                          
                          ⁢
                          t
                        
                        +
                        
                          
                            Δ
                            ⁢
                            
                              
                            
                            ⁢
                            π
                          
                          2
                        
                        -
                        
                          
                            
                              φ
                              0
                              -
                            
                            +
                            
                              φ
                              0
                              +
                            
                          
                          2
                        
                        +
                        δφ
                      
                      )
                    
                  
                
              
            
          
        
        ⁢
        
          

        
        ⁢
        
          
        
        ⁢
        
          where
          ,
          
            

          
          ⁢
          
            
          
          ⁢
          
            δ
            =
            
              
                (
                
                  
                    A
                    -
                  
                  -
                  
                    A
                    +
                  
                
                )
              
              /
              
                
                  (
                  
                    
                      A
                      -
                    
                    +
                    
                      A
                      +
                    
                  
                  )
                
                .
                
                  

                
                ⁢
                
                  
                
                ⁢
                
                  
                    
                      
                        δφ
                        =
                        
                        ⁢
                        
                          
                            tan
                            
                              -
                              1
                            
                          
                          (
                          
                            
                              
                                
                                  
                                    tan
                                    (
                                    
                                      
                                        φ
                                        0
                                      
                                      -
                                      
                                        
                                          
                                            φ
                                            0
                                            -
                                          
                                          +
                                          
                                            φ
                                            0
                                            +
                                          
                                        
                                        2
                                      
                                    
                                    )
                                  
                                  ·
                                
                              
                            
                            
                              
                                
                                  
                                    
                                      
                                        sin
                                        ⁡
                                        
                                          (
                                          
                                            Δ
                                            ⁢
                                            
                                              
                                            
                                            ⁢
                                            n
                                            ⁢
                                            
                                              
                                            
                                            ⁢
                                            π
                                          
                                          )
                                        
                                      
                                      ·
                                      
                                        sin
                                        (
                                        
                                          Δπ
                                          2
                                        
                                        )
                                      
                                    
                                    +
                                    
                                      δ
                                      ·
                                      
                                        cos
                                        ⁡
                                        
                                          (
                                          
                                            Δ
                                            ⁢
                                            
                                              
                                            
                                            ⁢
                                            n
                                            ⁢
                                            
                                              
                                            
                                            ⁢
                                            π
                                          
                                          )
                                        
                                      
                                      ·
                                      
                                        cos
                                        (
                                        
                                          Δπ
                                          2
                                        
                                        )
                                      
                                    
                                  
                                  
                                    
                                      
                                        cos
                                        ⁡
                                        
                                          (
                                          
                                            Δ
                                            ⁢
                                            
                                              
                                            
                                            ⁢
                                            n
                                            ⁢
                                            
                                              
                                            
                                            ⁢
                                            π
                                          
                                          )
                                        
                                      
                                      ·
                                      
                                        cos
                                        (
                                        
                                          Δπ
                                          2
                                        
                                        )
                                      
                                    
                                    +
                                    
                                      δ
                                      ·
                                      
                                        sin
                                        ⁡
                                        
                                          (
                                          
                                            Δ
                                            ⁢
                                            
                                              
                                            
                                            ⁢
                                            n
                                            ⁢
                                            
                                              
                                            
                                            ⁢
                                            π
                                          
                                          )
                                        
                                      
                                      ·
                                      
                                        sin
                                        (
                                        
                                          Δπ
                                          2
                                        
                                        )
                                      
                                    
                                  
                                
                              
                            
                          
                          )
                        
                      
                    
                  
                  
                    
                      
                        ≈
                        
                        ⁢
                        
                          
                            tan
                            
                              -
                              1
                            
                          
                          ⁡
                          
                            (
                            
                              
                                tan
                                (
                                
                                  
                                    φ
                                    0
                                  
                                  -
                                  
                                    
                                      
                                        φ
                                        0
                                        -
                                      
                                      +
                                      
                                        φ
                                        0
                                        +
                                      
                                    
                                    2
                                  
                                
                                )
                              
                              ·
                              
                                [
                                
                                  δ
                                  +
                                  
                                    
                                      tan
                                      (
                                      
                                        
                                          Δ
                                          ⁢
                                          
                                            
                                          
                                          ⁢
                                          π
                                        
                                        2
                                      
                                      )
                                    
                                    ·
                                    
                                      tan
                                      ⁡
                                      
                                        (
                                        
                                          Δ
                                          ⁢
                                          
                                            
                                          
                                          ⁢
                                          n
                                          ⁢
                                          
                                            
                                          
                                          ⁢
                                          π
                                        
                                        )
                                      
                                    
                                  
                                
                                ]
                              
                            
                            )
                          
                        
                      
                    
                  
                  
                    
                      
                        ≈
                        
                        ⁢
                        
                          
                            [
                            
                              δ
                              +
                              
                                
                                  tan
                                  (
                                  
                                    
                                      Δ
                                      ⁢
                                      
                                        
                                      
                                      ⁢
                                      π
                                    
                                    2
                                  
                                  )
                                
                                ·
                                
                                  tan
                                  ⁡
                                  
                                    (
                                    
                                      Δ
                                      ⁢
                                      
                                        
                                      
                                      ⁢
                                      n
                                      ⁢
                                      
                                        
                                      
                                      ⁢
                                      π
                                    
                                    )
                                  
                                
                              
                            
                            ]
                          
                          ·
                          
                            
                              (
                              
                                
                                  φ
                                  0
                                
                                -
                                
                                  
                                    
                                      φ
                                      0
                                      -
                                    
                                    +
                                    
                                      φ
                                      0
                                      +
                                    
                                  
                                  2
                                
                              
                              )
                            
                            .
                          
                        
                      
                    
                  
                
              
            
          
        
      
    
    
      
        (
        37
        )
      
    
  




Normally,




  
    Δ
    =
    1
  
  ,
  
    
      
        
          φ
          0
          -
        
        +
        
          φ
          0
          +
        
      
      2
    
    ≈
    
      φ
      0
    
  
  ,
  
    δ
    =
    
      
        
          
            A
            -
          
          -
          
            A
            +
          
        
        
          
            A
            -
          
          +
          
            A
            +
          
        
      
      ≈
      0
    
  
  ,




and δφ≈0. Therefore, the FM carrier phase is approximately





  
    
      
        ω
        m
      
      ⁢
      t
    
    +
    
      πΔ
      2
    
    +
    
      
        
          φ
          0
          +
        
        +
        
          φ
          0
          -
        
      
      2
    
  
  ,
  
    with
    ⁢
    
      
    
    ⁢
    
      
        
          φ
          0
          +
        
        +
        
          φ
          0
          -
        
      
      2
    
  




being the error term. The remodulated signal phase is determined by multiplying the FM carrier phase by the modulation rate, 2n, i.e.,





  
    
      
        
          2
          ⁢
          
            n
            ·
            
              (
              
                
                  
                    ω
                    m
                  
                  ⁢
                  t
                
                +
                
                  πΔ
                  2
                
                +
                
                  
                    
                      φ
                      0
                      +
                    
                    +
                    
                      φ
                      0
                      -
                    
                  
                  2
                
              
              )
            
          
        
        =
        
          
            
              ω
              0
            
            ⁢
            t
          
          +
          
            n
            ⁢
            
              
            
            ⁢
            πΔ
          
          +
          
            n
            ·
            
              (
              
                
                  φ
                  0
                  +
                
                +
                
                  φ
                  0
                  -
                
              
              )
            
          
        
      
    
    
      
        (
        38
        )
      
    
  




with the phase reference error being

Δφ=n·(φ0+−φ0−).  (39)

The following equations provide a phase error of the horizontal coil signal, as illustrated below by assuming that:




  
    
      
        
          φ
          k
          -
        
        =
        
          
            φ
            k
            +
          
          +
          
            δφ
            k
          
        
      
    
    
      
        
      
    
  
  
    
      
        
          A
          k
          -
        
        =
        
          
            A
            k
            +
          
          +
          
            δ
            ⁢
            
              
            
            ⁢
            
              A
              k
            
          
        
      
    
    
      
        
      
    
  
  
    
      
        
          
            ∂
            
              φ
              0
              -
            
          
          
            ∂
            
              φ
              j
            
          
        
        =
        
          
            
              
                ∑
                
                  k
                  =
                  0
                
                M
              
              ⁢
              
                
                  A
                  j
                  -
                
                ⁢
                
                  A
                  k
                  -
                
                ⁢
                
                  cos
                  ⁡
                  
                    (
                    
                      
                        φ
                        j
                        -
                      
                      -
                      
                        φ
                        k
                        -
                      
                    
                    )
                  
                
              
            
            
              
                ∑
                
                  j
                  =
                  0
                
                M
              
              ⁢
              
                
                  ∑
                  
                    k
                    =
                    0
                  
                  M
                
                ⁢
                
                  
                    A
                    j
                    -
                  
                  ⁢
                  
                    A
                    k
                    -
                  
                  ⁢
                  
                    cos
                    ⁡
                    
                      (
                      
                        
                          φ
                          j
                          -
                        
                        -
                        
                          φ
                          k
                          -
                        
                      
                      )
                    
                  
                
              
            
          
          ≈
          
            
              
                ∑
                
                  k
                  =
                  0
                
                M
              
              ⁢
              
                
                  A
                  j
                
                ⁢
                
                  A
                  k
                
                ⁢
                
                  cos
                  ⁡
                  
                    (
                    
                      
                        φ
                        j
                      
                      -
                      
                        φ
                        k
                      
                    
                    )
                  
                
              
            
            
              
                ∑
                
                  j
                  =
                  0
                
                M
              
              ⁢
              
                
                  ∑
                  
                    k
                    =
                    0
                  
                  M
                
                ⁢
                
                  
                    A
                    j
                  
                  ⁢
                  
                    A
                    k
                  
                  ⁢
                  
                    cos
                    ⁡
                    
                      (
                      
                        
                          φ
                          j
                        
                        -
                        
                          φ
                          k
                        
                      
                      )
                    
                  
                
              
            
          
        
      
    
    
      
        
      
    
  
  
    
      
        
          
            ∂
            
              φ
              0
              -
            
          
          
            ∂
            
              A
              j
            
          
        
        =
        
          
            
              
                ∑
                
                  k
                  =
                  0
                
                M
              
              ⁢
              
                
                  A
                  k
                  -
                
                ⁢
                
                  sin
                  ⁡
                  
                    (
                    
                      
                        φ
                        j
                        -
                      
                      -
                      
                        φ
                        k
                        -
                      
                    
                    )
                  
                
              
            
            
              
                ∑
                
                  j
                  =
                  0
                
                M
              
              ⁢
              
                
                  ∑
                  
                    k
                    =
                    0
                  
                  M
                
                ⁢
                
                  
                    A
                    j
                    -
                  
                  ⁢
                  
                    A
                    k
                    -
                  
                  ⁢
                  
                    cos
                    ⁡
                    
                      (
                      
                        
                          φ
                          j
                          -
                        
                        -
                        
                          φ
                          k
                          -
                        
                      
                      )
                    
                  
                
              
            
          
          ≈
          
            
              
                ∑
                
                  k
                  =
                  0
                
                M
              
              ⁢
              
                
                  A
                  k
                
                ⁢
                
                  sin
                  ⁡
                  
                    (
                    
                      
                        φ
                        j
                      
                      -
                      
                        φ
                        k
                      
                    
                    )
                  
                
              
            
            
              
                ∑
                
                  j
                  =
                  0
                
                M
              
              ⁢
              
                
                  ∑
                  
                    k
                    =
                    0
                  
                  M
                
                ⁢
                
                  
                    A
                    j
                  
                  ⁢
                  
                    A
                    k
                  
                  ⁢
                  
                    cos
                    ⁡
                    
                      (
                      
                        
                          φ
                          j
                        
                        -
                        
                          φ
                          k
                        
                      
                      )
                    
                  
                
              
            
          
        
      
    
    
      
        
      
    
  
  
    
      
        
          
            φ
            0
            +
          
          -
          
            φ
            0
            -
          
        
        ≈
        
          
            
              ∑
              
                j
                =
                0
              
              M
            
            ⁢
            
              
                
                  ∂
                  
                    φ
                    0
                    -
                  
                
                
                  ∂
                  
                    φ
                    j
                  
                
              
              ·
              
                δφ
                j
              
            
          
          +
          
            
              ∑
              
                j
                =
                0
              
              M
            
            ⁢
            
              
                
                  
                    ∂
                    
                      φ
                      0
                      -
                    
                  
                  
                    ∂
                    
                      A
                      j
                    
                  
                
                ·
                δ
              
              ⁢
              
                
              
              ⁢
              
                A
                j
              
            
          
        
      
    
    
      
        
      
    
  
  
    
      
        
          
        
        ⁢
        
          =
          
            
              
                
                  ∑
                  
                    j
                    =
                    0
                  
                  M
                
                ⁢
                
                  
                    ∑
                    
                      k
                      =
                      0
                    
                    M
                  
                  ⁢
                  
                    
                      A
                      j
                    
                    ⁢
                    
                      A
                      k
                    
                    ⁢
                    
                      
                        cos
                        ⁡
                        
                          (
                          
                            
                              φ
                              j
                            
                            -
                            
                              φ
                              k
                            
                          
                          )
                        
                      
                      ·
                      
                        δφ
                        j
                      
                    
                  
                
              
              
                
                  ∑
                  
                    j
                    =
                    0
                  
                  M
                
                ⁢
                
                  
                    ∑
                    
                      k
                      =
                      0
                    
                    M
                  
                  ⁢
                  
                    
                      A
                      j
                    
                    ⁢
                    
                      A
                      k
                    
                    ⁢
                    
                      cos
                      ⁡
                      
                        (
                        
                          
                            φ
                            j
                          
                          -
                          
                            φ
                            k
                          
                        
                        )
                      
                    
                  
                
              
            
            +
          
        
      
    
    
      
        
      
    
  
  
    
      
        
          
        
        ⁢
        
          
            
              ∑
              
                j
                =
                0
              
              M
            
            ⁢
            
              
                ∑
                
                  k
                  =
                  0
                
                M
              
              ⁢
              
                
                  A
                  k
                
                ⁢
                
                  
                    sin
                    ⁡
                    
                      (
                      
                        
                          φ
                          j
                        
                        -
                        
                          φ
                          k
                        
                      
                      )
                    
                  
                  ·
                  δ
                
                ⁢
                
                  
                
                ⁢
                
                  A
                  j
                
              
            
          
          
            
              ∑
              
                j
                =
                0
              
              M
            
            ⁢
            
              
                ∑
                
                  k
                  =
                  0
                
                M
              
              ⁢
              
                
                  A
                  j
                
                ⁢
                
                  A
                  k
                
                ⁢
                
                  cos
                  ⁡
                  
                    (
                    
                      
                        φ
                        j
                      
                      -
                      
                        φ
                        k
                      
                    
                    )
                  
                
              
            
          
        
      
    
    
      
        
      
    
  
  
    
      
        
          
        
        ⁢
        
          =
          
            
              
                
                  
                    
                      
                        ∑
                        
                          j
                          =
                          0
                        
                        M
                      
                      ⁢
                      
                        
                          ∑
                          
                            k
                            =
                            0
                          
                          M
                        
                        ⁢
                        
                          
                            A
                            j
                          
                          ⁢
                          
                            A
                            k
                          
                          ⁢
                          
                            
                              cos
                              ⁡
                              
                                (
                                
                                  
                                    φ
                                    j
                                  
                                  -
                                  
                                    φ
                                    k
                                  
                                
                                )
                              
                            
                            ·
                            
                              δφ
                              j
                            
                          
                        
                      
                    
                    +
                  
                
              
              
                
                  
                    
                      ∑
                      
                        j
                        =
                        0
                      
                      M
                    
                    ⁢
                    
                      
                        ∑
                        
                          k
                          =
                          0
                        
                        M
                      
                      ⁢
                      
                        
                          A
                          k
                        
                        ⁢
                        
                          
                            sin
                            ⁡
                            
                              (
                              
                                
                                  φ
                                  j
                                
                                -
                                
                                  φ
                                  k
                                
                              
                              )
                            
                          
                          ·
                          δ
                        
                        ⁢
                        
                          
                        
                        ⁢
                        
                          A
                          j
                        
                      
                    
                  
                
              
            
            
              
                ∑
                
                  j
                  =
                  0
                
                M
              
              ⁢
              
                
                  ∑
                  
                    k
                    =
                    0
                  
                  M
                
                ⁢
                
                  
                    A
                    j
                  
                  ⁢
                  
                    A
                    k
                  
                  ⁢
                  
                    cos
                    ⁡
                    
                      (
                      
                        
                          φ
                          j
                        
                        -
                        
                          φ
                          k
                        
                      
                      )
                    
                  
                
              
            
          
        
      
    
    
      
        
      
    
  
  
    
      with
    
    
      
        
      
    
  
  
    
      
        
          δ
          ⁢
          
            
          
          ⁢
          
            A
            j
          
        
        =
        
          
            
              
                z
                j
              
              ·
              δ
            
            ⁢
            
              
            
            ⁢
            
              I
              j
            
          
          
            
              
                (
                
                  x
                  -
                  
                    x
                    j
                  
                
                )
              
              2
            
            +
            
              z
              j
              2
            
          
        
      
    
    
      
        
      
    
  
  
    
      
        
          
        
        ⁢
        
          =
          
            
              
                z
                j
              
              ·
              
                (
                
                  
                    I
                    j
                    +
                  
                  -
                  
                    I
                    j
                    -
                  
                
                )
              
            
            
              
                
                  (
                  
                    x
                    -
                    
                      x
                      j
                    
                  
                  )
                
                2
              
              +
              
                z
                j
                2
              
            
          
        
      
    
    
      
        
      
    
  
  
    
      
        
          
        
        ⁢
        
          ≈
          
            
              
                
                  z
                  j
                
                ·
                
                  I
                  j
                
              
              
                
                  
                    (
                    
                      x
                      -
                      
                        x
                        j
                      
                    
                    )
                  
                  2
                
                +
                
                  z
                  j
                  2
                
              
            
            ·
            
              
                (
                
                  
                    I
                    j
                    +
                  
                  -
                  
                    I
                    j
                    -
                  
                
                )
              
              
                I
                j
              
            
          
        
      
    
    
      
        
      
    
  
  
    
      
        
          
        
        ⁢
        
          =
          
            
              A
              j
            
            ·
            
              
                
                  δ
                  ⁢
                  
                    
                  
                  ⁢
                  
                    I
                    j
                  
                
                
                  I
                  j
                
              
              .
            
          
        
      
    
    
      
        
      
    
  




Similarly the phase error of the vertical coil signal is




  
    
      
        
          
            
              φ
              
                v
                ⁢
                
                  
                
                ⁢
                0
              
              +
            
            -
            
              φ
              
                v
                ⁢
                
                  
                
                ⁢
                0
              
              -
            
          
          ≈
          
            
              
                
                  
                    
                      
                        
                          ∑
                          
                            j
                            =
                            0
                          
                          M
                        
                        ⁢
                        
                          
                            ∑
                            
                              k
                              =
                              0
                            
                            M
                          
                          ⁢
                          
                            
                              B
                              j
                            
                            ⁢
                            
                              B
                              k
                            
                            ⁢
                            
                              
                                cos
                                ⁡
                                
                                  (
                                  
                                    
                                      φ
                                      j
                                    
                                    -
                                    
                                      φ
                                      k
                                    
                                  
                                  )
                                
                              
                              ·
                              
                                δφ
                                j
                              
                            
                          
                        
                      
                      +
                    
                  
                
                
                  
                    
                      
                        ∑
                        
                          j
                          =
                          0
                        
                        M
                      
                      ⁢
                      
                        
                          ∑
                          
                            k
                            =
                            0
                          
                          M
                        
                        ⁢
                        
                          
                            B
                            k
                          
                          ⁢
                          
                            
                              cos
                              ⁡
                              
                                (
                                
                                  
                                    φ
                                    j
                                  
                                  -
                                  
                                    φ
                                    k
                                  
                                
                                )
                              
                            
                            ·
                            δ
                          
                          ⁢
                          
                            
                          
                          ⁢
                          
                            B
                            j
                          
                        
                      
                    
                  
                
              
              
                
                  ∑
                  
                    j
                    =
                    0
                  
                  M
                
                ⁢
                
                  
                    ∑
                    
                      k
                      =
                      0
                    
                    M
                  
                  ⁢
                  
                    
                      B
                      j
                    
                    ⁢
                    
                      B
                      k
                    
                    ⁢
                    
                      cos
                      ⁡
                      
                        (
                        
                          
                            φ
                            j
                          
                          -
                          
                            φ
                            k
                          
                        
                        )
                      
                    
                  
                
              
            
            .
            
              

            
            ⁢
            where
          
        
        ,
        
          
        
        ⁢
        
          
            δ
            ⁢
            
              
            
            ⁢
            
              B
              j
            
          
          ≈
          
            
              B
              j
            
            ·
            
              
                
                  δ
                  ⁢
                  
                    
                  
                  ⁢
                  
                    I
                    j
                  
                
                
                  I
                  j
                
              
              .
            
          
        
      
    
    
      
        (
        40
        )
      
    
  




By considering Δφ≠0, some embodiments of the present invention can derive the in-phase and quadrature components of the electromagnetic field from Equation (25) with phase ω0t+nπΔ+Δφ, leading to more accurate measurements of hnI(xj) and hnQ(xj), which allows for more accurate locating. The values of hnI(xj) and hnQ(xj) are shown as follows:




  
    
      
        
          
            
              
                
                  
                    a
                    
                      2
                      ⁢
                      n
                    
                  
                  ·
                  
                    cos
                    ⁡
                    
                      (
                      
                        
                          ω
                          0
                        
                        ⁢
                        t
                      
                      )
                    
                  
                
                +
                
                  
                    b
                    
                      2
                      ⁢
                      n
                    
                  
                  ·
                  
                    sin
                    ⁡
                    
                      (
                      
                        
                          ω
                          0
                        
                        ⁢
                        t
                      
                      )
                    
                  
                
              
              ≈
              
              ⁢
              
                
                  sin
                  ⁡
                  
                    (
                    
                      n
                      ⁢
                      
                        
                      
                      ⁢
                      π
                      ⁢
                      
                        
                      
                      ⁢
                      Δ
                    
                    )
                  
                
                
                  n
                  ⁢
                  
                    
                  
                  ⁢
                  π
                  ⁢
                  
                    
                  
                  ⁢
                  Δ
                
              
            
          
        
        
          
            
              
              ⁢
              
                {
                
                  
                    
                      
                        
                          
                            
                              
                                [
                                
                                  
                                    
                                      
                                        
                                          
                                            
                                              h
                                              n
                                              I
                                            
                                            ⁡
                                            
                                              (
                                              
                                                x
                                                
                                                  j
                                                  ⁢
                                                  
                                                    
                                                  
                                                
                                              
                                              )
                                            
                                          
                                          ·
                                          
                                            cos
                                            ⁡
                                            
                                              (
                                              
                                                Δ
                                                ⁢
                                                
                                                  
                                                
                                                ⁢
                                                φ
                                              
                                              )
                                            
                                          
                                        
                                        +
                                      
                                    
                                  
                                  
                                    
                                      
                                        
                                          
                                            h
                                            n
                                            Q
                                          
                                          ⁡
                                          
                                            (
                                            
                                              x
                                              j
                                            
                                            )
                                          
                                        
                                        ·
                                        
                                          sin
                                          ⁡
                                          
                                            (
                                            
                                              Δ
                                              ⁢
                                              
                                                
                                              
                                              ⁢
                                              φ
                                            
                                            )
                                          
                                        
                                      
                                    
                                  
                                
                                ]
                              
                              ·
                            
                          
                        
                        
                          
                            
                              
                                cos
                                ⁡
                                
                                  (
                                  
                                    
                                      
                                        ω
                                        0
                                      
                                      ⁢
                                      t
                                    
                                    +
                                    
                                      n
                                      ⁢
                                      
                                        
                                      
                                      ⁢
                                      πΔ
                                    
                                    +
                                    
                                      Δ
                                      ⁢
                                      
                                        
                                      
                                      ⁢
                                      φ
                                    
                                  
                                  )
                                
                              
                              -
                            
                          
                        
                      
                    
                  
                  
                    
                      
                        
                          
                            
                              
                                [
                                
                                  
                                    
                                      
                                        
                                          
                                            
                                              
                                                h
                                                n
                                                Q
                                              
                                              ⁡
                                              
                                                (
                                                
                                                  x
                                                  
                                                    j
                                                    ⁢
                                                    
                                                      
                                                    
                                                  
                                                
                                                )
                                              
                                            
                                            ·
                                            cos
                                          
                                          ⁢
                                          
                                            (
                                            
                                              Δ
                                              ⁢
                                              
                                                
                                              
                                              ⁢
                                              φ
                                            
                                            )
                                          
                                        
                                        -
                                      
                                    
                                  
                                  
                                    
                                      
                                        
                                          
                                            h
                                            n
                                            l
                                          
                                          ⁡
                                          
                                            (
                                            
                                              x
                                              j
                                            
                                            )
                                          
                                        
                                        ·
                                        
                                          sin
                                          ⁡
                                          
                                            (
                                            
                                              Δ
                                              ⁢
                                              
                                                
                                              
                                              ⁢
                                              φ
                                            
                                            )
                                          
                                        
                                      
                                    
                                  
                                
                                ]
                              
                              ·
                            
                          
                        
                        
                          
                            
                              sin
                              ⁡
                              
                                (
                                
                                  
                                    
                                      ω
                                      0
                                    
                                    ⁢
                                    t
                                  
                                  +
                                  
                                    n
                                    ⁢
                                    
                                      
                                    
                                    ⁢
                                    πΔ
                                  
                                  +
                                  
                                    Δ
                                    ⁢
                                    
                                      
                                    
                                    ⁢
                                    φ
                                  
                                
                                )
                              
                            
                          
                        
                      
                    
                  
                
                }
              
            
          
        
        
          
            
              =
              
              ⁢
              
                
                  sin
                  ⁡
                  
                    (
                    
                      n
                      ⁢
                      
                        
                      
                      ⁢
                      π
                      ⁢
                      
                        
                      
                      ⁢
                      Δ
                    
                    )
                  
                
                
                  n
                  ⁢
                  
                    
                  
                  ⁢
                  π
                  ⁢
                  
                    
                  
                  ⁢
                  Δ
                
              
            
          
        
        
          
            
              
              ⁢
              
                {
                
                  
                    
                      
                        
                          
                            
                              
                                h
                                n
                                I
                              
                              ⁡
                              
                                (
                                
                                  x
                                  
                                    j
                                    ⁢
                                    
                                      
                                    
                                  
                                
                                )
                              
                            
                            mod
                          
                          ·
                          
                            cos
                            ⁡
                            
                              (
                              
                                
                                  
                                    ω
                                    0
                                  
                                  ⁢
                                  t
                                
                                +
                                
                                  n
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  πΔ
                                
                                +
                                
                                  Δ
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  φ
                                
                              
                              )
                            
                          
                        
                        -
                      
                    
                  
                  
                    
                      
                        
                          
                            
                              h
                              n
                              Q
                            
                            ⁡
                            
                              (
                              
                                x
                                
                                  j
                                  ⁢
                                  
                                    
                                  
                                
                              
                              )
                            
                          
                          mod
                        
                        ·
                        
                          sin
                          ⁡
                          
                            (
                            
                              
                                
                                  ω
                                  0
                                
                                ⁢
                                t
                              
                              +
                              
                                n
                                ⁢
                                
                                  
                                
                                ⁢
                                πΔ
                              
                              +
                              
                                Δ
                                ⁢
                                
                                  
                                
                                ⁢
                                φ
                              
                            
                            )
                          
                        
                      
                    
                  
                
                }
              
            
          
        
      
    
    
      
        (
        41
        )
      
    
  




where, hnI(xj)mod and hnQ(xj)mod are defined as:





  
    
      
        
          
            
              
                h
                n
                I
              
              ⁡
              
                (
                
                  x
                  
                    j
                    ⁢
                    
                      
                    
                  
                
                )
              
            
            mod
          
          =
          
            
              
                sin
                ⁡
                
                  (
                  
                    n
                    ⁢
                    
                      
                    
                    ⁢
                    π
                    ⁢
                    
                      
                    
                    ⁢
                    Δ
                  
                  )
                
              
              
                n
                ⁢
                
                  
                
                ⁢
                π
                ⁢
                
                  
                
                ⁢
                Δ
              
            
            ·
            
              [
              
                
                  
                    
                      h
                      n
                      I
                    
                    ⁡
                    
                      (
                      
                        x
                        
                          j
                          ⁢
                          
                            
                          
                        
                      
                      )
                    
                  
                  ·
                  
                    cos
                    ⁡
                    
                      (
                      
                        Δ
                        ⁢
                        
                          
                        
                        ⁢
                        φ
                      
                      )
                    
                  
                
                +
                
                  
                    
                      h
                      n
                      Q
                    
                    ⁡
                    
                      (
                      
                        x
                        j
                      
                      )
                    
                  
                  ·
                  
                    sin
                    ⁡
                    
                      (
                      
                        Δ
                        ⁢
                        
                          
                        
                        ⁢
                        φ
                      
                      )
                    
                  
                
              
              ]
            
          
        
        ⁢
        
          

        
        ⁢
        
          
            
              
                h
                n
                Q
              
              ⁡
              
                (
                
                  x
                  
                    j
                    ⁢
                    
                      
                    
                  
                
                )
              
            
            mod
          
          =
          
            
              
                sin
                ⁡
                
                  (
                  
                    n
                    ⁢
                    
                      
                    
                    ⁢
                    π
                    ⁢
                    
                      
                    
                    ⁢
                    Δ
                  
                  )
                
              
              
                n
                ⁢
                
                  
                
                ⁢
                π
                ⁢
                
                  
                
                ⁢
                Δ
              
            
            ·
            
              [
              
                
                  
                    
                      h
                      n
                      Q
                    
                    ⁡
                    
                      (
                      
                        x
                        
                          j
                          ⁢
                          
                            
                          
                        
                      
                      )
                    
                  
                  ·
                  
                    cos
                    ⁡
                    
                      (
                      
                        Δ
                        ⁢
                        
                          
                        
                        ⁢
                        φ
                      
                      )
                    
                  
                
                -
                
                  
                    
                      h
                      n
                      I
                    
                    ⁡
                    
                      (
                      
                        x
                        j
                      
                      )
                    
                  
                  ·
                  
                    sin
                    ⁡
                    
                      (
                      
                        Δ
                        ⁢
                        
                          
                        
                        ⁢
                        φ
                      
                      )
                    
                  
                
              
              ]
            
          
        
        ⁢
        
          

        
        ⁢
        
          or
          ,
        
      
    
    
      
        (
        42
        )
      
    
  
  
    
      
        
          [
          
            
              
                
                  
                    
                      h
                      n
                      I
                    
                    ⁡
                    
                      (
                      
                        x
                        
                          j
                          ⁢
                          
                            
                          
                        
                      
                      )
                    
                  
                  mod
                
              
            
            
              
                
                  
                    
                      h
                      n
                      Q
                    
                    ⁡
                    
                      (
                      
                        x
                        
                          j
                          ⁢
                          
                            
                          
                        
                      
                      )
                    
                  
                  mod
                
              
            
          
          ]
        
        =
        
          
            
              
                sin
                ⁡
                
                  (
                  
                    n
                    ⁢
                    
                      
                    
                    ⁢
                    π
                    ⁢
                    
                      
                    
                    ⁢
                    Δ
                  
                  )
                
              
              
                n
                ⁢
                
                  
                
                ⁢
                π
                ⁢
                
                  
                
                ⁢
                Δ
              
            
            ·
            
              (
              
                
                  
                    
                      cos
                      ⁡
                      
                        (
                        
                          Δ
                          ⁢
                          
                            
                          
                          ⁢
                          φ
                        
                        )
                      
                    
                  
                  
                    
                      sin
                      ⁡
                      
                        (
                        Δφ
                        )
                      
                    
                  
                
                
                  
                    
                      -
                      
                        sin
                        ⁡
                        
                          (
                          Δφ
                          )
                        
                      
                    
                  
                  
                    
                      cos
                      ⁡
                      
                        (
                        Δφ
                        )
                      
                    
                  
                
              
              )
            
            ·
            
              
                [
                
                  
                    
                      
                        
                          h
                          n
                          I
                        
                        ⁡
                        
                          (
                          
                            x
                            
                              j
                              ⁢
                              
                                
                              
                            
                          
                          )
                        
                      
                    
                  
                  
                    
                      
                        
                          h
                          n
                          Q
                        
                        ⁡
                        
                          (
                          
                            x
                            
                              j
                              ⁢
                              
                                
                              
                            
                          
                          )
                        
                      
                    
                  
                
                ]
              
              ⁢
              
                

              
              [
              
                
                  
                    
                      
                        h
                        n
                        I
                      
                      ⁡
                      
                        (
                        
                          x
                          
                            j
                            ⁢
                            
                              
                            
                          
                        
                        )
                      
                    
                  
                
                
                  
                    
                      
                        h
                        n
                        Q
                      
                      ⁡
                      
                        (
                        
                          x
                          
                            j
                            ⁢
                            
                              
                            
                          
                        
                        )
                      
                    
                  
                
              
              ]
            
          
          =
          
            
              
                sin
                ⁡
                
                  (
                  
                    n
                    ⁢
                    
                      
                    
                    ⁢
                    π
                    ⁢
                    
                      
                    
                    ⁢
                    Δ
                  
                  )
                
              
              
                n
                ⁢
                
                  
                
                ⁢
                π
                ⁢
                
                  
                
                ⁢
                Δ
              
            
            ·
            
              (
              
                
                  
                    
                      cos
                      ⁡
                      
                        (
                        
                          Δ
                          ⁢
                          
                            
                          
                          ⁢
                          φ
                        
                        )
                      
                    
                  
                  
                    
                      -
                      
                        sin
                        ⁡
                        
                          (
                          Δφ
                          )
                        
                      
                    
                  
                
                
                  
                    
                      sin
                      ⁡
                      
                        (
                        Δφ
                        )
                      
                    
                  
                  
                    
                      cos
                      ⁡
                      
                        (
                        Δφ
                        )
                      
                    
                  
                
              
              )
            
            ·
            
              
                [
                
                  
                    
                      
                        
                          
                            h
                            n
                            I
                          
                          ⁡
                          
                            (
                            
                              x
                              
                                j
                                ⁢
                                
                                  
                                
                              
                            
                            )
                          
                        
                        mod
                      
                    
                  
                  
                    
                      
                        
                          
                            h
                            n
                            Q
                          
                          ⁡
                          
                            (
                            
                              x
                              
                                j
                                ⁢
                                
                                  
                                
                              
                            
                            )
                          
                        
                        mod
                      
                    
                  
                
                ]
              
              .
            
          
        
      
    
    
      
        (
        43
        )
      
    
  




For the vertical coil, the quadrature and in-phase fields are given by:




  
    
      
        
          [
          
            
              
                
                  
                    
                      h
                      v
                      I
                    
                    ⁡
                    
                      (
                      
                        x
                        j
                      
                      )
                    
                  
                  mod
                
              
            
            
              
                
                  
                    
                      h
                      v
                      Q
                    
                    ⁡
                    
                      (
                      
                        x
                        j
                      
                      )
                    
                  
                  mod
                
              
            
          
          ]
        
        =
        
          
        
        ⁢
        
          
            
              sin
              ⁡
              
                (
                
                  n
                  ⁢
                  
                    
                  
                  ⁢
                  π
                  ⁢
                  
                    
                  
                  ⁢
                  Δ
                
                )
              
            
            
              n
              ⁢
              
                
              
              ⁢
              π
              ⁢
              
                
              
              ⁢
              Δ
            
          
          ·
          
            
          
          ⁢
          
            (
            
              
                
                  
                    cos
                    ⁡
                    
                      (
                      
                        Δ
                        ⁢
                        
                          
                        
                        ⁢
                        
                          φ
                          v
                        
                      
                      )
                    
                  
                
                
                  
                    sin
                    ⁡
                    
                      (
                      
                        Δ
                        ⁢
                        
                          
                        
                        ⁢
                        
                          φ
                          v
                        
                      
                      )
                    
                  
                
              
              
                
                  
                    -
                    
                      sin
                      ⁡
                      
                        (
                        
                          Δ
                          ⁢
                          
                            
                          
                          ⁢
                          
                            φ
                            v
                          
                        
                        )
                      
                    
                  
                
                
                  
                    cos
                    ⁡
                    
                      (
                      
                        Δ
                        ⁢
                        
                          
                        
                        ⁢
                        
                          φ
                          v
                        
                      
                      )
                    
                  
                
              
            
            )
          
          ·
          
            [
            
              
                
                  
                    
                      h
                      v
                      I
                    
                    ⁡
                    
                      (
                      
                        x
                        j
                      
                      )
                    
                  
                
              
              
                
                  
                    
                      h
                      v
                      Q
                    
                    ⁡
                    
                      (
                      
                        x
                        j
                      
                      )
                    
                  
                
              
            
            ]
          
        
      
    
    
      
        (
        44
        )
      
    
  
  
    
      
        
          [
          
            
              
                
                  
                    h
                    v
                    I
                  
                  ⁡
                  
                    (
                    
                      x
                      j
                    
                    )
                  
                
              
            
            
              
                
                  
                    h
                    v
                    Q
                  
                  ⁡
                  
                    (
                    
                      x
                      j
                    
                    )
                  
                
              
            
          
          ]
        
        =
        
          
            
              sin
              ⁡
              
                (
                
                  n
                  ⁢
                  
                    
                  
                  ⁢
                  π
                  ⁢
                  
                    
                  
                  ⁢
                  Δ
                
                )
              
            
            
              n
              ⁢
              
                
              
              ⁢
              π
              ⁢
              
                
              
              ⁢
              Δ
            
          
          ·
          
            (
            
              
                
                  
                    cos
                    ⁡
                    
                      (
                      
                        Δ
                        ⁢
                        
                          
                        
                        ⁢
                        φ
                      
                      )
                    
                  
                
                
                  
                    -
                    
                      sin
                      ⁡
                      
                        (
                        
                          Δ
                          ⁢
                          
                            
                          
                          ⁢
                          φ
                        
                        )
                      
                    
                  
                
              
              
                
                  
                    sin
                    ⁡
                    
                      (
                      
                        Δ
                        ⁢
                        
                          
                        
                        ⁢
                        φ
                      
                      )
                    
                  
                
                
                  
                    cos
                    ⁡
                    
                      (
                      
                        Δ
                        ⁢
                        
                          
                        
                        ⁢
                        φ
                      
                      )
                    
                  
                
              
            
            )
          
          ·
          
            
              [
              
                
                  
                    
                      
                        
                          h
                          v
                          I
                        
                        ⁡
                        
                          (
                          
                            x
                            j
                          
                          )
                        
                      
                      mod
                    
                  
                
                
                  
                    
                      
                        
                          h
                          v
                          Q
                        
                        ⁡
                        
                          (
                          
                            x
                            j
                          
                          )
                        
                      
                      mod
                    
                  
                
              
              ]
            
            .
          
        
      
    
    
      
        
      
    
  




And for the top coil, the quadrature and in-phase fields are given by:




  
    
      
        
          [
          
            
              
                
                  
                    
                      h
                      t
                      I
                    
                    ⁡
                    
                      (
                      
                        x
                        j
                      
                      )
                    
                  
                  mod
                
              
            
            
              
                
                  
                    
                      h
                      t
                      Q
                    
                    ⁡
                    
                      (
                      
                        x
                        j
                      
                      )
                    
                  
                  mod
                
              
            
          
          ]
        
        =
        
          
            
              
                sin
                ⁡
                
                  (
                  
                    n
                    ⁢
                    
                      
                    
                    ⁢
                    π
                    ⁢
                    
                      
                    
                    ⁢
                    Δ
                  
                  )
                
              
              
                n
                ⁢
                
                  
                
                ⁢
                π
                ⁢
                
                  
                
                ⁢
                Δ
              
            
            ·
            
              (
              
                
                  
                    
                      cos
                      ⁡
                      
                        (
                        
                          Δ
                          ⁢
                          
                            
                          
                          ⁢
                          φ
                        
                        )
                      
                    
                  
                  
                    
                      sin
                      ⁡
                      
                        (
                        
                          Δ
                          ⁢
                          
                            
                          
                          ⁢
                          φ
                        
                        )
                      
                    
                  
                
                
                  
                    
                      -
                      
                        sin
                        ⁡
                        
                          (
                          
                            Δ
                            ⁢
                            
                              
                            
                            ⁢
                            φ
                          
                          )
                        
                      
                    
                  
                  
                    
                      cos
                      ⁡
                      
                        (
                        
                          Δ
                          ⁢
                          
                            
                          
                          ⁢
                          φ
                        
                        )
                      
                    
                  
                
              
              )
            
            ·
            
              
                [
                
                  
                    
                      
                        
                          h
                          t
                          I
                        
                        ⁡
                        
                          (
                          
                            x
                            j
                          
                          )
                        
                      
                    
                  
                  
                    
                      
                        
                          h
                          t
                          Q
                        
                        ⁡
                        
                          (
                          
                            x
                            j
                          
                          )
                        
                      
                    
                  
                
                ]
              
              ⁢
              
                

              
              ⁢
              
                
              
              [
              
                
                  
                    
                      
                        h
                        t
                        I
                      
                      ⁡
                      
                        (
                        
                          x
                          j
                        
                        )
                      
                    
                  
                
                
                  
                    
                      
                        h
                        t
                        Q
                      
                      ⁡
                      
                        (
                        
                          x
                          j
                        
                        )
                      
                    
                  
                
              
              ]
            
          
          =
          
            
              
                sin
                ⁡
                
                  (
                  
                    n
                    ⁢
                    
                      
                    
                    ⁢
                    π
                    ⁢
                    
                      
                    
                    ⁢
                    Δ
                  
                  )
                
              
              
                n
                ⁢
                
                  
                
                ⁢
                π
                ⁢
                
                  
                
                ⁢
                Δ
              
            
            ·
            
              (
              
                
                  
                    
                      cos
                      ⁡
                      
                        (
                        
                          Δ
                          ⁢
                          
                            
                          
                          ⁢
                          φ
                        
                        )
                      
                    
                  
                  
                    
                      -
                      
                        sin
                        ⁡
                        
                          (
                          
                            Δ
                            ⁢
                            
                              
                            
                            ⁢
                            φ
                          
                          )
                        
                      
                    
                  
                
                
                  
                    
                      sin
                      ⁡
                      
                        (
                        
                          Δ
                          ⁢
                          
                            
                          
                          ⁢
                          φ
                        
                        )
                      
                    
                  
                  
                    
                      cos
                      ⁡
                      
                        (
                        
                          Δ
                          ⁢
                          
                            
                          
                          ⁢
                          φ
                        
                        )
                      
                    
                  
                
              
              )
            
            ·
            
              
                [
                
                  
                    
                      
                        
                          
                            h
                            t
                            I
                          
                          ⁡
                          
                            (
                            
                              x
                              j
                            
                            )
                          
                        
                        mod
                      
                    
                  
                  
                    
                      
                        
                          
                            h
                            t
                            Q
                          
                          ⁡
                          
                            (
                            
                              x
                              j
                            
                            )
                          
                        
                        mod
                      
                    
                  
                
                ]
              
              .
            
          
        
      
    
    
      
        (
        45
        )
      
    
  




So the optimization algorithm for determining position can be formulated to minimize the following cost function:




  
    
      
        
          ∑
          j
        
        ⁢
        
          {
          
            
              
                
                  
                    
                      [
                      
                        
                          
                            
                              h
                              n
                              I
                            
                            ⁡
                            
                              (
                              
                                x
                                
                                  j
                                  ⁢
                                  
                                    
                                  
                                
                              
                              )
                            
                          
                          meas
                        
                        -
                        
                          
                            
                              h
                              n
                              I
                            
                            ⁡
                            
                              (
                              
                                x
                                
                                  j
                                  ⁢
                                  
                                    
                                  
                                
                              
                              )
                            
                          
                          mod
                        
                      
                      ]
                    
                    2
                  
                  +
                  
                    
                      [
                      
                        
                          
                            
                              h
                              n
                              Q
                            
                            ⁡
                            
                              (
                              
                                x
                                
                                  j
                                  ⁢
                                  
                                    
                                  
                                
                              
                              )
                            
                          
                          meas
                        
                        -
                        
                          
                            
                              h
                              n
                              Q
                            
                            ⁡
                            
                              (
                              
                                x
                                
                                  j
                                  ⁢
                                  
                                    
                                  
                                
                              
                              )
                            
                          
                          mod
                        
                      
                      ]
                    
                    2
                  
                  +
                
              
            
            
              
                
                  
                    
                      [
                      
                        
                          
                            
                              h
                              v
                              I
                            
                            ⁡
                            
                              (
                              
                                x
                                
                                  j
                                  ⁢
                                  
                                    
                                  
                                
                              
                              )
                            
                          
                          meas
                        
                        -
                        
                          
                            
                              h
                              v
                              I
                            
                            ⁡
                            
                              (
                              
                                x
                                
                                  j
                                  ⁢
                                  
                                    
                                  
                                
                              
                              )
                            
                          
                          mod
                        
                      
                      ]
                    
                    2
                  
                  +
                  
                    
                      [
                      
                        
                          
                            
                              h
                              v
                              Q
                            
                            ⁡
                            
                              (
                              
                                x
                                
                                  j
                                  ⁢
                                  
                                    
                                  
                                
                              
                              )
                            
                          
                          meas
                        
                        -
                        
                          
                            
                              h
                              v
                              Q
                            
                            ⁡
                            
                              (
                              
                                x
                                
                                  j
                                  ⁢
                                  
                                    
                                  
                                
                              
                              )
                            
                          
                          mod
                        
                      
                      ]
                    
                    2
                  
                  +
                
              
            
            
              
                
                  
                    
                      [
                      
                        
                          
                            
                              h
                              t
                              I
                            
                            ⁡
                            
                              (
                              
                                x
                                
                                  j
                                  ⁢
                                  
                                    
                                  
                                
                              
                              )
                            
                          
                          meas
                        
                        -
                        
                          
                            
                              h
                              t
                              I
                            
                            ⁡
                            
                              (
                              
                                x
                                
                                  j
                                  ⁢
                                  
                                    
                                  
                                
                              
                              )
                            
                          
                          mod
                        
                      
                      ]
                    
                    2
                  
                  +
                  
                    
                      [
                      
                        
                          
                            
                              h
                              t
                              Q
                            
                            ⁡
                            
                              (
                              
                                x
                                
                                  j
                                  ⁢
                                  
                                    
                                  
                                
                              
                              )
                            
                          
                          meas
                        
                        -
                        
                          
                            
                              h
                              t
                              Q
                            
                            ⁡
                            
                              (
                              
                                x
                                
                                  j
                                  ⁢
                                  
                                    
                                  
                                
                              
                              )
                            
                          
                          mod
                        
                      
                      ]
                    
                    2
                  
                
              
            
          
          }
        
      
    
    
      
        (
        46
        )
      
    
  




where, hnI(xj)meas, hnQ(xj)meas are the measured in-phase and quadrature field from horizontal coil 14; hvI(xj)meas, hvQ(xj)meas are the measured in-phase and quadrature field from vertical coil 19; htI(xj)meas, htQ(xj)meas are the measured in-phase and quadrature field from top coil 20. And hnI(xj)mod, hnQ(xj)mod, hvI(xj)mod, htQ(xj)mod, htI(xj)mod, htQ(xj)mod are functions of the kth line location (xk, zk), current Ik, phase φk, and the current amplitude variation εk, k=0, 1, . . . , M. In other words, the optimization algorithm compares the measured fields to the modulated fields to determine the value of the cost function so that, as provided below, locator 11 can determine a confidence level for the accuracy when locating the concealed line.

In general, the phase reference derived from the phase DPLL blocks 605 and 606 of FIG. 3 in the current signal select architecture would likely be biased due to the presence of bleedover lines. The phase measurement error term can be modeled as a function of locations, current and phase of each line, plus a current amplitude and phase variation term, namely δAj and δφj, wherein the phase measurement error term can be calculated according to the following:




  
    
      
        Δφ
        =
        
          n
          ·
          
            
              
                
                  
                    
                      
                        ∑
                        
                          j
                          =
                          0
                        
                        M
                      
                      ⁢
                      
                        
                          ∑
                          
                            k
                            =
                            0
                          
                          M
                        
                        ⁢
                        
                          
                            A
                            j
                          
                          ⁢
                          
                            A
                            k
                          
                          ⁢
                          
                            
                              cos
                              ⁡
                              
                                (
                                
                                  
                                    φ
                                    j
                                  
                                  -
                                  
                                    φ
                                    k
                                  
                                
                                )
                              
                            
                            ·
                            
                              δφ
                              j
                            
                          
                        
                      
                    
                    +
                  
                
              
              
                
                  
                    
                      ∑
                      
                        j
                        =
                        0
                      
                      M
                    
                    ⁢
                    
                      
                        ∑
                        
                          k
                          =
                          0
                        
                        M
                      
                      ⁢
                      
                        
                          A
                          k
                        
                        ⁢
                        
                          
                            sin
                            ⁡
                            
                              (
                              
                                
                                  φ
                                  j
                                
                                -
                                
                                  φ
                                  k
                                
                              
                              )
                            
                          
                          ·
                          δ
                        
                        ⁢
                        
                          
                        
                        ⁢
                        
                          A
                          j
                        
                      
                    
                  
                
              
            
            
              
                ∑
                
                  j
                  =
                  0
                
                M
              
              ⁢
              
                
                  ∑
                  
                    k
                    =
                    0
                  
                  M
                
                ⁢
                
                  
                    A
                    j
                  
                  ⁢
                  
                    A
                    k
                  
                  ⁢
                  
                    cos
                    ⁡
                    
                      (
                      
                        
                          φ
                          j
                        
                        -
                        
                          φ
                          k
                        
                      
                      )
                    
                  
                
              
            
          
        
      
    
    
      
        (
        47
        )
      
    
  
  
    
      
        
          Δφ
          v
        
        ≈
        
          n
          ·
          
            
              
                
                  
                    
                      
                        
                          ∑
                          
                            j
                            =
                            0
                          
                          M
                        
                        ⁢
                        
                          
                            ∑
                            
                              k
                              =
                              0
                            
                            M
                          
                          ⁢
                          
                            
                              B
                              j
                            
                            ⁢
                            
                              B
                              k
                            
                            ⁢
                            
                              
                                cos
                                ⁡
                                
                                  (
                                  
                                    
                                      φ
                                      j
                                    
                                    -
                                    
                                      φ
                                      k
                                    
                                  
                                  )
                                
                              
                              ·
                              
                                δφ
                                j
                              
                            
                          
                        
                      
                      +
                    
                  
                
                
                  
                    
                      
                        ∑
                        
                          j
                          =
                          0
                        
                        M
                      
                      ⁢
                      
                        
                          ∑
                          
                            k
                            =
                            0
                          
                          M
                        
                        ⁢
                        
                          
                            B
                            k
                          
                          ⁢
                          
                            
                              sin
                              ⁡
                              
                                (
                                
                                  
                                    φ
                                    j
                                  
                                  -
                                  
                                    φ
                                    k
                                  
                                
                                )
                              
                            
                            ·
                            δ
                          
                          ⁢
                          
                            
                          
                          ⁢
                          
                            B
                            j
                          
                        
                      
                    
                  
                
              
              
                
                  ∑
                  
                    j
                    =
                    0
                  
                  M
                
                ⁢
                
                  
                    ∑
                    
                      k
                      =
                      0
                    
                    M
                  
                  ⁢
                  
                    
                      B
                      j
                    
                    ⁢
                    
                      B
                      k
                    
                    ⁢
                    
                      cos
                      ⁡
                      
                        (
                        
                          
                            φ
                            j
                          
                          -
                          
                            φ
                            k
                          
                        
                        )
                      
                    
                  
                
              
            
            .
          
        
      
    
    
      
        (
        48
        )
      
    
  




By calculating the phase measurement errors as a function of the parameters of all of the hypothetic lines, locator 11 can determine an estimation of the confidence levels. Therefore, the walkover algorithm can be formulated as minimizing the following cost function.





  
    
      
        
          ∑
          j
        
        ⁢
        
          {
          
            
              
                
                  
                    
                      
                        
                          
                            [
                            
                              
                                
                                  
                                    h
                                    n
                                    I
                                  
                                  ⁡
                                  
                                    (
                                    
                                      x
                                      j
                                    
                                    )
                                  
                                
                                meas
                              
                              -
                              
                                
                                  
                                    h
                                    n
                                    I
                                  
                                  ⁡
                                  
                                    (
                                    
                                      x
                                      j
                                    
                                    )
                                  
                                
                                mod
                              
                            
                            ]
                          
                          2
                        
                        +
                        
                          
                            [
                            
                              
                                
                                  
                                    h
                                    n
                                    Q
                                  
                                  ⁡
                                  
                                    (
                                    
                                      x
                                      j
                                    
                                    )
                                  
                                
                                meas
                              
                              -
                              
                                
                                  
                                    h
                                    n
                                    Q
                                  
                                  ⁡
                                  
                                    (
                                    
                                      x
                                      j
                                    
                                    )
                                  
                                
                                mod
                              
                            
                            ]
                          
                          2
                        
                        +
                      
                    
                  
                  
                    
                      
                        
                          
                            [
                            
                              
                                
                                  
                                    h
                                    v
                                    I
                                  
                                  ⁡
                                  
                                    (
                                    
                                      x
                                      j
                                    
                                    )
                                  
                                
                                meas
                              
                              -
                              
                                
                                  
                                    h
                                    v
                                    I
                                  
                                  ⁡
                                  
                                    (
                                    
                                      x
                                      j
                                    
                                    )
                                  
                                
                                mod
                              
                            
                            ]
                          
                          2
                        
                        +
                        
                          
                            [
                            
                              
                                
                                  
                                    h
                                    v
                                    Q
                                  
                                  ⁡
                                  
                                    (
                                    
                                      x
                                      j
                                    
                                    )
                                  
                                
                                meas
                              
                              -
                              
                                
                                  
                                    h
                                    v
                                    Q
                                  
                                  ⁡
                                  
                                    (
                                    
                                      x
                                      j
                                    
                                    )
                                  
                                
                                mod
                              
                            
                            ]
                          
                          2
                        
                        +
                      
                    
                  
                
              
            
            
              
                
                  
                    
                      [
                      
                        
                          
                            
                              h
                              t
                              I
                            
                            ⁡
                            
                              (
                              
                                x
                                j
                              
                              )
                            
                          
                          meas
                        
                        -
                        
                          
                            
                              h
                              t
                              I
                            
                            ⁡
                            
                              (
                              
                                x
                                j
                              
                              )
                            
                          
                          mod
                        
                      
                      ]
                    
                    2
                  
                  +
                  
                    
                      [
                      
                        
                          
                            
                              h
                              t
                              Q
                            
                            ⁡
                            
                              (
                              
                                x
                                j
                              
                              )
                            
                          
                          meas
                        
                        -
                        
                          
                            
                              h
                              t
                              Q
                            
                            ⁡
                            
                              (
                              
                                x
                                j
                              
                              )
                            
                          
                          mod
                        
                      
                      ]
                    
                    2
                  
                
              
            
          
          }
        
      
    
    
      
        (
        49
        )
      
    
  




FIG. 4A illustrates an exemplary geometry of a walkover locate, Unknown parameters (x0, z0, and I0) at point 33 can be estimated, with the parameters (xj, hn) as the measurement set (with the measured parameter hn being complex). The walkover procedure starts at arbitrary reference position 30 and proceeds, for example, along the path indicated, across the region that has significant detectible signal strength, which can encompass the target line 31 and all bleedover lines. One of the bleedover lines is represented by line 32 in FIG. 4A. FIG. 4B shows the real part of the aggregate electromagnetic field strength as a function of walkover distance (i.e., the distance from target line 33). Curve 36 is the measured magnitude of the electromagnetic field strength, which is a summation of the real parts of the electromagnetic field strength 37 (illustrated in its negative position to distinguish from curve 36) originating from main line 31 and field strength 35 originating from bleedover line 32. Only the signal represented by curve 36 is measurable; signals 37 and 35 are indirectly obtained via the optimization.
Some embodiments of the invention use a forward facing electromagnetic field sensor, as in coil 24 of FIG. 2, to form a third dimension of field measurements. Although long-haul lines are generally deployed in rights-of-way that have clear line directions, there can be curvatures and road crossings, for which a front-facing coil is an appropriate method to detect if the operator is walking in a transverse (right angle) direction to the line, rather than at a more oblique angle. The data from coil 24 is generally not used in the optimization, but some embodiments of the invention may extend the field model represented by Equation (11) to allow off-axis oblique walkovers.
Some embodiments of the optimization procedures performed in model optimization 23 (FIG. 2) tend to be more sensitive to both measurement errors and initial conditions as the number of EM field sources increases, but optimization can be performed with numerous lines. Buried underground cable vault systems tend to hold many lines; in some cases up to 60 lines may be allocated within a 2 square meter duct. However, due to the importance of physical proximity, bleedover coupling tends to occur in a relatively small subset of lines within the duct. Typically, in addition to the target line, usually only one or two lines in a large duct are found to be carrying significant bleedover current (e.g., more than 10% of the current in the primary target line). Thus, a numerical optimization approach, through gradient progression, solves for the set of unknown line positions, phases, and currents, compatible with the problem of line location within a duct.
One complication in the optimization procedure performed in model optimization 23 for solving the vectoring problem is additional field distortion caused by non-zero conduction (σ) of the soil. In the simple case, where only a single line is present and the soil conductivity is uniform around the line, correction for conductivity does not significantly affect the outcome (at worst, the depth is underestimated a bit). However, when multiple lines contribute to the measured electromagnetic field or when the soil conductivity is not uniform, an optimization can have some problems. Nonzero conductivity causes the field originating from target line 15 to have a “ghost” quadrature component, and non-uniformity of the conductivity can cause some spatial distortion in the field from each line. Although these distortions can introduce a bias into the vectoring estimate, the measured signal from target line 15 can be allowed in some embodiments of the model to have a quadrature component (non-zero phase) in the optimization, minimizing this potential source of error by allowing a phase rotation to offset the quadrature component that might be present due to this situation.
Estimation of Confidence Intervals
The Levenberg-Marquardt optimization problem, described below, treats each source (target line 15 and numerous, for example three, strong bleedover lines) as having unknown depth, centerline, and current. The soil conductivity σ can be assumed fixed and equal to zero, although in some embodiments conductivity can also be part of the variable set included in the optimization. Using the notation above, the optimization variables are zp, xp, and Ipi+jIpq, where p ranges from 0 to N, and index 0 refers to target line 15. The superscripts i and q denote the in-phase and quadrature components of the current, respectively, such that the phase of the current in that line is equal to




  
    
      tan
      
        -
        1
      
    
    ⁡
    
      (
      
        
          I
          p
          q
        
        
          I
          p
          i
        
      
      )
    
  
  .




In addition, the angle parameter θ models the phase transfer function on target line 15 that affects all measurements uniformly, such that the observed phase on each line (with respect to the reference phase imparted at the transmitter) is





  
    
      tan
      
        -
        1
      
    
    ⁡
    
      (
      
        
          I
          p
          q
        
        
          I
          p
          i
        
      
      )
    
  
  +
  
    θ
    .
  




This constant phase rotation of the complex plane accounts for the unknown phase delay along target line 15 from transmitter 10 to the measurement position.

A constraint is placed on every iteration of the Levenberg-Marquardt optimization algorithm. Once per iteration the optimization method results in new current, depth, and centerline estimates, as described below, for target line 15 and all bleedover lines 16. The predicted currents are forced to adhere to the equation




  
    
      
        
          
            ∑
            
              p
              =
              1
            
            k
          
          ⁢
          
            
              
                I
                p
                
                  i
                  2
                
              
              +
              
                jI
                p
                
                  q
                  2
                
              
            
          
        
        ≤
        
          
            
              I
              0
              
                i
                2
              
            
            +
            
              jI
              0
              
                q
                2
              
            
          
        
      
    
    
      
        (
        50
        )
      
    
  




such that the absolute value of the sum of bleedover and return currents cannot be greater than the outgoing current in the targeted line. In all practical cases the magnitude sum on the left side of this inequality is not close to the magnitude of the target line current, since the walkover field measurements see only distinct conductors, and not the current that is returning through earth ground.

Because there is no a priori information regarding how many bleedover lines 16 carry significant current (that would distort the field emanating from target line 15), a hypothesis test procedure is used to determine the most likely scenario. To accomplish a target line locate, several (for example four) scenarios are run using the Levenberg-Marquardt optimization method. A first scenario uses a model of the field in which there are no bleedover sources 16 present. Other scenarios add the effects of increasing numbers of bleedover lines 16. In total there are 5, 9, 13, 17, etc. (corresponding to 1, 2, 3, 4, etc. bleedover lines 16, respectively) unknowns for the modeled scenarios, respectively. The optimization of model optimization 23 can provide a result as long as the number of measurements (xn, hn) is equal to or greater then the number of unknowns. To test which of the hypothetical bleedover scenarios best represents the data, a cost function can be formed using the mean square error of the predicted field (after optimization) compared to the measured data. This is used as a quality metric for that bleedover model. Appropriate choice of a minimum acceptable fit criterion (using the cost function) is an effective way to eliminate walkover locates which are dominated by noise and interference, and thus might result in “false positive” centerline and depth prediction errors.
The hypothesis test procedure can also be sensitive to over-determination of the model, when more bleedover sources 16 are modeled than actually exist in a particular scenario. Although these extra bleedover lines 16 increase the freedom that the optimization has to achieve a good fit, they can also result in a potential centerline prediction bias of target line 15. Penalties are levied on the cost function as the number of bleedover lines 16 increases, so that the least complex solution that still achieves a targeted model performance is chosen by the hypothesis test procedure.
At the end of the optimization process, and subsequent hypothesis test, the chosen result includes predictions of centerline, depth, current, phase, and error for target line 15 and each of the bleedover lines 16. Only the parameters associated with target line 15 are of interest, and the accuracy of these predictions must be stated with confidence bounds. A confidence bound (or interval) for these model parameters is developed below.
The parameter set for the optimization cost function is defined as:

a=[x0, z0, I0, φ0, dI0, dφ0, x1, z1, I1, φ1dI1, dφ1, . . . , xM, zM, Im, φM, dIM, dφM]

The optimization cost function that is minimized to estimate the model parameters is the sum-of-squares of the model output errors,





  
    
      
        
          
            
              
                S
                ⁡
                
                  (
                  a
                  )
                
              
              =
              
              ⁢
              
                
                  ∑
                  
                    j
                    =
                    1
                  
                  N
                
                ⁢
                
                  
                    
                    
                      
                        y
                        j
                      
                      -
                      
                        y
                        ⁡
                        
                          (
                          
                            
                              x
                              j
                            
                            ;
                            a
                          
                          )
                        
                      
                    
                    
                  
                  2
                
              
            
          
        
        
          
            
              =
              
              ⁢
              
                
                  ∑
                  j
                
                ⁢
                
                  {
                  
                    
                      
                        
                          
                            
                              
                                
                                  
                                    
                                      
                                        
                                          [
                                          
                                            
                                              
                                                
                                                  h
                                                  n
                                                  I
                                                
                                                ⁡
                                                
                                                  (
                                                  
                                                    x
                                                    j
                                                  
                                                  )
                                                
                                              
                                              meas
                                            
                                            -
                                            
                                              
                                                
                                                  h
                                                  n
                                                  I
                                                
                                                ⁡
                                                
                                                  (
                                                  
                                                    x
                                                    j
                                                  
                                                  )
                                                
                                              
                                              mod
                                            
                                          
                                          ]
                                        
                                        2
                                      
                                      +
                                    
                                  
                                
                                
                                  
                                    
                                      
                                        
                                          [
                                          
                                            
                                              
                                                h
                                                n
                                                Q
                                              
                                              ⁢
                                              
                                                
                                                  (
                                                  
                                                    x
                                                    j
                                                  
                                                  )
                                                
                                                meas
                                              
                                            
                                            -
                                            
                                              
                                                
                                                  h
                                                  n
                                                  Q
                                                
                                                ⁡
                                                
                                                  (
                                                  
                                                    x
                                                    j
                                                  
                                                  )
                                                
                                              
                                              mod
                                            
                                          
                                          ]
                                        
                                        2
                                      
                                      +
                                    
                                  
                                
                              
                            
                          
                          
                            
                              
                                
                                  
                                    
                                      
                                        
                                          [
                                          
                                            
                                              
                                                
                                                  h
                                                  v
                                                  I
                                                
                                                ⁡
                                                
                                                  (
                                                  
                                                    x
                                                    j
                                                  
                                                  )
                                                
                                              
                                              meas
                                            
                                            -
                                            
                                              
                                                
                                                  h
                                                  v
                                                  I
                                                
                                                ⁡
                                                
                                                  (
                                                  
                                                    x
                                                    j
                                                  
                                                  )
                                                
                                              
                                              mod
                                            
                                          
                                          ]
                                        
                                        2
                                      
                                      +
                                    
                                  
                                
                                
                                  
                                    
                                      
                                        
                                          [
                                          
                                            
                                              
                                                h
                                                v
                                                Q
                                              
                                              ⁢
                                              
                                                
                                                  (
                                                  
                                                    x
                                                    j
                                                  
                                                  )
                                                
                                                meas
                                              
                                            
                                            -
                                            
                                              
                                                
                                                  h
                                                  v
                                                  Q
                                                
                                                ⁡
                                                
                                                  (
                                                  
                                                    x
                                                    j
                                                  
                                                  )
                                                
                                              
                                              mod
                                            
                                          
                                          ]
                                        
                                        2
                                      
                                      +
                                    
                                  
                                
                              
                            
                          
                        
                      
                    
                    
                      
                        
                          
                            
                              
                                
                                  
                                    [
                                    
                                      
                                        
                                          
                                            h
                                            v
                                            I
                                          
                                          ⁡
                                          
                                            (
                                            
                                              x
                                              j
                                            
                                            )
                                          
                                        
                                        meas
                                      
                                      -
                                      
                                        
                                          
                                            h
                                            n
                                            I
                                          
                                          ⁡
                                          
                                            (
                                            
                                              x
                                              j
                                            
                                            )
                                          
                                        
                                        mod
                                      
                                    
                                    ]
                                  
                                  2
                                
                                +
                              
                            
                          
                          
                            
                              
                                
                                  [
                                  
                                    
                                      
                                        h
                                        n
                                        Q
                                      
                                      ⁢
                                      
                                        
                                          (
                                          
                                            x
                                            j
                                          
                                          )
                                        
                                        meas
                                      
                                    
                                    -
                                    
                                      
                                        
                                          h
                                          v
                                          Q
                                        
                                        ⁡
                                        
                                          (
                                          
                                            x
                                            j
                                          
                                          )
                                        
                                      
                                      mod
                                    
                                  
                                  ]
                                
                                2
                              
                            
                          
                        
                      
                    
                  
                  }
                
              
            
          
        
      
    
    
      
        (
        51
        )
      
    
  




where yr, hn, etc. are the measurements.

Associated with the model is the curvature matrix, composed of the second derivatives of the cost function with respect to the model parameters,




  
    
      
        α
        =
        
          
            [
            
              
                1
                2
              
              ⁢
              
                
                  
                    ∂
                    2
                  
                  ⁢
                  
                    S
                    ⁡
                    
                      (
                      a
                      )
                    
                  
                
                
                  
                    ∂
                    
                      a
                      k
                    
                  
                  ⁢
                  
                    ∂
                    
                      a
                      l
                    
                  
                
              
            
            ]
          
          ≈
          
            [
            
              
                
                  ∂
                  
                    S
                    ⁡
                    
                      (
                      a
                      )
                    
                  
                
                
                  ∂
                  
                    a
                    k
                  
                
              
              ·
              
                
                  ∂
                  
                    S
                    ⁡
                    
                      (
                      a
                      )
                    
                  
                
                
                  ∂
                  
                    a
                    l
                  
                
              
            
            ]
          
        
      
    
    
      
        (
        52
        )
      
    
  




often approximated with only the first derivatives of the model as





  
    α
    kl
  
  ⁢
  
    
      ∑
      
        j
        =
        1
      
      N
    
    ⁢
    
      (
      
        
          
            
              
                
                  
                    
                      
                        
                          
                            
                              
                                ∂
                                
                                  
                                    
                                      h
                                      n
                                      I
                                    
                                    ⁡
                                    
                                      (
                                      
                                        x
                                        j
                                      
                                      )
                                    
                                  
                                  mod
                                
                              
                              
                                ∂
                                
                                  a
                                  k
                                
                              
                            
                            ·
                            
                              
                                ∂
                                
                                  
                                    
                                      h
                                      
                                        n
                                        ⁢
                                        
                                          
                                        
                                      
                                      I
                                    
                                    ⁡
                                    
                                      (
                                      
                                        x
                                        j
                                      
                                      )
                                    
                                  
                                  mod
                                
                              
                              
                                ∂
                                
                                  a
                                  l
                                
                              
                            
                          
                          +
                        
                      
                    
                    
                      
                        
                          
                            
                              
                                ∂
                                
                                  
                                    
                                      h
                                      n
                                      Q
                                    
                                    ⁡
                                    
                                      (
                                      
                                        x
                                        j
                                      
                                      )
                                    
                                  
                                  mod
                                
                              
                              
                                ∂
                                
                                  a
                                  k
                                
                              
                            
                            ·
                            
                              
                                ∂
                                
                                  
                                    
                                      h
                                      n
                                      Q
                                    
                                    ⁡
                                    
                                      (
                                      
                                        x
                                        j
                                      
                                      )
                                    
                                  
                                  mod
                                
                              
                              
                                ∂
                                
                                  a
                                  l
                                
                              
                            
                          
                          +
                        
                      
                    
                  
                
              
              
                
                  
                    
                      
                        
                          
                            
                              
                                ∂
                                
                                  
                                    
                                      h
                                      v
                                      I
                                    
                                    ⁡
                                    
                                      (
                                      
                                        x
                                        j
                                      
                                      )
                                    
                                  
                                  mod
                                
                              
                              
                                ∂
                                
                                  a
                                  k
                                
                              
                            
                            ·
                            
                              
                                ∂
                                
                                  
                                    
                                      h
                                      v
                                      I
                                    
                                    ⁡
                                    
                                      (
                                      
                                        x
                                        j
                                      
                                      )
                                    
                                  
                                  mod
                                
                              
                              
                                ∂
                                
                                  a
                                  l
                                
                              
                            
                          
                          +
                        
                      
                    
                    
                      
                        
                          
                            
                              
                                ∂
                                
                                  
                                    
                                      h
                                      
                                        v
                                        ⁢
                                        
                                          
                                        
                                      
                                      Q
                                    
                                    ⁡
                                    
                                      (
                                      
                                        x
                                        j
                                      
                                      )
                                    
                                  
                                  mod
                                
                              
                              
                                ∂
                                
                                  a
                                  k
                                
                              
                            
                            ·
                            
                              
                                ∂
                                
                                  
                                    
                                      h
                                      v
                                      Q
                                    
                                    ⁡
                                    
                                      (
                                      
                                        x
                                        j
                                      
                                      )
                                    
                                  
                                  mod
                                
                              
                              
                                ∂
                                
                                  a
                                  l
                                
                              
                            
                          
                          +
                        
                      
                    
                  
                
              
            
          
        
        
          
            
              
                
                  
                    
                      
                        
                          ∂
                          
                            
                              
                                h
                                t
                                I
                              
                              ⁡
                              
                                (
                                
                                  x
                                  j
                                
                                )
                              
                            
                            mod
                          
                        
                        
                          ∂
                          
                            a
                            k
                          
                        
                      
                      ·
                      
                        
                          ∂
                          
                            
                              
                                h
                                t
                                I
                              
                              ⁡
                              
                                (
                                
                                  x
                                  j
                                
                                )
                              
                            
                            mod
                          
                        
                        
                          ∂
                          
                            a
                            l
                          
                        
                      
                    
                    +
                  
                
              
              
                
                  
                    
                      
                        ∂
                        
                          
                            
                              h
                              t
                              Q
                            
                            ⁡
                            
                              (
                              
                                x
                                j
                              
                              )
                            
                          
                          mod
                        
                      
                      
                        ∂
                        
                          a
                          k
                        
                      
                    
                    ·
                    
                      
                        ∂
                        
                          
                            
                              h
                              l
                              Q
                            
                            ⁡
                            
                              (
                              
                                x
                                j
                              
                              )
                            
                          
                          mod
                        
                      
                      
                        ∂
                        
                          a
                          l
                        
                      
                    
                  
                
              
            
          
        
      
      )
    
  




The covariance matrix of the model parameters at the optimum is important for estimating a confidence interval. This matrix is the inverse of the curvature matrix,

C=α−1  (54)

The primary parameters estimated by the optimization are the centerline and depth of target line 15, and the estimated values of those parameters can be annotated by a confidence interval.
Under the following assumptions, the confidence interval of each individual model parameter can be estimated based on sum of squares function S(a). Sum of squares function S(a) is a function of the parameter elements of a only. The data provide the numerical coefficients in S(a). In the parameter space, the function S(a) can be represented by the contours of a surface. If the model were linear in a, the surface contours would be ellipsoidal and would have a single global minimum, S(â) , at the location defined by the least square estimator â. If the model is nonlinear, the contours are not ellipsoidal but tend to be irregular perhaps with several local minimal points. An exact confidence contour is defined by taking S(a)=constant, which, in this high-dimension nonlinear example, is not feasible. Under the assumption that the linearized form of the model is valid around â, the final estimate of a, the ellipsoidal confidence region is obtained by the following formula:

(a−â)·(α)a=â·(a−â)≦p·s2·F(p,n−p,1−α)

s2=S(â)/(n−p)  (55)

where, p is the total number of parameters; n is the total number of measurements; and 100(1−α)% is the confidence level (for instance, α=0.05 represents 95% confidence); F(p,n−p,1−α) is the F-distribution function.

The embodiments described above can be used to precisely locate a target line in areas where bleedover can cause large errors in positioning. Although some extra time is associated with setting up the walkover (the transect should be associated with a reference point away from the centerline (point 30 in FIG. 4A)), the benefit in positioning accuracy is large. However, when bleedover is minimal, it is not always necessary to perform the walkover locate using optimization methods to determine the target line position. In fact, traditional peak and null centerline detection methods can be just as precise in some scenarios, Thus there is a benefit in having a “bleedover detection” method built in to the locate receiver in order to assist the user in determining when a walkover locate should be performed.
FIGS. 5A & 5B illustrate a component of graphical display that is presented on the locator display 12, in the standard line locate modes. Pointer 71 shows the user the detected line and signal direction, The amount of deflection of the pointer (off of vertical line 72) is based on the measured signal strength of the forward facing coil 24 compared to the signal strength of the horizontal coil 14. In effect, this gives a qualitative measure of whether the tracked line has an impending curvature. If the locate receiver is facing directly forward, and the line extends straight ahead, then Pointer 71 is directed straight upward (toward the upper half plane of the circle).
Pointer 71 is also used to denote the direction of the signal in the cable or pipe, as discussed in the '376 application. When the aggregate signal phase is positive, the pointer movement is in the upper half plane, else it is in the lower half plane, as in pointer 74 of FIG. 5B. The aggregate signal phase can be taken from the horizontal coil quantity




  
    
      tan
      
        -
        1
      
    
    ⁡
    
      (
      
        
          
            Im
            ⁡
            
              [
              
                h
                n
              
              ]
            
          
          hpll
        
        
          
            Re
            ⁡
            
              [
              
                h
                n
              
              ]
            
          
          hpll
        
      
      )
    
  
  ,




or the equivalent from the vertical coil (whichever has greater signal magnitude),

The shaded backgrounds 73 and 76 of FIGS. 5A & 5B represents the level of bleedover in the measured signal, The aggregate signal phase is used to determine the level of bleedover, which is a qualitative indication of the degree of distortion in the electromagnetic field. When the phase is zero, the level of the shaded background is zero (there is no shading). When the phase is 90°, indicating that the signals from bleedover lines predominate the measurement, the background is completely shaded. The update rate of the bleedover detection icon is on the order of 30 Hz, or more quickly than the field changes while an operator is walking down the line during a locate. Thus, the user is effectively warned about bleedover situations and can take explicit action to invoke the walkover locate precise positioning mode.
Global Optimization
To set up the Levenberg-Marquardt non-linear optimization problem, measurements of the parameters (x, hn, hv, ht) are made in line locator 11. The h values are measured simultaneously using horizontal (peak) coil 14, vertical (null) coil 19, and top coil 20. In some embodiments, a demodulation approach such as that described in the '239 application can be utilized. In this way it is possible to fit both the horizontal model, the vertical model, and/or the top model using optimization 23 and select the best-behaved model from the statistics that are inherent in the optimization process.
Due to the nature of nonlinearity, a global solution is not guaranteed. More often than not, a local minimum might lead to a false line location. Here, some embodiments consistent with the present invention use a “grid and search” method to estimate the true line location, as provided in FIG. 6, based on the following:

    
    
        1) Based on the measurement, figure out an initial estimate of the 1st line centerline position (e.g., center dot 60);
        2) Surround this initial estimate with a square box the size of, for example, 6′′×6′′ (default size, subject to user definition).
        3) Surround this box both horizontally and vertically with many of the same size, to a larger box with user defined size.
        4) Within each small box, the center coordinates can be the initial centerline positions and steps of the 1st and 2nd line, and the box can be the boundary conditions of the centerlines of these two lines. An optimization is performed for each box and a local solution can be found within each box.
    
    


The above steps can essentially generate many local solutions, with the “true” solution buried in them. The next step can be to pick the “true” solution and exclude the false solutions. For each local solution, the following steps are performed:

    
    
        1) Compute Fit based on the following equation
    
    






  
    
      
        
          
            
              
                S
                ⁡
                
                  (
                  a
                  )
                
              
              =
              
              ⁢
              
                
                  ∑
                  
                    j
                    =
                    1
                  
                  N
                
                ⁢
                
                  
                    
                    
                      
                        y
                        j
                      
                      -
                      
                        y
                        ⁡
                        
                          (
                          
                            
                              x
                              j
                            
                            ;
                            a
                          
                          )
                        
                      
                    
                    
                  
                  2
                
              
            
          
        
        
          
            
              =
              
              ⁢
              
                
                  ∑
                  j
                
                ⁢
                
                  {
                  
                    
                      
                        
                          
                            
                              
                                
                                  
                                    [
                                    
                                      
                                        
                                          
                                            h
                                            n
                                            I
                                          
                                          ⁡
                                          
                                            (
                                            
                                              x
                                              j
                                            
                                            )
                                          
                                        
                                        meas
                                      
                                      -
                                      
                                        
                                          
                                            h
                                            n
                                            I
                                          
                                          ⁡
                                          
                                            (
                                            
                                              x
                                              j
                                            
                                            )
                                          
                                        
                                        mod
                                      
                                    
                                    ]
                                  
                                  2
                                
                                +
                                
                                  
                                    [
                                    
                                      
                                        
                                          
                                            h
                                            n
                                            Q
                                          
                                          ⁡
                                          
                                            (
                                            
                                              x
                                              j
                                            
                                            )
                                          
                                        
                                        meas
                                      
                                      -
                                      
                                        
                                          
                                            h
                                            n
                                            Q
                                          
                                          ⁡
                                          
                                            (
                                            
                                              x
                                              
                                                j
                                                ⁢
                                                
                                                  
                                                
                                              
                                            
                                            )
                                          
                                        
                                        mod
                                      
                                    
                                    ]
                                  
                                  2
                                
                                +
                              
                            
                          
                          
                            
                              
                                
                                  
                                    [
                                    
                                      
                                        
                                          
                                            h
                                            v
                                            I
                                          
                                          ⁡
                                          
                                            (
                                            
                                              x
                                              j
                                            
                                            )
                                          
                                        
                                        meas
                                      
                                      -
                                      
                                        
                                          
                                            h
                                            v
                                            I
                                          
                                          ⁡
                                          
                                            (
                                            
                                              x
                                              j
                                            
                                            )
                                          
                                        
                                        mod
                                      
                                    
                                    ]
                                  
                                  2
                                
                                +
                                
                                  
                                    [
                                    
                                      
                                        
                                          
                                            h
                                            v
                                            Q
                                          
                                          ⁡
                                          
                                            (
                                            
                                              x
                                              j
                                            
                                            )
                                          
                                        
                                        meas
                                      
                                      -
                                      
                                        
                                          
                                            h
                                            v
                                            Q
                                          
                                          ⁡
                                          
                                            (
                                            
                                              x
                                              j
                                            
                                            )
                                          
                                        
                                        mod
                                      
                                    
                                    ]
                                  
                                  2
                                
                                +
                              
                            
                          
                        
                      
                    
                    
                      
                        
                          
                            
                              [
                              
                                
                                  
                                    
                                      h
                                      v
                                      I
                                    
                                    ⁡
                                    
                                      (
                                      
                                        x
                                        j
                                      
                                      )
                                    
                                  
                                  meas
                                
                                -
                                
                                  
                                    
                                      h
                                      n
                                      I
                                    
                                    ⁡
                                    
                                      (
                                      
                                        x
                                        j
                                      
                                      )
                                    
                                  
                                  mod
                                
                              
                              ]
                            
                            2
                          
                          +
                          
                            
                              [
                              
                                
                                  
                                    
                                      h
                                      n
                                      Q
                                    
                                    ⁡
                                    
                                      (
                                      
                                        x
                                        j
                                      
                                      )
                                    
                                  
                                  
                                    mea
                                    ⁢
                                    
                                      
                                    
                                    ⁢
                                    s
                                  
                                
                                -
                                
                                  
                                    
                                      h
                                      v
                                      Q
                                    
                                    ⁡
                                    
                                      (
                                      
                                        x
                                        j
                                      
                                      )
                                    
                                  
                                  mod
                                
                              
                              ]
                            
                            2
                          
                        
                      
                    
                  
                  }
                
              
            
          
        
      
    
    
      
        
      
    
  
  
    
      
        
          
        
        ⁢
        
          
            fsMagSum
            =
            
              
                ∑
                j
              
              ⁢
              
                [
                
                  
                    
                      
                        
                          
                            
                              H
                              n
                              I
                            
                            ⁡
                            
                              (
                              
                                x
                                j
                              
                              )
                            
                          
                          2
                        
                        +
                        
                          
                            
                              h
                              n
                              Q
                            
                            ⁡
                            
                              (
                              
                                x
                                j
                              
                              )
                            
                          
                          2
                        
                        +
                        
                          
                            
                              h
                              v
                              I
                            
                            ⁡
                            
                              (
                              
                                x
                                j
                              
                              )
                            
                          
                          2
                        
                        +
                      
                    
                  
                  
                    
                      
                        
                          
                            
                              h
                              v
                              Q
                            
                            ⁡
                            
                              (
                              
                                x
                                j
                              
                              )
                            
                          
                          2
                        
                        +
                        
                          
                            
                              h
                              t
                              I
                            
                            ⁡
                            
                              (
                              
                                x
                                j
                              
                              )
                            
                          
                          2
                        
                        +
                        
                          
                            
                              h
                              t
                              Q
                            
                            ⁡
                            
                              (
                              
                                x
                                j
                              
                              )
                            
                          
                          2
                        
                      
                    
                  
                
                ]
              
            
          
          ⁢
          
            

          
          ⁢
          
            
          
          ⁢
          
            Fit
            =
            
              
                fsMagSum
                
                  (
                  
                    
                      S
                      ⁡
                      
                        (
                        a
                        )
                      
                    
                    
                      (
                      
                        n
                        -
                        p
                      
                      )
                    
                  
                  )
                
              
            
          
        
      
    
    
      
        
      
    
  




    
    
        
        
            where,
            
                n is total number of measurements, and
                p is the total number of parameters.
            
            
        
        
        2) Check the estimated current and phase relationship and compute the effective Fit, effFit.
        
            The primary (target) line parameter vector is [x0, z0, I0, φ0, dI0, dφ0]. Assuming that the transfer function of the current along the target line is a stable minimum phase system (no RHP poles and zeros), the phase at a certain frequency f can roughly be related to the slope of the gain Bode plot. For determining φ0 based on a Bode plot:

φ0=90°·N(f0)

            where, N(f0) is the slope of the gain Bode plot at frequency f0. Assuming that the current gain Bode plot started roll-off at frequency f0/100:
        
        
    
    






  
    
      
        I
        0
      
      
        I
        
          ma
          ⁢
          
            
          
          ⁢
          x
        
      
    
    ≥
    
      
        (
        
          
            f
            0
          
          
            
              f
              0
            
            100
          
        
        )
      
      
        N
        ⁡
        
          (
          
            f
            0
          
          )
        
      
    
  
  =
  
    100
    
      N
      ⁡
      
        (
        
          f
          0
        
        )
      
    
  




    
    
        
        
            To create an upper bound for I0, it is assumed that the current Bode plot started roll-off at half locating frequency,
        
        
    
    






  
    
      
        I
        0
      
      
        I
        
          ma
          ⁢
          
            
          
          ⁢
          x
        
      
    
    ≤
    
      
        (
        
          
            f
            0
          
          
            
              f
              0
            
            /
            2
          
        
        )
      
      
        N
        ⁡
        
          (
          
            f
            0
          
          )
        
      
    
  
  =
  
    
      2
      
        N
        ⁡
        
          (
          
            f
            0
          
          )
        
      
    
    .
  




    
    
        
        
            Therefore, the current estimate should fall into the following range:

Imax100N(f0)≦I0≦Imax·2N(f0 )

N(f0=φ0/90°.

            If the estimated current violates the above range, the Fit can be penalized in the following way:
        
        
    
    





  
    
      factor
      ⁢
      
        
      
      ⁢
      1
    
    =
    
      min
      ⁡
      
        (
        
          1
          ,
          
            
              I
              0
            
            
              
                I
                
                  m
                  ⁢
                  
                    
                  
                  ⁢
                  ax
                
              
              ·
              
                100
                
                  N
                  ⁡
                  
                    (
                    
                      f
                      0
                    
                    )
                  
                
              
            
          
          ,
          
            
              
                I
                
                  ma
                  ⁢
                  
                    
                  
                  ⁢
                  x
                
              
              ·
              
                2
                
                  N
                  ⁡
                  
                    (
                    
                      f
                      0
                    
                    )
                  
                
              
            
            
              I
              0
            
          
        
        )
      
    
  




  
    effFit
    =
    
      factor
      ⁢
      
        
      
      ⁢
      
        1
        ·
        
          Fit
          .
        
      
    
  



    
    
        3) Check the estimated current attenuation at the side tones and update the effective Fit, effFit
        
            Define I0+, I0− as the current amplitude at the two side tone frequencies of the primary line in Signal-Select mode. dI0 can be described in the following equation:
        
        
    
    





  
    
      dI
      0
    
    =
    
      
        
          
            I
            0
            +
          
          -
          
            I
            0
            -
          
        
        
          I
          0
          -
        
      
      =
      
        
          
            I
            0
            +
          
          
            I
            0
            -
          
        
        -
        1
      
    
  




  
    
      
        I
        0
        +
      
      
        I
        0
        -
      
    
    =
    
      1
      +
      
        
          dI
          0
        
        .
      
    
  



    
    
        
        
            The ratio of the frequencies at the side tones, where n=8, is (2n+1)/(2n−1)=17/15, giving the following:
        
        
    
    






  
    
      I
      0
      +
    
    
      I
      0
      -
    
  
  =
  
    
      1
      +
      
        dI
        0
      
    
    =
    
      
        (
        
          17
          15
        
        )
      
      
        N
        ⁡
        
          (
          
            f
            0
          
          )
        
      
    
  




    
    
        
        
            Introducing another factor2 to penalize the effFit if the above relationship is violated:
        
        
    
    





  
    
      factor
      ⁢
      
        
      
      ⁢
      2
    
    =
    
      1
      -
      
        
        
          1
          +
          
            dI
            0
          
          -
          
            
              (
              
                17
                15
              
              )
            
            
              N
              ⁡
              
                (
                
                  f
                  0
                
                )
              
            
          
        
        
      
    
  




  
    effFit
    =
    
      factor
      ⁢
      
        
      
      ⁢
      
        2
        ·
        effFit
      
    
  



    
    
        4) Introduce BLEEDFAC to penalize the total number of lines in the Fit:

effFit=effFit/BleedFac
        
            where, BleedFac=1 if there is no bleedover line; BleedFac=21/4 for one bleedover line; BleedFac=41/4 for two bleedover line; BleedFac=81/4 for three bleedover lines.
        
        
        5) Check if any of the secondary lines are “too close” to the primary line. If the distance between the primary and secondary lines is less than 1.5′′, that solution can be treated as a false location and excluded.
        6) Check the uncertainty bound box of the solution (centerline and depth). If estimated centerline error is greater than 24′′, or the diagonal of the error box (centerline and depth) is greater than 36′′, the solution can be excluded.
        7) After all, the solution with the largest effFit is the one to be picked.
    
    


The method described above has been verified through a lot of field data collected at places where the field is distorted by the other lines with bleedover and/or return current. So far, this method is able to identify the true line location with certain confidence intervals.
In general, the current phase measurement of the target line in Signal-Select mode can be biased due to the presence of bleedover lines. The error term however can be modeled as a function of locations, current and phase of each line, plus a current amplitude and phase variations. An enhanced electromagnetic field model including this phase measurement error is used in the precise locating method. A global optimization approach is proposed to find the true line location among many false locations.
All Appendices (Appendix 1 and Appendix 2) are considered part of this disclosure and are herein incorporated by reference in their entirety. The embodiments described herein are examples only of the invention. Other embodiments of the invention that are within the scope and spirit of this disclosure will be apparent to those skilled in the art from consideration of the specification and practice of the invention disclosed herein. It is intended that the specification and examples be considered as exemplary only and not limiting. The scope of the invention, therefore, is limited only by the following claims.
Appendix I-Computation of Fourier Series Coefficients
The Fourier series coefficients akand bk are calculated for Equation (21)according to the following steps:




  
    
      
        
          
            (
            
              
                T
                1
              
              +
              
                T
                2
              
            
            )
          
          ·
          
            a
            
              2
              ⁢
              n
            
          
        
        =
        
          
            ∫
            0
            
              
                T
                1
              
              +
              
                T
                2
              
            
          
          ⁢
          
            
              2
              ·
              
                I
                ⁡
                
                  (
                  t
                  )
                
              
              ·
              
                cos
                ⁡
                
                  (
                  
                    2
                    ⁢
                    
                      n
                      ·
                      
                        ω
                        m
                      
                    
                    ⁢
                    t
                  
                  )
                
              
            
            ⁢
            
              d
              t
            
          
        
      
    
    
      
        
      
    
  
  
    
      
        
          
        
        ⁢
        
          =
          
            
              
                ∫
                0
                
                  T
                  1
                
              
              ⁢
              
                
                  2
                  ·
                  
                    I
                    tx
                  
                  ·
                  
                    cos
                    ⁡
                    
                      (
                      
                        
                          ω
                          1
                        
                        ⁢
                        t
                      
                      )
                    
                  
                  ·
                  
                    cos
                    ⁡
                    
                      (
                      
                        
                          ω
                          0
                        
                        ⁢
                        t
                      
                      )
                    
                  
                
                ⁢
                
                  d
                  t
                
              
            
            +
          
        
      
    
    
      
        
      
    
  
  
    
      
        
          
        
        ⁢
        
          
            ∫
            
              T
              1
            
            
              
                T
                1
              
              +
              
                T
                2
              
            
          
          ⁢
          
            
              2
              ·
              
                I
                tx
              
              ·
              
                cos
                ⁡
                
                  (
                  
                    
                      ω
                      2
                    
                    ⁡
                    
                      (
                      
                        t
                        -
                        
                          T
                          1
                        
                      
                      )
                    
                  
                  )
                
              
              ·
              
                cos
                ⁡
                
                  (
                  
                    
                      ω
                      0
                    
                    ⁢
                    t
                  
                  )
                
              
            
            ⁢
            
              d
              t
            
          
        
      
    
    
      
        
      
    
  
  
    
      
        
          
        
        ⁢
        
          =
          
            
              
                ∫
                0
                
                  T
                  1
                
              
              ⁢
              
                
                  
                    I
                    tx
                  
                  ·
                  
                    cos
                    ⁡
                    
                      [
                      
                        
                          (
                          
                            
                              ω
                              1
                            
                            +
                            
                              ω
                              0
                            
                          
                          )
                        
                        ⁢
                        t
                      
                      ]
                    
                  
                
                ⁢
                
                  d
                  t
                
              
            
            +
          
        
      
    
    
      
        
      
    
  
  
    
      
        
          
        
        ⁢
        
          
            
              ∫
              0
              
                T
                1
              
            
            ⁢
            
              
                
                  I
                  tx
                
                ·
                
                  cos
                  ⁡
                  
                    [
                    
                      
                        (
                        
                          
                            ω
                            1
                          
                          -
                          
                            ω
                            0
                          
                        
                        )
                      
                      ⁢
                      t
                    
                    ]
                  
                
              
              ⁢
              
                d
                t
              
            
          
          +
        
      
    
    
      
        
      
    
  
  
    
      
        
          
        
        ⁢
        
          
            
              ∫
              
                T
                1
              
              
                
                  T
                  1
                
                +
                
                  T
                  
                    2
                    ⁢
                    
                      
                    
                  
                
              
            
            ⁢
            
              
                
                  I
                  tx
                
                ·
                
                  cos
                  ⁡
                  
                    [
                    
                      
                        
                          (
                          
                            
                              ω
                              2
                            
                            +
                            
                              ω
                              0
                            
                          
                          )
                        
                        ⁢
                        t
                      
                      -
                      
                        
                          ω
                          2
                        
                        ⁢
                        
                          T
                          1
                        
                      
                    
                    ]
                  
                
              
              ⁢
              
                d
                t
              
            
          
          +
        
      
    
    
      
        
      
    
  
  
    
      
        
          
        
        ⁢
        
          
            ∫
            
              T
              1
            
            
              
                T
                1
              
              +
              
                T
                2
              
            
          
          ⁢
          
            
              
                I
                tx
              
              ·
              
                cos
                ⁡
                
                  [
                  
                    
                      
                        (
                        
                          
                            ω
                            2
                          
                          -
                          
                            ω
                            0
                          
                        
                        )
                      
                      ⁢
                      t
                    
                    -
                    
                      
                        ω
                        2
                      
                      ⁢
                      
                        T
                        1
                      
                    
                  
                  ]
                
              
            
            ⁢
            
              d
              t
            
          
        
      
    
    
      
        
      
    
  
  
    
      
        
          
        
        ⁢
        
          =
          
            
              
                
                  I
                  tx
                
                
                  
                    ω
                    1
                  
                  +
                  
                    ω
                    0
                  
                
              
              ·
              
                sin
                ⁡
                
                  (
                  
                    
                      -
                      2
                    
                    ⁢
                    n
                    ⁢
                    
                      
                    
                    ⁢
                    π
                    ⁢
                    
                      
                    
                    ⁢
                    Δ
                  
                  )
                
              
            
            +
            
              
                
                  I
                  tx
                
                
                  
                    ω
                    1
                  
                  -
                  
                    ω
                    0
                  
                
              
              ·
              
                sin
                ⁡
                
                  (
                  
                    2
                    ⁢
                    n
                    ⁢
                    
                      
                    
                    ⁢
                    π
                    ⁢
                    
                      
                    
                    ⁢
                    Δ
                  
                  )
                
              
            
            +
          
        
      
    
    
      
        
      
    
  
  
    
      
        
          
        
        ⁢
        
          
            
              
                I
                tx
              
              
                
                  ω
                  2
                
                +
                
                  ω
                  0
                
              
            
            ·
            
              sin
              ⁡
              
                (
                
                  2
                  ⁢
                  n
                  ⁢
                  
                    
                  
                  ⁢
                  π
                  ⁢
                  
                    
                  
                  ⁢
                  Δ
                
                )
              
            
          
          +
          
            
              
                I
                tx
              
              
                
                  ω
                  2
                
                -
                
                  ω
                  0
                
              
            
            ·
            
              [
              
                -
                
                  sin
                  ⁡
                  
                    (
                    
                      2
                      ⁢
                      n
                      ⁢
                      
                        
                      
                      ⁢
                      π
                      ⁢
                      
                        
                      
                      ⁢
                      Δ
                    
                    )
                  
                
              
              ]
            
          
        
      
    
    
      
        
      
    
  
  
    
      
        
          
        
        ⁢
        
          =
          
            
              
                
                  I
                  tx
                
                
                  
                    ω
                    0
                  
                  ·
                  Δ
                
              
              ·
              
                sin
                ⁡
                
                  (
                  
                    2
                    ⁢
                    n
                    ⁢
                    
                      
                    
                    ⁢
                    π
                    ⁢
                    
                      
                    
                    ⁢
                    Δ
                  
                  )
                
              
            
            +
            
              
                
                  I
                  tx
                
                
                  
                    -
                    
                      ω
                      0
                    
                  
                  ·
                  Δ
                
              
              ·
              
                [
                
                  -
                  
                    sin
                    ⁡
                    
                      (
                      
                        2
                        ⁢
                        
                          
                        
                        ⁢
                        n
                        ⁢
                        
                          
                        
                        ⁢
                        π
                        ⁢
                        
                          
                        
                        ⁢
                        Δ
                      
                      )
                    
                  
                
                ]
              
            
          
        
      
    
    
      
        
      
    
  
  
    
      
        
          
        
        ⁢
        
          =
          
            
              
                2
                ⁢
                
                  I
                  tx
                
              
              
                
                  ω
                  0
                
                ·
                Δ
              
            
            ·
            
              sin
              ⁡
              
                (
                
                  2
                  ⁢
                  n
                  ⁢
                  
                    
                  
                  ⁢
                  π
                  ⁢
                  
                    
                  
                  ⁢
                  Δ
                
                )
              
            
          
        
      
    
    
      
        
      
    
  




leading to





  
    
      
        
          
            a
            
              2
              ⁢
              n
            
          
          ≈
          
            
              2
              ⁢
              
                I
                tx
              
              ⁢
              
                sin
                ⁡
                
                  (
                  
                    2
                    ⁢
                    n
                    ⁢
                    
                      
                    
                    ⁢
                    π
                    ⁢
                    
                      
                    
                    ⁢
                    Δ
                  
                  )
                
              
            
            
              
                
                  (
                  
                    
                      T
                      1
                    
                    +
                    
                      T
                      2
                    
                  
                  )
                
                ·
                
                  ω
                  0
                
              
              ⁢
              Δ
            
          
        
        =
        
          
            
              I
              tx
            
            ⁢
            
              
                sin
                ⁡
                
                  (
                  
                    n
                    ⁢
                    
                      
                    
                    ⁢
                    π
                    ⁢
                    
                      
                    
                    ⁢
                    Δ
                  
                  )
                
              
              ·
              
                cos
                ⁡
                
                  (
                  
                    n
                    ⁢
                    
                      
                    
                    ⁢
                    π
                    ⁢
                    
                      
                    
                    ⁢
                    Δ
                  
                  )
                
              
            
          
          
            n
            ⁢
            
              
            
            ⁢
            π
            ⁢
            
              
            
            ⁢
            Δ
          
        
      
    
    
      
        
      
    
  
  
    
      
        
          
            (
            
              
                T
                1
              
              +
              
                T
                2
              
            
            )
          
          ·
          
            b
            
              2
              ⁢
              n
            
          
        
        =
        
          
            ∫
            0
            
              
                T
                1
              
              +
              
                T
                2
              
            
          
          ⁢
          
            
              2
              ·
              
                I
                ⁡
                
                  (
                  t
                  )
                
              
              ·
              
                sin
                ⁡
                
                  (
                  
                    2
                    ⁢
                    
                      n
                      ⁢
                      
                        
                      
                      ·
                      
                        ω
                        m
                      
                    
                    ⁢
                    t
                  
                  )
                
              
            
            ⁢
            
              d
              t
            
          
        
      
    
    
      
        
      
    
  
  
    
      
        
          
        
        ⁢
        
          =
          
            
              
                ∫
                0
                
                  T
                  1
                
              
              ⁢
              
                
                  2
                  ·
                  
                    I
                    tx
                  
                  ·
                  
                    cos
                    ⁡
                    
                      (
                      
                        
                          ω
                          1
                        
                        ⁢
                        t
                      
                      )
                    
                  
                  ·
                  
                    sin
                    ⁡
                    
                      (
                      
                        
                          ω
                          0
                        
                        ⁢
                        t
                      
                      )
                    
                  
                
                ⁢
                
                  d
                  t
                
              
            
            +
          
        
      
    
    
      
        
      
    
  
  
    
      
        
          
        
        ⁢
        
          
            ∫
            
              T
              1
            
            
              
                T
                1
              
              +
              
                T
                2
              
            
          
          ⁢
          
            
              2
              ·
              
                I
                x
              
              ·
              
                cos
                ⁡
                
                  (
                  
                    
                      ω
                      2
                    
                    ⁡
                    
                      (
                      
                        t
                        -
                        
                          T
                          1
                        
                      
                      )
                    
                  
                  )
                
              
              ·
              
                sin
                ⁡
                
                  (
                  
                    
                      ω
                      0
                    
                    ⁢
                    t
                  
                  )
                
              
            
            ⁢
            
              d
              t
            
          
        
      
    
    
      
        
      
    
  
  
    
      
        
          
        
        ⁢
        
          =
          
            
              
                ∫
                0
                
                  T
                  1
                
              
              ⁢
              
                
                  
                    I
                    tx
                  
                  ·
                  
                    sin
                    ⁡
                    
                      [
                      
                        
                          (
                          
                            
                              ω
                              1
                            
                            +
                            
                              ω
                              0
                            
                          
                          )
                        
                        ⁢
                        t
                      
                      ]
                    
                  
                
                ⁢
                
                  d
                  t
                
              
            
            -
          
        
      
    
    
      
        
      
    
  
  
    
      
        
          
        
        ⁢
        
          
            
              ∫
              0
              
                T
                1
              
            
            ⁢
            
              
                
                  I
                  tx
                
                ·
                
                  sin
                  ⁡
                  
                    [
                    
                      
                        (
                        
                          
                            ω
                            1
                          
                          -
                          
                            ω
                            0
                          
                        
                        )
                      
                      ⁢
                      t
                    
                    ]
                  
                
              
              ⁢
              
                d
                t
              
            
          
          +
        
      
    
    
      
        
      
    
  
  
    
      
        
          
        
        ⁢
        
          
            
              ∫
              
                T
                1
              
              
                
                  T
                  1
                
                +
                
                  T
                  2
                
              
            
            ⁢
            
              
                
                  I
                  tx
                
                ·
                
                  sin
                  ⁡
                  
                    [
                    
                      
                        
                          (
                          
                            
                              ω
                              2
                            
                            +
                            
                              ω
                              0
                            
                          
                          )
                        
                        ⁢
                        t
                      
                      -
                      
                        
                          ω
                          2
                        
                        ⁢
                        
                          T
                          1
                        
                      
                    
                    ]
                  
                
              
              ⁢
              
                d
                t
              
            
          
          -
        
      
    
    
      
        
      
    
  
  
    
      
        
          
        
        ⁢
        
          
            ∫
            T
            
              
                T
                1
              
              +
              
                T
                2
              
            
          
          ⁢
          
            
              
                I
                tx
              
              ·
              
                sin
                ⁡
                
                  [
                  
                    
                      
                        (
                        
                          
                            ω
                            2
                          
                          -
                          
                            ω
                            0
                          
                        
                        )
                      
                      ⁢
                      t
                    
                    -
                    
                      
                        ω
                        2
                      
                      ⁢
                      
                        T
                        1
                      
                    
                  
                  ]
                
              
            
            ⁢
            
              d
              t
            
          
        
      
    
    
      
        
      
    
  
  
    
      
        
          
        
        ⁢
        
          =
          
            
              
                
                  I
                  tx
                
                
                  
                    ω
                    1
                  
                  +
                  
                    ω
                    0
                  
                
              
              ·
              
                [
                
                  1
                  -
                  
                    cos
                    ⁡
                    
                      (
                      
                        2
                        ⁢
                        n
                        ⁢
                        
                          
                        
                        ⁢
                        π
                        ⁢
                        
                          
                        
                        ⁢
                        Δ
                      
                      )
                    
                  
                
                ]
              
            
            +
          
        
      
    
    
      
        
      
    
  
  
    
      
        
          
        
        ⁢
        
          
            
              
                I
                tx
              
              
                
                  ω
                  1
                
                -
                
                  ω
                  0
                
              
            
            ·
            
              [
              
                
                  cos
                  ⁡
                  
                    (
                    
                      2
                      ⁢
                      n
                      ⁢
                      
                        
                      
                      ⁢
                      π
                      ⁢
                      
                        
                      
                      ⁢
                      Δ
                    
                    )
                  
                
                -
                1
              
              ]
            
          
          +
        
      
    
    
      
        
      
    
  
  
    
      
        
          
        
        ⁢
        
          
            
              
                I
                tx
              
              
                
                  ω
                  2
                
                +
                
                  ω
                  0
                
              
            
            ·
            
              [
              
                
                  cos
                  ⁡
                  
                    (
                    
                      2
                      ⁢
                      n
                      ⁢
                      
                        
                      
                      ⁢
                      π
                      ⁢
                      
                        
                      
                      ⁢
                      Δ
                    
                    )
                  
                
                -
                1
              
              ]
            
          
          +
        
      
    
    
      
        
      
    
  
  
    
      
        
          
        
        ⁢
        
          
            
              I
              tx
            
            
              
                ω
                2
              
              -
              
                ω
                0
              
            
          
          ·
          
            [
            
              1
              -
              
                cos
                ⁡
                
                  (
                  
                    2
                    ⁢
                    n
                    ⁢
                    
                      
                    
                    ⁢
                    π
                    ⁢
                    
                      
                    
                    ⁢
                    Δ
                  
                  )
                
              
            
            ]
          
        
      
    
    
      
        
      
    
  
  
    
      
        
          
        
        ⁢
        
          =
          
            
              
                
                  I
                  tx
                
                
                  
                    ω
                    0
                  
                  ·
                  Δ
                
              
              ·
              
                [
                
                  
                    cos
                    ⁡
                    
                      (
                      
                        2
                        ⁢
                        n
                        ⁢
                        
                          
                        
                        ⁢
                        π
                        ⁢
                        
                          
                        
                        ⁢
                        Δ
                      
                      )
                    
                  
                  -
                  1
                
                ]
              
            
            +
          
        
      
    
    
      
        
      
    
  
  
    
      
        
          
        
        ⁢
        
          
            
              I
              
                t
                ⁢
                
                  
                
                ⁢
                x
              
            
            
              
                -
                
                  ω
                  2
                
              
              ·
              Δ
            
          
          ·
          
            [
            
              1
              -
              
                cos
                ⁡
                
                  (
                  
                    2
                    ⁢
                    n
                    ⁢
                    
                      
                    
                    ⁢
                    π
                    ⁢
                    
                      
                    
                    ⁢
                    Δ
                  
                  )
                
              
            
            ]
          
        
      
    
    
      
        
      
    
  
  
    
      
        
          
        
        ⁢
        
          =
          
            
              
                2
                ⁢
                
                  I
                  tx
                
              
              
                
                  ω
                  0
                
                ·
                Δ
              
            
            ·
            
              [
              
                
                  cos
                  ⁡
                  
                    (
                    
                      2
                      ⁢
                      n
                      ⁢
                      
                        
                      
                      ⁢
                      π
                      ⁢
                      
                        
                      
                      ⁢
                      Δ
                    
                    )
                  
                
                -
                1
              
              ]
            
          
        
      
    
    
      
        
      
    
  
  
    
      
        
          
        
        ⁢
        
          =
          
            
              
                -
                4
              
              ⁢
              
                I
                tx
              
              ⁢
              
                
                  sin
                  2
                
                ⁡
                
                  (
                  
                    n
                    ⁢
                    
                      
                    
                    ⁢
                    π
                    ⁢
                    
                      
                    
                    ⁢
                    Δ
                  
                  )
                
              
            
            
              
                ω
                0
              
              ·
              Δ
            
          
        
      
    
    
      
        
      
    
  




leads to





  
    
      b
      
        2
        ⁢
        n
      
    
    ≈
    
      
        
          -
          4
        
        ⁢
        
          I
          tx
        
        ⁢
        
          
            sin
            2
          
          ⁡
          
            (
            
              n
              ⁢
              
                
              
              ⁢
              π
              ⁢
              
                
              
              ⁢
              Δ
            
            )
          
        
      
      
        
          (
          
            
              T
              1
            
            +
            
              T
              2
            
          
          )
        
        ⁢
        
          
            ω
            0
          
          ·
          Δ
        
      
    
  
  =
  
    
      
        
          -
          
            I
            tx
          
        
        ⁢
        
          
            sin
            2
          
          ⁡
          
            (
            
              n
              ⁢
              
                
              
              ⁢
              π
              ⁢
              
                
              
              ⁢
              Δ
            
            )
          
        
      
      
        n
        ⁢
        
          
        
        ⁢
        π
        ⁢
        
          
        
        ⁢
        Δ
      
    
    .
  




Therefore, the main tone can be expressed as:





  
    
      
        
          
            
              
                
                  a
                  
                    2
                    ⁢
                    n
                  
                
                ·
                
                  cos
                  ⁡
                  
                    (
                    
                      
                        ω
                        0
                      
                      ⁢
                      t
                    
                    )
                  
                
              
              +
              
                
                  
                    b
                    
                      2
                      ⁢
                      n
                    
                  
                  ·
                  sin
                
                ⁢
                
                  (
                  
                    
                      ω
                      0
                    
                    ⁢
                    t
                  
                  )
                
              
            
            ≈
            
              
                
                  
                    
                      I
                      tx
                    
                    ⁢
                    
                      
                        sin
                        ⁡
                        
                          (
                          
                            n
                            ⁢
                            
                              
                            
                            ⁢
                            πΔ
                          
                          )
                        
                      
                      ·
                      
                        cos
                        ⁡
                        
                          (
                          
                            n
                            ⁢
                            
                              
                            
                            ⁢
                            π
                            ⁢
                            
                              
                            
                            ⁢
                            Δ
                          
                          )
                        
                      
                    
                  
                  
                    n
                    ⁢
                    
                      
                    
                    ⁢
                    π
                    ⁢
                    
                      
                    
                    ⁢
                    Δ
                  
                
                ·
                
                  
                
                ⁢
                
                  cos
                  ⁡
                  
                    (
                    
                      
                        ω
                        0
                      
                      ⁢
                      t
                    
                    )
                  
                
              
              +
              
                
                  
                    
                      -
                      
                        I
                        tx
                      
                    
                    ⁢
                    
                      
                        sin
                        2
                      
                      ⁡
                      
                        (
                        
                          n
                          ⁢
                          
                            
                          
                          ⁢
                          π
                          ⁢
                          
                            
                          
                          ⁢
                          Δ
                        
                        )
                      
                    
                  
                  
                    n
                    ⁢
                    
                      
                    
                    ⁢
                    π
                    ⁢
                    
                      
                    
                    ⁢
                    Δ
                  
                
                ·
                
                  sin
                  ⁡
                  
                    (
                    
                      
                        ω
                        0
                      
                      ⁢
                      t
                    
                    )
                  
                
                ·
              
            
          
          =
          
            
              
                
                  I
                  tx
                
                ⁢
                
                  sin
                  ⁡
                  
                    (
                    
                      n
                      ⁢
                      
                        
                      
                      ⁢
                      π
                      ⁢
                      
                        
                      
                      ⁢
                      Δ
                    
                    )
                  
                
              
              
                n
                ⁢
                
                  
                
                ⁢
                π
                ⁢
                
                  
                
                ⁢
                Δ
              
            
            ·
            
              cos
              ⁡
              
                (
                
                  
                    
                      ω
                      0
                    
                    ⁢
                    t
                  
                  +
                  
                    n
                    ⁢
                    
                      
                    
                    ⁢
                    π
                    ⁢
                    
                      
                    
                    ⁢
                    Δ
                  
                
                )
              
            
          
        
        ⁢
        
          

        
        ⁢
        
          
        
        ⁢
        
          
            
              (
              
                
                  T
                  1
                
                +
                
                  T
                  2
                
              
              )
            
            ·
            
              a
              
                
                  
                    2
                    ⁢
                    n
                  
                  +
                  1
                
                ⁢
                
                  
                
              
            
          
          =
          
            
              ∫
              0
              
                
                  T
                  1
                
                +
                
                  T
                  2
                
              
            
            ⁢
            
              
                2
                ·
                
                  I
                  ⁡
                  
                    (
                    t
                    )
                  
                
                ·
                
                  cos
                  ⁡
                  
                    (
                    
                      
                        ω
                        0
                        +
                      
                      ⁢
                      t
                    
                    )
                  
                
              
              ⁢
              
                d
                t
              
            
          
        
      
    
    
      
        
      
    
  
  
    
      
        
          
        
        ⁢
        
          =
          
            
              
                ∫
                0
                
                  T
                  1
                
              
              ⁢
              
                
                  2
                  ·
                  
                    I
                    tx
                  
                  ·
                  
                    cos
                    ⁡
                    
                      (
                      
                        
                          ω
                          1
                        
                        ⁢
                        t
                      
                      )
                    
                  
                  ·
                  
                    cos
                    ⁡
                    
                      (
                      
                        
                          ω
                          0
                          +
                        
                        ⁢
                        t
                      
                      )
                    
                  
                
                ⁢
                
                  d
                  t
                
              
            
            +
          
        
      
    
    
      
        
      
    
  
  
    
      
        
          
        
        ⁢
        
          
            ∫
            
              T
              1
            
            
              
                T
                1
              
              +
              
                T
                2
              
            
          
          ⁢
          
            
              2
              ·
              
                I
                tx
              
              ·
              
                cos
                ⁡
                
                  (
                  
                    
                      ω
                      2
                    
                    ⁡
                    
                      (
                      
                        t
                        -
                        
                          T
                          1
                        
                      
                      )
                    
                  
                  )
                
              
              ·
              
                cos
                ⁡
                
                  (
                  
                    
                      ω
                      0
                      +
                    
                    ⁢
                    t
                  
                  )
                
              
            
            ⁢
            
              d
              t
            
          
        
      
    
    
      
        
      
    
  
  
    
      
        
          
        
        ⁢
        
          =
          
            
              
                ∫
                0
                
                  T
                  1
                
              
              ⁢
              
                
                  
                    I
                    tx
                  
                  ·
                  
                    cos
                    ⁡
                    
                      [
                      
                        
                          (
                          
                            
                              ω
                              1
                            
                            +
                            
                              ω
                              0
                              +
                            
                          
                          )
                        
                        ⁢
                        t
                      
                      ]
                    
                  
                
                ⁢
                
                  d
                  t
                
              
            
            +
          
        
      
    
    
      
        
      
    
  
  
    
      
        
          
        
        ⁢
        
          
            
              ∫
              0
              
                T
                1
              
            
            ⁢
            
              
                
                  I
                  tx
                
                ·
                
                  cos
                  ⁡
                  
                    [
                    
                      
                        (
                        
                          
                            ω
                            1
                          
                          -
                          
                            ω
                            0
                            +
                          
                        
                        )
                      
                      ⁢
                      t
                    
                    ]
                  
                
              
              ⁢
              
                d
                t
              
            
          
          +
        
      
    
    
      
        
      
    
  
  
    
      
        
          
        
        ⁢
        
          
            
              ∫
              
                T
                1
              
              
                
                  T
                  1
                
                +
                
                  T
                  2
                
              
            
            ⁢
            
              
                
                  I
                  tx
                
                ·
                
                  cos
                  ⁡
                  
                    [
                    
                      
                        
                          (
                          
                            
                              ω
                              2
                            
                            +
                            
                              ω
                              0
                              +
                            
                          
                          )
                        
                        ⁢
                        t
                      
                      -
                      
                        
                          ω
                          2
                        
                        ⁢
                        
                          T
                          2
                        
                      
                    
                    ]
                  
                
              
              ⁢
              
                d
                t
              
            
          
          +
        
      
    
    
      
        
      
    
  
  
    
      
        
          
        
        ⁢
        
          
            ∫
            T
            
              
                T
                1
              
              +
              
                T
                2
              
            
          
          ⁢
          
            
              
                I
                tx
              
              ·
              
                cos
                ⁡
                
                  [
                  
                    
                      
                        (
                        
                          
                            ω
                            2
                          
                          -
                          
                            ω
                            0
                            +
                          
                        
                        )
                      
                      ⁢
                      t
                    
                    -
                    
                      
                        ω
                        2
                      
                      ⁢
                      
                        T
                        1
                      
                    
                  
                  ]
                
              
            
            ⁢
            
              d
              t
            
          
        
      
    
    
      
        
      
    
  
  
    
      
        
          
        
        ⁢
        
          =
          
            
              
                
                  I
                  tx
                
                
                  
                    ω
                    1
                  
                  +
                  
                    ω
                    0
                    +
                  
                
              
              ·
              
                sin
                ⁡
                
                  (
                  
                    
                      ω
                      0
                      +
                    
                    ⁢
                    
                      T
                      1
                    
                  
                  )
                
              
            
            +
            
              
                
                  I
                  tx
                
                
                  
                    ω
                    1
                  
                  -
                  
                    ω
                    0
                    +
                  
                
              
              ·
              
                sin
                ⁡
                
                  (
                  
                    
                      -
                      
                        ω
                        0
                        +
                      
                    
                    ⁢
                    
                      T
                      1
                    
                  
                  )
                
              
            
            +
          
        
      
    
    
      
        
      
    
  
  
    
      
        
          
        
        ⁢
        
          
            
              
                I
                tx
              
              
                
                  ω
                  2
                
                +
                
                  ω
                  0
                  +
                
              
            
            ·
            
              [
              
                -
                
                  sin
                  ⁡
                  
                    (
                    
                      
                        ω
                        0
                        +
                      
                      ⁢
                      
                        T
                        1
                      
                    
                    )
                  
                
              
              ]
            
          
          +
          
            
              
                I
                tx
              
              
                
                  ω
                  2
                
                -
                
                  ω
                  0
                  +
                
              
            
            ·
            
              sin
              ⁡
              
                (
                
                  
                    ω
                    0
                    +
                  
                  ⁢
                  
                    T
                    1
                  
                
                )
              
            
          
        
      
    
    
      
        
      
    
  
  
    
      
        
          
        
        ⁢
        
          =
          
            
              
                
                  2
                  ⁢
                  n
                  ⁢
                  
                    
                  
                  ⁢
                  
                    I
                    tx
                  
                
                
                  
                    (
                    
                      
                        2
                        ⁢
                        n
                        ⁢
                        
                          
                        
                        ⁢
                        Δ
                      
                      -
                      1
                    
                    )
                  
                  ⁢
                  
                    ω
                    0
                  
                
              
              ·
              
                sin
                ⁡
                
                  (
                  
                    
                      -
                      
                        ω
                        0
                        +
                      
                    
                    ⁢
                    
                      T
                      1
                    
                  
                  )
                
              
            
            +
            
              
                
                  
                    -
                    2
                  
                  ⁢
                  n
                  ⁢
                  
                    
                  
                  ⁢
                  
                    I
                    tx
                  
                
                
                  
                    (
                    
                      
                        2
                        ⁢
                        n
                        ⁢
                        
                          
                        
                        ⁢
                        Δ
                      
                      +
                      1
                    
                    )
                  
                  ⁢
                  
                    ω
                    0
                  
                
              
              ·
            
          
        
      
    
    
      
        
      
    
  
  
    
      
        
          
        
        ⁢
        
          sin
          ⁡
          
            (
            
              
                ω
                0
                +
              
              ⁢
              
                T
                1
              
            
            )
          
        
      
    
    
      
        
      
    
  
  
    
      
        
          
        
        ⁢
        
          =
          
            
              2
              ⁢
              
                
                  nI
                  tx
                
                ·
                4
              
              ⁢
              n
              ⁢
              
                
              
              ⁢
              
                Δ
                ⁢
                
                  
                
                ·
                
                  sin
                  ⁡
                  
                    (
                    
                      
                        ω
                        0
                        +
                      
                      ⁢
                      
                        T
                        1
                      
                    
                    )
                  
                
              
            
            
              
                [
                
                  1
                  -
                  
                    
                      (
                      
                        2
                        ⁢
                        n
                        ⁢
                        
                          
                        
                        ⁢
                        Δ
                      
                      )
                    
                    2
                  
                
                ]
              
              ·
              
                ω
                0
              
            
          
        
      
    
    
      
        
      
    
  




leads to





  
    
      
        
          
            a
            
              
                2
                ⁢
                n
              
              +
              1
            
          
          ≈
          
            
              2
              ⁢
              
                
                  nI
                  tx
                
                ·
                4
              
              ⁢
              n
              ⁢
              
                
              
              ⁢
              
                Δ
                ·
                
                  sin
                  ⁡
                  
                    (
                    
                      
                        ω
                        0
                        +
                      
                      ⁢
                      
                        T
                        1
                      
                    
                    )
                  
                
              
            
            
              
                
                  [
                  
                    1
                    -
                    
                      
                        (
                        
                          2
                          ⁢
                          n
                          ⁢
                          
                            
                          
                          ⁢
                          Δ
                        
                        )
                      
                      2
                    
                  
                  ]
                
                ·
                
                  (
                  
                    
                      T
                      1
                    
                    +
                    
                      T
                      2
                    
                  
                  )
                
              
              ⁢
              
                ω
                0
              
            
          
        
        =
        
          
            
              
                
                  2
                  ⁢
                  n
                  ⁢
                  
                    
                  
                  ⁢
                  
                    
                      Δ
                      /
                      π
                    
                    ·
                    
                      I
                      tx
                    
                    ·
                    
                      sin
                      ⁡
                      
                        (
                        
                          
                            ω
                            0
                            +
                          
                          ⁢
                          
                            T
                            1
                          
                        
                        )
                      
                    
                  
                
                
                  [
                  
                    1
                    -
                    
                      
                        (
                        
                          2
                          ⁢
                          n
                          ⁢
                          
                            
                          
                          ⁢
                          Δ
                        
                        )
                      
                      2
                    
                  
                  ]
                
              
              .
              
                

              
              ⁢
              
                (
                
                  
                    T
                    1
                  
                  +
                  
                    T
                    2
                  
                
                )
              
            
            ·
            
              b
              
                
                  2
                  ⁢
                  n
                
                +
                1
              
            
          
          =
          
            
              ∫
              0
              
                
                  T
                  1
                
                +
                
                  T
                  2
                
              
            
            ⁢
            
              
                2
                ·
                
                  I
                  ⁡
                  
                    (
                    t
                    )
                  
                
                ·
                
                  sin
                  ⁡
                  
                    (
                    
                      
                        ω
                        0
                        +
                      
                      ⁢
                      t
                    
                    )
                  
                
              
              ⁢
              
                d
                t
              
            
          
        
      
    
    
      
        
      
    
  
  
    
      
        
          
        
        ⁢
        
          =
          
            
              
                ∫
                0
                
                  T
                  1
                
              
              ⁢
              
                
                  2
                  ·
                  
                    I
                    tx
                  
                  ·
                  
                    cos
                    ⁡
                    
                      (
                      
                        
                          ω
                          1
                        
                        ⁢
                        t
                      
                      )
                    
                  
                  ·
                  
                    sin
                    ⁡
                    
                      (
                      
                        
                          ω
                          0
                          +
                        
                        ⁢
                        t
                      
                      )
                    
                  
                
                ⁢
                
                  d
                  t
                
              
            
            +
          
        
      
    
    
      
        
      
    
  
  
    
      
        
          
        
        ⁢
        
          
            ∫
            
              T
              1
            
            
              
                T
                1
              
              +
              
                T
                2
              
            
          
          ⁢
          
            
              2
              ·
              
                l
                n
              
              ·
              
                cos
                ⁡
                
                  (
                  
                    
                      ω
                      2
                    
                    ⁡
                    
                      (
                      
                        t
                        -
                        
                          T
                          1
                        
                      
                      )
                    
                  
                  )
                
              
              ·
              
                sin
                ⁡
                
                  (
                  
                    
                      ω
                      0
                      +
                    
                    ⁢
                    t
                  
                  )
                
              
            
            ⁢
            
              
            
            ⁢
            
              d
              t
            
          
        
      
    
    
      
        
      
    
  
  
    
      
        
          
        
        ⁢
        
          =
          
            
              
                ∫
                0
                
                  T
                  1
                
              
              ⁢
              
                
                  
                    I
                    tx
                  
                  ·
                  
                    sin
                    ⁡
                    
                      [
                      
                        
                          (
                          
                            
                              ω
                              1
                            
                            +
                            
                              ω
                              0
                              +
                            
                          
                          )
                        
                        ⁢
                        t
                      
                      ]
                    
                  
                
                ⁢
                
                  d
                  t
                
              
            
            -
          
        
      
    
    
      
        
      
    
  
  
    
      
        
          
        
        ⁢
        
          
            
              ∫
              0
              
                T
                1
              
            
            ⁢
            
              
                
                  I
                  tx
                
                ·
                
                  sin
                  ⁡
                  
                    [
                    
                      
                        (
                        
                          
                            ω
                            1
                          
                          -
                          
                            ω
                            0
                            +
                          
                        
                        )
                      
                      ⁢
                      t
                    
                    ]
                  
                
              
              ⁢
              
                d
                t
              
            
          
          +
        
      
    
    
      
        
      
    
  
  
    
      
        
          
        
        ⁢
        
          
            
              ∫
              
                T
                1
              
              
                
                  T
                  1
                
                +
                
                  T
                  2
                
              
            
            ⁢
            
              
                
                  I
                  tx
                
                ·
                
                  sin
                  ⁡
                  
                    [
                    
                      
                        
                          (
                          
                            
                              ω
                              2
                            
                            +
                            
                              ω
                              0
                              +
                            
                          
                          )
                        
                        ⁢
                        t
                      
                      -
                      
                        
                          ω
                          2
                        
                        ⁢
                        
                          T
                          1
                        
                      
                    
                    ]
                  
                
              
              ⁢
              
                d
                t
              
            
          
          -
        
      
    
    
      
        
      
    
  
  
    
      
        
          
        
        ⁢
        
          
            ∫
            
              T
              1
            
            
              
                T
                1
              
              +
              
                T
                2
              
            
          
          ⁢
          
            
              
                I
                tx
              
              ·
              
                sin
                ⁡
                
                  [
                  
                    
                      
                        (
                        
                          
                            ω
                            2
                          
                          -
                          
                            ω
                            0
                            +
                          
                        
                        )
                      
                      ⁢
                      t
                    
                    -
                    
                      
                        ω
                        2
                      
                      ⁢
                      
                        T
                        1
                      
                    
                  
                  ]
                
              
            
            ⁢
            
              d
              t
            
          
        
      
    
    
      
        
      
    
  
  
    
      
        
          
        
        ⁢
        
          =
          
            
              
                
                  I
                  tx
                
                
                  
                    ω
                    1
                  
                  +
                  
                    ω
                    0
                    +
                  
                
              
              ·
              
                [
                
                  1
                  -
                  
                    cos
                    ⁡
                    
                      (
                      
                        
                          ω
                          0
                          +
                        
                        ⁢
                        
                          T
                          1
                        
                      
                      )
                    
                  
                
                ]
              
            
            +
            
              
                
                  I
                  tx
                
                
                  
                    ω
                    1
                  
                  -
                  
                    ω
                    0
                    +
                  
                
              
              ·
            
          
        
      
    
    
      
        
      
    
  
  
    
      
        
          
        
        ⁢
        
          
            [
            
              
                cos
                ⁡
                
                  (
                  
                    
                      -
                      
                        ω
                        0
                        +
                      
                    
                    ⁢
                    
                      T
                      1
                    
                  
                  )
                
              
              -
              1
            
            ]
          
          +
          
            
              
                I
                tx
              
              
                
                  ω
                  2
                
                +
                
                  ω
                  0
                  +
                
              
            
            ·
            
              [
              
                
                  cos
                  ⁡
                  
                    (
                    
                      
                        ω
                        0
                        +
                      
                      ⁢
                      
                        T
                        1
                      
                    
                    )
                  
                
                -
                1
              
              ]
            
          
          +
        
      
    
    
      
        
      
    
  
  
    
      
        
          
        
        ⁢
        
          
            
              I
              tx
            
            
              
                ω
                2
              
              -
              
                ω
                0
                +
              
            
          
          ·
          
            [
            
              1
              -
              
                cos
                ⁡
                
                  (
                  
                    
                      -
                      
                        ω
                        0
                        +
                      
                    
                    ⁢
                    
                      T
                      1
                    
                  
                  )
                
              
            
            ]
          
        
      
    
    
      
        
      
    
  
  
    
      
        
          
        
        ⁢
        
          ≈
          
            
              
                
                  2
                  ⁢
                  
                    nI
                    tx
                  
                
                
                  
                    (
                    
                      
                        2
                        ⁢
                        n
                        ⁢
                        
                          
                        
                        ⁢
                        Δ
                      
                      -
                      1
                    
                    )
                  
                  ⁢
                  
                    ω
                    0
                  
                
              
              ·
              
                [
                
                  
                    cos
                    ⁡
                    
                      (
                      
                        
                          ω
                          0
                          +
                        
                        ⁢
                        
                          T
                          1
                        
                      
                      )
                    
                  
                  -
                  1
                
                ]
              
            
            +
            
              
                
                  
                    -
                    2
                  
                  ⁢
                  n
                  ⁢
                  
                    
                  
                  ⁢
                  
                    I
                    tx
                  
                
                
                  
                    (
                    
                      
                        2
                        ⁢
                        n
                        ⁢
                        
                          
                        
                        ⁢
                        Δ
                      
                      +
                      1
                    
                    )
                  
                  ⁢
                  
                    ω
                    0
                  
                
              
              ·
            
          
        
      
    
    
      
        
      
    
  
  
    
      
                                            
        ⁢
        
          [
          
            1
            -
            
              cos
              ⁡
              
                (
                
                  
                    ω
                    0
                    +
                  
                  ⁢
                  
                    T
                    1
                  
                
                )
              
            
          
          ]
        
      
    
    
      
        
      
    
  
  
    
      
        
          
        
        ⁢
        
          =
          
            
              
                2
                ⁢
                n
                ⁢
                
                  
                
                ⁢
                
                  
                    I
                    tx
                  
                  ·
                  4
                
                ⁢
                n
                ⁢
                
                  
                
                ⁢
                Δ
              
              
                
                  ω
                  0
                
                ·
                
                  (
                  
                    
                      
                        (
                        
                          2
                          ⁢
                          n
                          ⁢
                          
                            
                          
                          ⁢
                          Δ
                        
                        )
                      
                      2
                    
                    -
                    1
                  
                  )
                
              
            
            ·
            
              [
              
                
                  cos
                  ⁡
                  
                    (
                    
                      
                        ω
                        0
                        +
                      
                      ⁢
                      
                        T
                        1
                      
                    
                    )
                  
                
                -
                1
              
              ]
            
          
        
      
    
    
      
        
      
    
  




leads to





  
    
      
        
          
            b
            
              
                2
                ⁢
                n
              
              +
              1
            
          
          ≈
          
            
              
                2
                ⁢
                
                  
                    nI
                    tx
                  
                  ·
                  4
                
                ⁢
                n
                ⁢
                
                  
                
                ⁢
                Δ
              
              
                
                  (
                  
                    
                      T
                      1
                    
                    +
                    
                      T
                      
                        2
                        ⁢
                        
                          
                        
                      
                    
                  
                  )
                
                ⁢
                
                  
                    ω
                    0
                  
                  ·
                  
                    (
                    
                      
                        
                          (
                          
                            2
                            ⁢
                            n
                            ⁢
                            
                              
                            
                            ⁢
                            Δ
                          
                          )
                        
                        2
                      
                      -
                      1
                    
                    )
                  
                
              
            
            ·
            
              [
              
                
                  cos
                  ⁡
                  
                    (
                    
                      
                        ω
                        0
                        +
                      
                      ⁢
                      
                        T
                        1
                      
                    
                    )
                  
                
                -
                1
              
              ]
            
          
        
        =
        
          
            
              
                
                  I
                  tx
                
                ·
                4
              
              ⁢
              
                
              
              ⁢
              n
              ⁢
              
                
              
              ⁢
              
                Δ
                /
                π
              
            
            
              (
              
                1
                -
                
                  
                    (
                    
                      1
                      ⁢
                      n
                      ⁢
                      
                        
                      
                      ⁢
                      Δ
                    
                    )
                  
                  2
                
              
              ⁢
              
                
              
              )
            
          
          ·
          
            
              
                sin
                2
              
              ⁡
              
                (
                
                  
                    
                      ω
                      0
                      +
                    
                    ⁢
                    T
                  
                  2
                
                )
              
            
            .
          
        
      
    
    
      
        
      
    
  
  
    
      
        
          
            (
            
              
                T
                1
              
              +
              
                T
                2
              
            
            )
          
          ·
          
            a
            
              
                2
                ⁢
                n
              
              -
              1
            
          
        
        =
        
          
            ∫
            0
            
              
                T
                1
              
              +
              
                T
                2
              
            
          
          ⁢
          
            
              2
              ·
              
                I
                ⁡
                
                  (
                  t
                  )
                
              
              ·
              
                cos
                ⁡
                
                  (
                  
                    
                      ω
                      0
                      -
                    
                    ⁢
                    t
                  
                  )
                
              
            
            ⁢
            
              d
              t
            
          
        
      
    
    
      
        
      
    
  
  
    
      
        
          
        
        ⁢
        
          =
          
            
              
                ∫
                0
                
                  T
                  1
                
              
              ⁢
              
                
                  2
                  ·
                  
                    I
                    tx
                  
                  ·
                  
                    cos
                    ⁡
                    
                      (
                      
                        
                          ω
                          1
                        
                        ⁢
                        t
                      
                      )
                    
                  
                  ·
                  
                    cos
                    ⁡
                    
                      (
                      
                        
                          ω
                          0
                          -
                        
                        ⁢
                        t
                      
                      )
                    
                  
                
                ⁢
                
                  d
                  t
                
              
            
            +
          
        
      
    
    
      
        
      
    
  
  
    
      
        
          
        
        ⁢
        
          
            ∫
            
              T
              1
            
            
              
                T
                1
              
              +
              
                T
                2
              
            
          
          ⁢
          
            
              2
              ·
              
                I
                tx
              
              ·
              
                cos
                ⁡
                
                  (
                  
                    
                      ω
                      2
                    
                    ⁡
                    
                      (
                      
                        t
                        -
                        
                          T
                          1
                        
                      
                      )
                    
                  
                  )
                
              
              ·
              
                cos
                ⁡
                
                  (
                  
                    
                      ω
                      0
                      -
                    
                    ⁢
                    t
                  
                  )
                
              
            
            ⁢
            
              d
              t
            
          
        
      
    
    
      
        
      
    
  
  
    
      
        
          
        
        ⁢
        
          =
          
            
              
                ∫
                0
                
                  T
                  1
                
              
              ⁢
              
                
                  
                    I
                    tx
                  
                  ·
                  
                    cos
                    ⁡
                    
                      [
                      
                        
                          (
                          
                            
                              ω
                              1
                            
                            +
                            
                              ω
                              0
                              -
                            
                          
                          )
                        
                        ⁢
                        t
                      
                      ]
                    
                  
                
                ⁢
                
                  d
                  t
                
              
            
            +
          
        
      
    
    
      
        
      
    
  
  
    
      
        
          
        
        ⁢
        
          
            
              ∫
              0
              
                T
                1
              
            
            ⁢
            
              
                
                  I
                  tx
                
                ·
                
                  cos
                  ⁡
                  
                    [
                    
                      
                        (
                        
                          
                            ω
                            1
                          
                          -
                          
                            ω
                            0
                            -
                          
                        
                        )
                      
                      ⁢
                      t
                    
                    ]
                  
                
              
              ⁢
              
                d
                t
              
            
          
          +
        
      
    
    
      
        
      
    
  
  
    
      
        
          
        
        ⁢
        
          
            
              ∫
              
                T
                1
              
              
                
                  T
                  1
                
                +
                
                  T
                  2
                
              
            
            ⁢
            
              
                
                  I
                  tx
                
                ·
                
                  cos
                  ⁡
                  
                    [
                    
                      
                        
                          (
                          
                            
                              ω
                              2
                            
                            +
                            
                              ω
                              0
                              -
                            
                          
                          )
                        
                        ⁢
                        t
                      
                      -
                      
                        
                          ω
                          2
                        
                        ⁢
                        
                          T
                          1
                        
                      
                    
                    ]
                  
                
              
              ⁢
              
                d
                t
              
            
          
          +
        
      
    
    
      
        
      
    
  
  
    
      
        
          
        
        ⁢
        
          
            ∫
            T
            
              
                T
                1
              
              +
              
                T
                2
              
            
          
          ⁢
          
            
              
                I
                tx
              
              ·
              
                cos
                ⁡
                
                  [
                  
                    
                      
                        (
                        
                          
                            ω
                            2
                          
                          -
                          
                            ω
                            0
                            -
                          
                        
                        )
                      
                      ⁢
                      t
                    
                    -
                    
                      
                        ω
                        2
                      
                      ⁢
                      
                        T
                        1
                      
                    
                  
                  ]
                
              
            
            ⁢
            
              d
              t
            
          
        
      
    
    
      
        
      
    
  
  
    
      
        
          
        
        ⁢
        
          =
          
            
              
                
                  I
                  tx
                
                
                  
                    ω
                    1
                  
                  +
                  
                    ω
                    0
                    -
                  
                
              
              ·
              
                sin
                ⁡
                
                  (
                  
                    
                      ω
                      0
                      -
                    
                    ⁢
                    
                      T
                      1
                    
                  
                  )
                
              
            
            +
            
              
                
                  I
                  tx
                
                
                  
                    
                      ω
                      1
                    
                    -
                    
                      ω
                      0
                      -
                    
                  
                  ⁢
                  
                    
                  
                
              
              ·
              
                sin
                ⁡
                
                  (
                  
                    
                      -
                      
                        ω
                        0
                        -
                      
                    
                    ⁢
                    
                      T
                      1
                    
                  
                  )
                
              
            
            +
          
        
      
    
    
      
        
      
    
  
  
    
      
        
          
        
        ⁢
        
          
            
              
                I
                tx
              
              
                
                  ω
                  2
                
                +
                
                  ω
                  0
                  -
                
              
            
            ·
            
              [
              
                -
                
                  sin
                  ⁡
                  
                    (
                    
                      
                        ω
                        0
                        -
                      
                      ⁢
                      
                        T
                        1
                      
                    
                    )
                  
                
              
              ]
            
          
          +
          
            
              
                I
                tx
              
              
                
                  ω
                  1
                
                -
                
                  ω
                  0
                  -
                
              
            
            ·
            
              sin
              ⁡
              
                (
                
                  
                    -
                    
                      ω
                      0
                      -
                    
                  
                  ⁢
                  
                    T
                    1
                  
                
                )
              
            
          
        
      
    
    
      
        
      
    
  
  
    
      
        
          
        
        ⁢
        
          ≈
          
            
              
                
                  2
                  ⁢
                  
                    nI
                    tx
                  
                
                
                  
                    (
                    
                      
                        2
                        ⁢
                        n
                        ⁢
                        
                          
                        
                        ⁢
                        Δ
                      
                      +
                      1
                    
                    )
                  
                  ⁢
                  
                    ω
                    0
                  
                
              
              ·
              
                sin
                ⁡
                
                  (
                  
                    
                      -
                      
                        ω
                        0
                        -
                      
                    
                    ⁢
                    
                      T
                      1
                    
                  
                  )
                
              
            
            +
            
              
                
                  2
                  ⁢
                  
                    nI
                    tx
                  
                
                
                  
                    (
                    
                      1
                      -
                      
                        2
                        ⁢
                        n
                        ⁢
                        
                          
                        
                        ⁢
                        Δ
                      
                    
                    )
                  
                  ⁢
                  
                    ω
                    0
                  
                
              
              ·
              
                sin
                ⁡
                
                  (
                  
                    
                      ω
                      0
                      -
                    
                    ⁢
                    
                      T
                      1
                    
                  
                  )
                
              
            
          
        
      
    
    
      
        
      
    
  
  
    
      
        
          
        
        ⁢
        
          =
          
            
              2
              ⁢
              
                
                  nI
                  tx
                
                ·
                4
              
              ⁢
              n
              ⁢
              
                
              
              ⁢
              
                Δ
                ⁢
                
                  
                
                ·
                
                  sin
                  ⁡
                  
                    (
                    
                      
                        ω
                        0
                        -
                      
                      ⁢
                      
                        T
                        1
                      
                    
                    )
                  
                
              
            
            
              
                [
                
                  1
                  -
                  
                    
                      (
                      
                        2
                        ⁢
                        n
                        ⁢
                        
                          
                        
                        ⁢
                        Δ
                      
                      )
                    
                    2
                  
                
                ]
              
              ·
              
                ω
                0
              
            
          
        
      
    
    
      
        
      
    
  




leads to





  
    
      
        
          
            a
            
              
                2
                ⁢
                n
              
              -
              1
            
          
          ≈
          
            
              2
              ⁢
              
                
                  nI
                  tx
                
                ·
                4
              
              ⁢
              n
              ⁢
              
                
              
              ⁢
              
                Δ
                ·
                
                  sin
                  ⁡
                  
                    (
                    
                      
                        ω
                        0
                        -
                      
                      ⁢
                      
                        T
                        1
                      
                    
                    )
                  
                
              
            
            
              
                
                  [
                  
                    1
                    -
                    
                      
                        (
                        
                          2
                          ⁢
                          n
                          ⁢
                          
                            
                          
                          ⁢
                          Δ
                        
                        )
                      
                      2
                    
                  
                  ]
                
                ·
                
                  (
                  
                    
                      T
                      1
                    
                    +
                    
                      T
                      2
                    
                  
                  )
                
              
              ⁢
              
                ω
                0
              
            
          
        
        =
        
          
            
              
                
                  I
                  tx
                
                ·
                2
              
              ⁢
              n
              ⁢
              
                
              
              ⁢
              
                Δ
                /
                π
              
            
            
              [
              
                1
                -
                
                  
                    (
                    
                      2
                      ⁢
                      n
                      ⁢
                      
                        
                      
                      ⁢
                      Δ
                    
                    )
                  
                  2
                
              
              ]
            
          
          ·
          
            
              sin
              ⁡
              
                (
                
                  
                    ω
                    0
                    -
                  
                  ⁢
                  
                    T
                    1
                  
                
                )
              
            
            .
          
        
      
    
    
      
        
      
    
  
  
    
      
        
          
            (
            
              
                T
                1
              
              +
              
                T
                2
              
            
            )
          
          ·
          
            b
            
              
                2
                ⁢
                n
              
              -
              1
            
          
        
        =
        
          
            ∫
            0
            
              
                T
                1
              
              +
              
                T
                2
              
            
          
          ⁢
          
            
              2
              ·
              
                I
                ⁡
                
                  (
                  t
                  )
                
              
              ·
              
                sin
                ⁡
                
                  (
                  
                    
                      ω
                      0
                      -
                    
                    ⁢
                    t
                  
                  )
                
              
            
            ⁢
            
              d
              t
            
          
        
      
    
    
      
        
      
    
  
  
    
      
        
          
        
        ⁢
        
          =
          
            
              
                ∫
                0
                
                  T
                  1
                
              
              ⁢
              
                
                  2
                  ·
                  
                    I
                    tx
                  
                  ·
                  
                    cos
                    ⁡
                    
                      (
                      
                        
                          ω
                          1
                        
                        ⁢
                        t
                      
                      )
                    
                  
                  ·
                  
                    sin
                    ⁡
                    
                      (
                      
                        
                          ω
                          0
                          -
                        
                        ⁢
                        t
                      
                      )
                    
                  
                
                ⁢
                
                  d
                  t
                
              
            
            +
          
        
      
    
    
      
        
      
    
  
  
    
      
        
          
        
        ⁢
        
          
            ∫
            
              T
              1
            
            
              
                T
                1
              
              +
              
                T
                2
              
            
          
          ⁢
          
            
              2
              ·
              
                I
                tx
              
              ·
              
                cos
                ⁡
                
                  (
                  
                    
                      ω
                      2
                    
                    ⁡
                    
                      (
                      
                        t
                        -
                        
                          T
                          1
                        
                      
                      )
                    
                  
                  )
                
              
              ·
              
                sin
                ⁡
                
                  (
                  
                    
                      ω
                      0
                      -
                    
                    ⁢
                    t
                  
                  )
                
              
            
            ⁢
            
              d
              t
            
          
        
      
    
    
      
        
      
    
  
  
    
      
        
          
        
        ⁢
        
          =
          
            
              
                ∫
                0
                
                  T
                  1
                
              
              ⁢
              
                
                  
                    I
                    tx
                  
                  ·
                  
                    sin
                    ⁡
                    
                      [
                      
                        
                          (
                          
                            
                              ω
                              1
                            
                            +
                            
                              ω
                              0
                              -
                            
                          
                          )
                        
                        ⁢
                        t
                      
                      ]
                    
                  
                
                ⁢
                
                  d
                  t
                
              
            
            -
          
        
      
    
    
      
        
      
    
  
  
    
      
        
          
        
        ⁢
        
          
            
              ∫
              0
              
                T
                1
              
            
            ⁢
            
              
                
                  I
                  tx
                
                ·
                
                  sin
                  ⁡
                  
                    [
                    
                      
                        (
                        
                          
                            ω
                            1
                          
                          -
                          
                            ω
                            0
                            -
                          
                        
                        )
                      
                      ⁢
                      t
                    
                    ]
                  
                
              
              ⁢
              
                d
                t
              
            
          
          +
        
      
    
    
      
        
      
    
  
  
    
      
        
          
        
        ⁢
        
          
            
              ∫
              
                T
                1
              
              
                
                  T
                  1
                
                +
                
                  T
                  2
                
              
            
            ⁢
            
              
                
                  I
                  tx
                
                ·
                
                  sin
                  ⁡
                  
                    [
                    
                      
                        
                          (
                          
                            
                              ω
                              2
                            
                            +
                            
                              ω
                              0
                              -
                            
                          
                          )
                        
                        ⁢
                        t
                      
                      -
                      
                        
                          ω
                          2
                        
                        ⁢
                        
                          T
                          1
                        
                      
                    
                    ]
                  
                
              
              ⁢
              
                d
                t
              
            
          
          -
        
      
    
    
      
        
      
    
  
  
    
      
        
          
        
        ⁢
        
          
            ∫
            
              T
              1
            
            
              
                T
                1
              
              +
              
                T
                2
              
            
          
          ⁢
          
            
              
                I
                tx
              
              ·
              
                sin
                ⁡
                
                  [
                  
                    
                      
                        (
                        
                          
                            ω
                            2
                          
                          -
                          
                            ω
                            0
                            -
                          
                        
                        )
                      
                      ⁢
                      t
                    
                    -
                    
                      
                        ω
                        2
                      
                      ⁢
                      
                        T
                        1
                      
                    
                  
                  ]
                
              
            
            ⁢
            
              d
              t
            
          
        
      
    
    
      
        
      
    
  
  
    
      
        
          
        
        ⁢
        
          =
          
            
              
                
                  I
                  tx
                
                
                  
                    ω
                    1
                  
                  +
                  
                    ω
                    0
                    -
                  
                
              
              ·
              
                [
                
                  1
                  -
                  
                    cos
                    ⁡
                    
                      (
                      
                        
                          ω
                          0
                          -
                        
                        ⁢
                        
                          T
                          1
                        
                      
                      )
                    
                  
                
                ]
              
            
            +
            
              
                
                  I
                  tx
                
                
                  
                    ω
                    1
                  
                  -
                  
                    ω
                    0
                    -
                  
                
              
              ·
            
          
        
      
    
    
      
        
      
    
  
  
    
      
                                           
        ⁢
        
          
            [
            
              
                cos
                ⁡
                
                  (
                  
                    
                      -
                      
                        ω
                        0
                        -
                      
                    
                    ⁢
                    
                      T
                      1
                    
                  
                  )
                
              
              -
              1
            
            ]
          
          +
          
            
              
                I
                tx
              
              
                
                  ω
                  2
                
                +
                
                  ω
                  0
                  -
                
              
            
            ·
            
              [
              
                
                  cos
                  ⁡
                  
                    (
                    
                      
                        ω
                        0
                        -
                      
                      ⁢
                      
                        T
                        1
                      
                    
                    )
                  
                
                -
                1
              
              ]
            
          
          +
        
      
    
    
      
        
      
    
  
  
    
      
        
          
        
        ⁢
        
          
            
              I
              tx
            
            
              
                ω
                2
              
              -
              
                ω
                0
                -
              
            
          
          ·
          
            [
            
              1
              -
              
                cos
                ⁡
                
                  (
                  
                    
                      -
                      
                        ω
                        0
                        -
                      
                    
                    ⁢
                    
                      T
                      1
                    
                  
                  )
                
              
            
            ]
          
        
      
    
    
      
        
      
    
  
  
    
      
        
          
        
        ⁢
        
          ≈
          
            
              
                
                  2
                  ⁢
                  n
                  ⁢
                  
                    
                  
                  ⁢
                  
                    I
                    tx
                  
                
                
                  
                    (
                    
                      
                        2
                        ⁢
                        n
                        ⁢
                        
                          
                        
                        ⁢
                        Δ
                      
                      +
                      1
                    
                    )
                  
                  ⁢
                  
                    ω
                    0
                  
                
              
              ·
              
                [
                
                  
                    cos
                    ⁡
                    
                      (
                      
                        
                          ω
                          0
                          -
                        
                        ⁢
                        
                          T
                          1
                        
                      
                      )
                    
                  
                  -
                  1
                
                ]
              
            
            +
            
              
                
                  2
                  ⁢
                  
                    nI
                    tx
                  
                
                
                  
                    (
                    
                      1
                      -
                      
                        2
                        ⁢
                        n
                        ⁢
                        
                          
                        
                        ⁢
                        Δ
                      
                    
                    )
                  
                  ⁢
                  
                    ω
                    0
                  
                
              
              ·
            
          
        
      
    
    
      
        
      
    
  
  
    
      
                                            
        ⁢
        
          [
          
            1
            -
            
              cos
              ⁡
              
                (
                
                  
                    ω
                    0
                    -
                  
                  ⁢
                  
                    T
                    1
                  
                
                )
              
            
          
          ]
        
      
    
    
      
        
      
    
  
  
    
      
        
          
        
        ⁢
        
          =
          
            
              
                2
                ⁢
                n
                ⁢
                
                  
                
                ⁢
                
                  
                    I
                    tx
                  
                  ·
                  4
                
                ⁢
                n
                ⁢
                
                  
                
                ⁢
                Δ
              
              
                
                  ω
                  0
                
                ·
                
                  (
                  
                    1
                    -
                    
                      
                        (
                        
                          2
                          ⁢
                          n
                          ⁢
                          
                            
                          
                          ⁢
                          Δ
                        
                        )
                      
                      
                        2
                        ⁢
                        
                          
                        
                      
                    
                  
                  )
                
              
            
            ·
            
              [
              
                1
                -
                
                  cos
                  ⁡
                  
                    (
                    
                      
                        ω
                        0
                        -
                      
                      ⁢
                      
                        T
                        1
                      
                    
                    )
                  
                
              
              ]
            
          
        
      
    
    
      
        
      
    
  




leads to





  
    b
    
      
        2
        ⁢
        n
      
      -
      1
    
  
  =
  
    
      
        
          2
          ⁢
          n
          ⁢
          
            
          
          ⁢
          
            
              I
              tx
            
            ·
            4
          
          ⁢
          
            
          
          ⁢
          n
          ⁢
          
            
          
          ⁢
          Δ
        
        
          
            (
            
              
                T
                1
              
              +
              
                T
                2
              
            
            )
          
          ⁢
          
            
              ω
              0
            
            ·
            
              (
              
                1
                -
                
                  
                    (
                    
                      2
                      ⁢
                      n
                      ⁢
                      
                        
                      
                      ⁢
                      Δ
                    
                    )
                  
                  2
                
              
              )
            
          
        
      
      ·
      
        [
        
          1
          -
          
            cos
            ⁡
            
              (
              
                
                  ω
                  0
                  -
                
                ⁢
                
                  
                
                ⁢
                
                  T
                  1
                
              
              )
            
          
        
        ]
      
    
    =
    
      
        
          
            
              I
              tx
            
            ·
            4
          
          ⁢
          n
          ⁢
          
            
          
          ⁢
          
            Δ
            /
            π
          
        
        
          (
          
            1
            -
            
              
                (
                
                  2
                  ⁢
                  n
                  ⁢
                  
                    
                  
                  ⁢
                  Δ
                
                )
              
              2
            
          
          )
        
      
      ·
      
        
          
            sin
            2
          
          ⁡
          
            (
            
              
                
                  ω
                  0
                  -
                
                ⁢
                T
              
              2
            
            )
          
        
        .
      
    
  




So the two tones can be expressed as





  
    
      
        
          a
          
            
              2
              ⁢
              n
            
            -
            1
          
        
        ·
        
          cos
          ⁡
          
            (
            
              
                ω
                0
                -
              
              ⁢
              t
            
            )
          
        
      
      +
      
        
          b
          
            
              2
              ⁢
              n
            
            -
            1
          
        
        ·
        
          sin
          ⁡
          
            (
            
              
                ω
                0
                -
              
              ⁢
              t
            
            )
          
        
      
    
    ≈
    
      
        
          
            2
            ⁢
            n
            ⁢
            
              
            
            ⁢
            
              
                Δ
                /
                π
              
              ·
              
                I
                tx
              
              ·
              
                sin
                ⁡
                
                  (
                  
                    
                      ω
                      0
                      -
                    
                    ⁢
                    
                      T
                      1
                    
                  
                  )
                
              
            
          
          
            [
            
              1
              -
              
                
                  (
                  
                    2
                    ⁢
                    n
                    ⁢
                    
                      
                    
                    ⁢
                    Δ
                  
                  )
                
                2
              
            
            ]
          
        
        ·
        
          cos
          ⁡
          
            (
            
              
                ω
                0
                -
              
              ⁢
              t
            
            )
          
        
      
      +
      
        
          
            
              
                I
                tx
              
              ·
              4
            
            ⁢
            n
            ⁢
            
              
            
            ⁢
            
              Δ
              /
              π
            
          
          
            (
            
              1
              -
              
                
                  (
                  
                    2
                    ⁢
                    n
                    ⁢
                    
                      
                    
                    ⁢
                    Δ
                  
                  )
                
                2
              
            
            )
          
        
        ·
        
          
            sin
            2
          
          ⁡
          
            (
            
              
                
                  ω
                  0
                  -
                
                ⁢
                T
              
              2
            
            )
          
        
        ·
        
          sin
          ⁡
          
            (
            
              
                ω
                0
                -
              
              ⁢
              t
            
            )
          
        
      
    
  
  =
  
    
      
        
          
            
              
                
                  I
                  tx
                
                ·
                4
              
              ⁢
              n
              ⁢
              
                
              
              ⁢
              
                Δ
                /
                π
              
            
            
              (
              
                1
                -
                
                  
                    (
                    
                      2
                      ⁢
                      n
                      ⁢
                      
                        
                      
                      ⁢
                      Δ
                    
                    )
                  
                  2
                
              
              )
            
          
          ·
          
            sin
            ⁡
            
              (
              
                
                  
                    ω
                    0
                    -
                  
                  ⁢
                  T
                
                2
              
              )
            
          
          ·
          
            cos
            ⁡
            
              (
              
                
                  
                    ω
                    0
                    -
                  
                  ⁢
                  t
                
                -
                
                  
                    
                      ω
                      0
                      -
                    
                    ⁢
                    T
                  
                  2
                
              
              )
            
          
        
        ⁢
        
          

        
        ⁢
        
          
            a
            
              
                2
                ⁢
                n
              
              +
              1
            
          
          ·
          
            cos
            ⁡
            
              (
              
                
                  ω
                  0
                  +
                
                ⁢
                t
              
              )
            
          
        
      
      +
      
        
          b
          
            
              2
              ⁢
              n
            
            +
            1
          
        
        ·
        
          sin
          ⁡
          
            (
            
              
                ω
                0
                +
              
              ⁢
              t
            
            )
          
        
      
    
    =
    
      
        
          
            
              2
              ⁢
              n
              ⁢
              
                
              
              ⁢
              
                
                  Δ
                  /
                  π
                
                ·
                
                  I
                  tx
                
                ·
                
                  sin
                  ⁡
                  
                    (
                    
                      
                        ω
                        0
                        +
                      
                      ⁢
                      
                        T
                        1
                      
                    
                    )
                  
                
              
            
            
              [
              
                1
                -
                
                  
                    (
                    
                      2
                      ⁢
                      n
                      ⁢
                      
                        
                      
                      ⁢
                      Δ
                    
                    )
                  
                  2
                
              
              ]
            
          
          ·
          
            cos
            ⁡
            
              (
              
                
                  ω
                  0
                  +
                
                ⁢
                t
              
              )
            
          
        
        +
        
          
            
              
                
                  I
                  tx
                
                ·
                4
              
              ⁢
              n
              ⁢
              
                
              
              ⁢
              
                Δ
                /
                π
              
            
            
              (
              
                1
                -
                
                  
                    (
                    
                      2
                      ⁢
                      n
                      ⁢
                      
                        
                      
                      ⁢
                      Δ
                    
                    )
                  
                  2
                
              
              )
            
          
          ·
          
            
              sin
              2
            
            ⁡
            
              (
              
                
                  
                    ω
                    0
                    +
                  
                  ⁢
                  T
                
                2
              
              )
            
          
          ·
          
            sin
            ⁡
            
              (
              
                
                  ω
                  0
                  +
                
                ⁢
                t
              
              )
            
          
        
      
      =
      
        
          
            
              
                I
                tx
              
              ·
              4
            
            ⁢
            n
            ⁢
            
              
            
            ⁢
            
              Δ
              /
              π
            
          
          
            (
            
              1
              -
              
                
                  (
                  
                    2
                    ⁢
                    n
                    ⁢
                    
                      
                    
                    ⁢
                    Δ
                  
                  )
                
                2
              
            
            )
          
        
        ·
        
          sin
          ⁡
          
            (
            
              
                
                  ω
                  0
                  +
                
                ⁢
                T
              
              2
            
            )
          
        
        ·
        
          cos
          ⁡
          
            (
            
              
                
                  ω
                  0
                  +
                
                ⁢
                t
              
              -
              
                
                  
                    ω
                    0
                    +
                  
                  ⁢
                  
                    T
                    1
                  
                
                2
              
            
            )
          
        
      
    
  




Appendix II-Phase Error Derivation
Appendix II provides the following detailed analysis for calculating the phase error derivation according to Equation (37).




  
    
      
        (
        
          1
          +
          δ
        
        )
      
      ·
      
        [
        
          
            cos
            ⁡
            
              (
              
                
                  
                    ω
                    m
                  
                  ⁢
                  t
                
                +
                
                  
                    ω
                    0
                    -
                  
                  ⁢
                  
                    T
                    1
                  
                
                +
                
                  n
                  ⁢
                  
                    
                  
                  ⁢
                  π
                  ⁢
                  
                    
                  
                  ⁢
                  Δ
                
                +
                
                  φ
                  0
                
                -
                
                  φ
                  0
                  -
                
              
              )
            
          
          -
          
            cos
            ⁡
            
              (
              
                
                  
                    ω
                    m
                  
                  ⁢
                  t
                
                +
                
                  n
                  ⁢
                  
                    
                  
                  ⁢
                  π
                  ⁢
                  
                    
                  
                  ⁢
                  Δ
                
                +
                
                  φ
                  0
                
                -
                
                  φ
                  0
                  -
                
              
              )
            
          
        
        ]
      
    
    +
    
      
        (
        
          1
          -
          δ
        
        )
      
      ·
      
        [
        
          
            
              
                
                  cos
                  ⁢
                  
                    (
                    
                      
                        
                          -
                          
                            ω
                            m
                          
                        
                        ⁢
                        t
                      
                      +
                      
                        
                          ω
                          0
                          +
                        
                        ⁢
                        
                          T
                          1
                        
                      
                      +
                      
                        n
                        ⁢
                        
                          
                        
                        ⁢
                        π
                        ⁢
                        
                          
                        
                        ⁢
                        Δ
                      
                      +
                      
                        φ
                        0
                      
                      -
                      
                        φ
                        0
                        +
                      
                    
                    )
                  
                
                -
              
            
          
          
            
              
                cos
                ⁡
                
                  (
                  
                    
                      
                        -
                        
                          ω
                          m
                        
                      
                      ⁢
                      t
                    
                    +
                    
                      n
                      ⁢
                      
                        
                      
                      ⁢
                      π
                      ⁢
                      
                        
                      
                      ⁢
                      Δ
                    
                    +
                    
                      φ
                      0
                    
                    -
                    
                      φ
                      0
                      +
                    
                  
                  )
                
              
            
          
        
        ]
      
    
  
  =
  
     
    
      
        [
        
          
            
              
                
                  cos
                  ⁢
                  
                    (
                    
                      
                        
                          ω
                          m
                        
                        ⁢
                        t
                      
                      +
                      
                        
                          ω
                          0
                          -
                        
                        ⁢
                        
                          T
                          1
                        
                      
                      +
                      
                        n
                        ⁢
                        
                          
                        
                        ⁢
                        π
                        ⁢
                        
                          
                        
                        ⁢
                        Δ
                      
                      +
                      
                        φ
                        0
                      
                      -
                      
                        φ
                        0
                        -
                      
                    
                    )
                  
                
                +
              
            
          
          
            
              
                cos
                ⁡
                
                  (
                  
                    
                      
                        -
                        
                          ω
                          m
                        
                      
                      ⁢
                      t
                    
                    +
                    
                      
                        ω
                        0
                        +
                      
                      ⁢
                      
                        T
                        1
                      
                    
                    +
                    
                      n
                      ⁢
                      
                        
                      
                      ⁢
                      π
                      ⁢
                      
                        
                      
                      ⁢
                      Δ
                    
                    +
                    
                      φ
                      0
                    
                    -
                    
                      φ
                      0
                      +
                    
                  
                  )
                
              
            
          
        
        ]
      
      -
      
         
        
          
            
              [
              
                
                  
                    
                      
                        cos
                        ⁢
                        
                          (
                          
                            
                              
                                ω
                                m
                              
                              ⁢
                              
                                
                              
                              ⁢
                              t
                            
                            +
                            
                              n
                              ⁢
                              
                                
                              
                              ⁢
                              π
                              ⁢
                              
                                
                              
                              ⁢
                              Δ
                            
                            +
                            
                              φ
                              0
                            
                            -
                            
                              φ
                              0
                              -
                            
                          
                          )
                        
                      
                      +
                    
                  
                
                
                  
                    
                      cos
                      ⁡
                      
                        (
                        
                          
                            
                              -
                              
                                ω
                                m
                              
                            
                            ⁢
                            t
                          
                          +
                          
                            n
                            ⁢
                            
                              
                            
                            ⁢
                            π
                            ⁢
                            
                              
                            
                            ⁢
                            Δ
                          
                          +
                          
                            φ
                            0
                          
                          -
                          
                            φ
                            0
                            +
                          
                        
                        )
                      
                    
                  
                
              
              ]
            
            +
            
              δ
              ·
              
                [
                
                  
                    
                      
                        
                          cos
                          ⁢
                          
                            (
                            
                              
                                
                                  ω
                                  m
                                
                                ⁢
                                t
                              
                              +
                              
                                
                                  ω
                                  0
                                  -
                                
                                ⁢
                                
                                  T
                                  1
                                
                              
                              +
                              
                                n
                                ⁢
                                
                                  
                                
                                ⁢
                                π
                                ⁢
                                
                                  
                                
                                ⁢
                                Δ
                              
                              +
                              
                                φ
                                0
                              
                              -
                              
                                φ
                                0
                                -
                              
                            
                            )
                          
                        
                        -
                      
                    
                  
                  
                    
                      
                        cos
                        ⁡
                        
                          (
                          
                            
                              
                                -
                                
                                  ω
                                  m
                                
                              
                              ⁢
                              t
                            
                            +
                            
                              
                                ω
                                0
                                +
                              
                              ⁢
                              
                                T
                                1
                              
                            
                            +
                            
                              n
                              ⁢
                              
                                
                              
                              ⁢
                              π
                              ⁢
                              
                                
                              
                              ⁢
                              Δ
                            
                            +
                            
                              φ
                              0
                            
                            -
                            
                              φ
                              0
                              +
                            
                          
                          )
                        
                      
                    
                  
                
                ]
              
            
            -
            
              
                δ
                ·
                
                  [
                  
                    
                      cos
                      ⁡
                      
                        (
                        
                          
                            
                              ω
                              m
                            
                            ⁢
                            t
                          
                          +
                          
                            n
                            ⁢
                            
                              
                            
                            ⁢
                            π
                            ⁢
                            
                              
                            
                            ⁢
                            Δ
                          
                          +
                          
                            φ
                            0
                          
                          -
                          
                            φ
                            0
                            -
                          
                        
                        )
                      
                    
                    -
                    
                      cos
                      ⁡
                      
                        (
                        
                          
                            
                              -
                              
                                ω
                                m
                              
                            
                            ⁢
                            t
                          
                          +
                          
                            n
                            ⁢
                            
                              
                            
                            ⁢
                            π
                            ⁢
                            
                              
                            
                            ⁢
                            Δ
                          
                          +
                          
                            φ
                            0
                          
                          -
                          
                            φ
                            0
                            +
                          
                        
                        )
                      
                    
                  
                  ]
                
              
              ⁢
              
                

              
              ⁢
              
                
                  cos
                  ⁡
                  
                    (
                    
                      
                        
                          ω
                          m
                        
                        ⁢
                        t
                      
                      +
                      
                        
                          ω
                          0
                          -
                        
                        ⁢
                        
                          T
                          1
                        
                      
                      +
                      
                        n
                        ⁢
                        
                          
                        
                        ⁢
                        π
                        ⁢
                        
                          
                        
                        ⁢
                        Δ
                      
                      +
                      
                        φ
                        0
                      
                      -
                      
                        φ
                        0
                        -
                      
                    
                    )
                  
                
                +
                
                  cos
                  ⁡
                  
                    (
                    
                      
                        
                          -
                          
                            ω
                            m
                          
                        
                        ⁢
                        t
                      
                      +
                      
                        
                          ω
                          0
                          -
                        
                        ⁢
                        
                          T
                          1
                        
                      
                      +
                      
                        n
                        ⁢
                        
                          
                        
                        ⁢
                        π
                        ⁢
                        
                          
                        
                        ⁢
                        Δ
                      
                      +
                      
                        φ
                        0
                      
                      -
                      
                        φ
                        0
                        +
                      
                    
                    )
                  
                
              
            
          
          =
          
            
              2
              ⁢
              
                
                  cos
                  ⁡
                  
                    (
                    
                      
                        
                          ω
                          0
                        
                        ⁢
                        
                          T
                          1
                        
                      
                      +
                      
                        n
                        ⁢
                        
                          
                        
                        ⁢
                        π
                        ⁢
                        
                          
                        
                        ⁢
                        Δ
                      
                      +
                      
                        φ
                        0
                      
                      -
                      
                        
                          
                            φ
                            0
                            -
                          
                          +
                          
                            φ
                            0
                            +
                          
                        
                        2
                      
                    
                    )
                  
                
                ·
                
                  cos
                  ⁡
                  
                    (
                    
                      
                        
                          ω
                          m
                        
                        ⁢
                        t
                      
                      -
                      
                        
                          ω
                          m
                        
                        ⁢
                        
                          T
                          1
                        
                      
                      -
                      
                        
                          
                            φ
                            0
                            -
                          
                          -
                          
                            φ
                            0
                            +
                          
                        
                        2
                      
                    
                    )
                  
                
              
              ⁢
              
                

              
              ⁢
              
                
                  cos
                  ⁡
                  
                    (
                    
                      
                        
                          ω
                          m
                        
                        ⁢
                        t
                      
                      +
                      
                        n
                        ⁢
                        
                          
                        
                        ⁢
                        π
                        ⁢
                        
                          
                        
                        ⁢
                        Δ
                      
                      +
                      
                        φ
                        0
                      
                      -
                      
                        φ
                        0
                        -
                      
                    
                    )
                  
                
                +
                
                  cos
                  ⁡
                  
                    (
                    
                      
                        
                          -
                          
                            ω
                            m
                          
                        
                        ⁢
                        t
                      
                      +
                      
                        n
                        ⁢
                        
                          
                        
                        ⁢
                        π
                        ⁢
                        
                          
                        
                        ⁢
                        Δ
                      
                      +
                      
                        φ
                        0
                      
                      -
                      
                        φ
                        0
                        +
                      
                    
                    )
                  
                
              
            
            =
            
              
                2
                ⁢
                
                  
                    cos
                    ⁡
                    
                      (
                      
                        
                          n
                          ⁢
                          
                            
                          
                          ⁢
                          π
                          ⁢
                          
                            
                          
                          ⁢
                          Δ
                        
                        +
                        
                          φ
                          0
                        
                        -
                        
                          
                            
                              φ
                              0
                              -
                            
                            +
                            
                              φ
                              0
                              +
                            
                          
                          2
                        
                      
                      )
                    
                  
                  ·
                  
                    cos
                    ⁡
                    
                      (
                      
                        
                          
                            ω
                            m
                          
                          ⁢
                          
                            
                          
                          ⁢
                          t
                        
                        -
                        
                          
                            
                              φ
                              0
                              -
                            
                            -
                            
                              φ
                              0
                              +
                            
                          
                          2
                        
                      
                      )
                    
                  
                
                ⁢
                
                  

                
                ⁢
                
                  
                    cos
                    ⁡
                    
                      (
                      
                        
                          
                            ω
                            m
                          
                          ⁢
                          t
                        
                        +
                        
                          
                            ω
                            0
                            -
                          
                          ⁢
                          
                            T
                            1
                          
                        
                        +
                        
                          n
                          ⁢
                          
                            
                          
                          ⁢
                          π
                          ⁢
                          
                            
                          
                          ⁢
                          Δ
                        
                        +
                        
                          φ
                          0
                        
                        -
                        
                          φ
                          0
                          -
                        
                      
                      )
                    
                  
                  -
                  
                    cos
                    ⁡
                    
                      (
                      
                        
                          
                            -
                            
                              ω
                              m
                            
                          
                          ⁢
                          t
                        
                        +
                        
                          
                            ω
                            0
                            +
                          
                          ⁢
                          
                            T
                            1
                          
                        
                        +
                        
                          n
                          ⁢
                          
                            
                          
                          ⁢
                          π
                          ⁢
                          
                            
                          
                          ⁢
                          Δ
                        
                        +
                        
                          φ
                          0
                        
                        -
                        
                          φ
                          0
                          +
                        
                      
                      )
                    
                  
                
              
              =
              
                
                  
                    
                      -
                      2
                    
                    ⁢
                    
                      
                        sin
                        ⁡
                        
                          (
                          
                            
                              
                                ω
                                0
                              
                              ⁢
                              
                                T
                                1
                              
                            
                            +
                            
                              n
                              ⁢
                              
                                
                              
                              ⁢
                              π
                              ⁢
                              
                                
                              
                              ⁢
                              Δ
                            
                            +
                            
                              φ
                              0
                            
                            -
                            
                              
                                
                                  φ
                                  0
                                  -
                                
                                +
                                
                                  φ
                                  0
                                  +
                                
                              
                              2
                            
                          
                          )
                        
                      
                      ·
                      
                        sin
                        ⁡
                        
                          (
                          
                            
                              
                                ω
                                m
                              
                              ⁢
                              t
                            
                            -
                            
                              
                                ω
                                m
                              
                              ⁢
                              
                                T
                                1
                              
                            
                            -
                            
                              
                                
                                  φ
                                  0
                                  -
                                
                                -
                                
                                  φ
                                  0
                                  +
                                
                              
                              2
                            
                          
                          )
                        
                      
                    
                    ⁢
                    
                      

                    
                    ⁢
                    
                      cos
                      ⁡
                      
                        (
                        
                          
                            
                              ω
                              m
                            
                            ⁢
                            t
                          
                          +
                          
                            n
                            ⁢
                            
                              
                            
                            ⁢
                            π
                            ⁢
                            
                              
                            
                            ⁢
                            Δ
                          
                          +
                          
                            φ
                            0
                          
                          -
                          
                            φ
                            0
                            -
                          
                        
                        )
                      
                    
                  
                  -
                  
                    cos
                    ⁡
                    
                      (
                      
                        
                          
                            -
                            
                              ω
                              m
                            
                          
                          ⁢
                          t
                        
                        +
                        
                          n
                          ⁢
                          
                            
                          
                          ⁢
                          π
                          ⁢
                          
                            
                          
                          ⁢
                          Δ
                        
                        +
                        
                          φ
                          0
                        
                        -
                        
                          φ
                          0
                          +
                        
                      
                      )
                    
                  
                
                =
                
                  2
                  ⁢
                  
                    
                      sin
                      ⁡
                      
                        (
                        
                          
                            n
                            ⁢
                            
                              
                            
                            ⁢
                            π
                            ⁢
                            
                              
                            
                            ⁢
                            Δ
                          
                          +
                          
                            φ
                            0
                          
                          -
                          
                            
                              
                                φ
                                0
                                -
                              
                              +
                              
                                φ
                                0
                                +
                              
                            
                            2
                          
                        
                        )
                      
                    
                    ·
                    
                      
                        sin
                        ⁡
                        
                          (
                          
                            
                              
                                ω
                                m
                              
                              ⁢
                              t
                            
                            -
                            
                              
                                
                                  φ
                                  0
                                  -
                                
                                -
                                
                                  φ
                                  0
                                  +
                                
                              
                              2
                            
                          
                          )
                        
                      
                      .
                    
                  
                
              
            
          
        
      
    
  




Since ω0T1+nπΔ=(1−Δ)·2nπ+nπΔ=2nπ−Δnπ,then





  
    
      [
      
        
          
            
              
                cos
                ⁢
                
                  (
                  
                    
                      
                        ω
                        m
                      
                      ⁢
                      t
                    
                    +
                    
                      
                        ω
                        0
                        -
                      
                      ⁢
                      
                        T
                        1
                      
                    
                    ⁢
                    
                      
                    
                    +
                    
                      n
                      ⁢
                      
                        
                      
                      ⁢
                      π
                      ⁢
                      
                        
                      
                      ⁢
                      Δ
                    
                    +
                    
                      φ
                      0
                    
                    -
                    
                      φ
                      0
                      -
                    
                  
                  )
                
              
              +
            
          
        
        
          
            
              cos
              ⁡
              
                (
                
                  
                    
                      -
                      
                        ω
                        m
                      
                    
                    ⁢
                    t
                  
                  +
                  
                    
                      ω
                      0
                      +
                    
                    ⁢
                    
                      T
                      1
                    
                  
                  +
                  
                    n
                    ⁢
                    
                      
                    
                    ⁢
                    π
                    ⁢
                    
                      
                    
                    ⁢
                    Δ
                  
                  +
                  
                    φ
                    0
                  
                  -
                  
                    φ
                    0
                    +
                  
                
                )
              
            
          
        
      
      ]
    
    -
    
      [
      
        
          
            
              
                cos
                ⁢
                
                  (
                  
                    
                      
                        ω
                        m
                      
                      ⁢
                      t
                    
                    +
                    
                      n
                      ⁢
                      
                        
                      
                      ⁢
                      π
                      ⁢
                      
                        
                      
                      ⁢
                      Δ
                    
                    +
                    
                      φ
                      0
                    
                    -
                    
                      φ
                      0
                      -
                    
                  
                  )
                
              
              +
            
          
        
        
          
            
              cos
              ⁡
              
                (
                
                  
                    
                      -
                      
                        ω
                        m
                      
                    
                    ⁢
                    t
                  
                  +
                  
                    n
                    ⁢
                    
                      
                    
                    ⁢
                    π
                    ⁢
                    
                      
                    
                    ⁢
                    Δ
                  
                  +
                  
                    φ
                    0
                  
                  -
                  
                    φ
                    0
                    +
                  
                
                )
              
            
          
        
      
      ]
    
  
  =
  
    
      
        2
        ⁢
        
          
            cos
            ⁡
            
              (
              
                
                  
                    ω
                    0
                  
                  ⁢
                  
                    T
                    1
                  
                
                +
                
                  n
                  ⁢
                  
                    
                  
                  ⁢
                  π
                  ⁢
                  
                    
                  
                  ⁢
                  Δ
                
                +
                
                  φ
                  0
                
                -
                
                  
                    
                      φ
                      0
                      -
                    
                    +
                    
                      φ
                      0
                      +
                    
                  
                  2
                
              
              )
            
          
          ·
          
            cos
            ⁡
            
              (
              
                
                  
                    ω
                    m
                  
                  ⁢
                  t
                
                -
                
                  
                    ω
                    m
                  
                  ⁢
                  
                    T
                    1
                  
                
                -
                
                  
                    
                      φ
                      0
                      -
                    
                    -
                    
                      φ
                      0
                      +
                    
                  
                  2
                
              
              )
            
          
        
      
      -
      
        2
        ⁢
        
          
        
        ⁢
        
          
            cos
            ⁡
            
              (
              
                
                  n
                  ⁢
                  
                    
                  
                  ⁢
                  π
                  ⁢
                  
                    
                  
                  ⁢
                  Δ
                
                +
                
                  φ
                  0
                
                -
                
                  
                    
                      φ
                      0
                      -
                    
                    +
                    
                      φ
                      0
                      +
                    
                  
                  2
                
              
              )
            
          
          ·
          
            cos
            ⁡
            
              (
              
                
                  
                    ω
                    m
                  
                  ⁢
                  t
                
                -
                
                  
                    
                      φ
                      0
                      -
                    
                    -
                    
                      φ
                      0
                      +
                    
                  
                  2
                
              
              )
            
          
        
      
    
    =
    
      
        
          2
          ⁢
          
            
              cos
              ⁡
              
                (
                
                  
                    n
                    ⁢
                    
                      
                    
                    ⁢
                    π
                    ⁢
                    
                      
                    
                    ⁢
                    Δ
                  
                  -
                  
                    Δ
                    ⁢
                    
                      
                    
                    ⁢
                    n
                    ⁢
                    
                      
                    
                    ⁢
                    π
                  
                  +
                  
                    φ
                    0
                  
                  -
                  
                    
                      
                        φ
                        0
                        -
                      
                      +
                      
                        φ
                        0
                        +
                      
                    
                    2
                  
                
                )
              
            
            ·
            
              cos
              ⁡
              
                (
                
                  
                    
                      ω
                      m
                    
                    ⁢
                    t
                  
                  -
                  
                    
                      ω
                      m
                    
                    ⁢
                    
                      T
                      1
                    
                  
                  -
                  
                    
                      
                        φ
                        0
                        -
                      
                      -
                      
                        φ
                        0
                        +
                      
                    
                    2
                  
                
                )
              
            
          
        
        -
        
          2
          ⁢
          
            
              cos
              ⁡
              
                (
                
                  
                    n
                    ⁢
                    
                      
                    
                    ⁢
                    π
                    ⁢
                    
                      
                    
                    ⁢
                    Δ
                  
                  +
                  
                    φ
                    0
                  
                  -
                  
                    
                      
                        φ
                        0
                        -
                      
                      +
                      
                        φ
                        0
                        +
                      
                    
                    2
                  
                
                )
              
            
            ·
            
              cos
              ⁡
              
                (
                
                  
                    
                      ω
                      m
                    
                    ⁢
                    t
                  
                  -
                  
                    
                      
                        φ
                        0
                        -
                      
                      -
                      
                        φ
                        0
                        +
                      
                    
                    2
                  
                
                )
              
            
          
        
      
      =
      
        
          
            2
            ⁢
            
              
                cos
                ⁡
                
                  (
                  
                    
                      
                        -
                        Δ
                      
                      ⁢
                      
                        
                      
                      ⁢
                      n
                      ⁢
                      
                        
                      
                      ⁢
                      π
                    
                    +
                    
                      φ
                      0
                    
                    -
                    
                      
                        
                          φ
                          0
                          -
                        
                        +
                        
                          φ
                          0
                          +
                        
                      
                      2
                    
                  
                  )
                
              
              ·
              
                cos
                ⁡
                
                  (
                  
                    
                      
                        ω
                        m
                      
                      ⁢
                      t
                    
                    -
                    
                      
                        ω
                        m
                      
                      ⁢
                      
                        T
                        1
                      
                    
                    -
                    
                      
                        
                          φ
                          0
                          -
                        
                        -
                        
                          φ
                          0
                          +
                        
                      
                      2
                    
                  
                  )
                
              
            
          
          -
          
            2
            ⁢
            
              
                cos
                ⁡
                
                  (
                  
                    
                      n
                      ⁢
                      
                        
                      
                      ⁢
                      π
                      ⁢
                      
                        
                      
                      ⁢
                      Δ
                    
                    +
                    
                      φ
                      0
                    
                    -
                    
                      
                        
                          φ
                          0
                          -
                        
                        +
                        
                          φ
                          0
                          +
                        
                      
                      2
                    
                  
                  )
                
              
              ·
              
                cos
                ⁡
                
                  (
                  
                    
                      
                        ω
                        m
                      
                      ⁢
                      t
                    
                    -
                    
                      
                        
                          φ
                          0
                          -
                        
                        -
                        
                          φ
                          0
                          +
                        
                      
                      2
                    
                  
                  )
                
              
            
          
        
        =
        
          2
          ⁢
          
            
              cos
              ⁡
              
                (
                
                  
                    
                      ω
                      m
                    
                    ⁢
                    t
                  
                  -
                  
                    
                      ω
                      m
                    
                    ⁢
                    
                      T
                      1
                    
                  
                  -
                  
                    
                      
                        φ
                        0
                        -
                      
                      -
                      
                        φ
                        0
                        +
                      
                    
                    2
                  
                
                )
              
            
            ·
            
               
              
                
                  
                    [
                    
                      
                        
                          
                            
                              
                                cos
                                ⁡
                                
                                  (
                                  
                                    Δ
                                    ⁢
                                    
                                      
                                    
                                    ⁢
                                    n
                                    ⁢
                                    
                                      
                                    
                                    ⁢
                                    π
                                  
                                  )
                                
                              
                              ·
                              
                                cos
                                ⁡
                                
                                  (
                                  
                                    
                                      φ
                                      0
                                    
                                    -
                                    
                                      
                                        
                                          φ
                                          0
                                          -
                                        
                                        +
                                        
                                          φ
                                          0
                                          +
                                        
                                      
                                      2
                                    
                                  
                                  )
                                
                              
                            
                            +
                          
                        
                      
                      
                        
                          
                            
                              sin
                              ⁡
                              
                                (
                                
                                  Δ
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  n
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  π
                                
                                )
                              
                            
                            ·
                            
                              sin
                              ⁡
                              
                                (
                                
                                  
                                    φ
                                    0
                                  
                                  -
                                  
                                    
                                      
                                        φ
                                        0
                                        -
                                      
                                      +
                                      
                                        φ
                                        0
                                        +
                                      
                                    
                                    2
                                  
                                
                                )
                              
                            
                          
                        
                      
                    
                    ]
                  
                  -
                  
                    2
                    ⁢
                    
                      
                        cos
                        ⁡
                        
                          (
                          
                            
                              
                                
                                  
                                    
                                      ω
                                      m
                                    
                                    ⁢
                                    t
                                  
                                  -
                                
                              
                            
                            
                              
                                
                                  
                                    
                                      φ
                                      0
                                      -
                                    
                                    -
                                    
                                      φ
                                      0
                                      +
                                    
                                  
                                  2
                                
                              
                            
                          
                          )
                        
                      
                      ·
                      
                        [
                        
                          
                            
                              
                                
                                  
                                    cos
                                    ⁡
                                    
                                      (
                                      
                                        Δ
                                        ⁢
                                        
                                          
                                        
                                        ⁢
                                        n
                                        ⁢
                                        
                                          
                                        
                                        ⁢
                                        π
                                      
                                      )
                                    
                                  
                                  ·
                                  
                                    cos
                                    ⁡
                                    
                                      (
                                      
                                        
                                          φ
                                          0
                                        
                                        -
                                        
                                          
                                            
                                              φ
                                              0
                                              -
                                            
                                            +
                                            
                                              φ
                                              0
                                              +
                                            
                                          
                                          2
                                        
                                      
                                      )
                                    
                                  
                                
                                -
                              
                            
                          
                          
                            
                              
                                
                                  sin
                                  ⁡
                                  
                                    (
                                    
                                      Δ
                                      ⁢
                                      
                                        
                                      
                                      ⁢
                                      n
                                      ⁢
                                      
                                        
                                      
                                      ⁢
                                      π
                                    
                                    )
                                  
                                
                                ·
                                
                                  sin
                                  ⁡
                                  
                                    (
                                    
                                      
                                        φ
                                        0
                                      
                                      -
                                      
                                        
                                          
                                            φ
                                            0
                                            -
                                          
                                          +
                                          
                                            φ
                                            0
                                            +
                                          
                                        
                                        2
                                      
                                    
                                    )
                                  
                                
                              
                            
                          
                        
                        ]
                      
                    
                  
                
                =
                
                  
                    2
                    ⁢
                    
                      
                        cos
                        ⁡
                        
                          (
                          
                            Δ
                            ⁢
                            
                              
                            
                            ⁢
                            n
                            ⁢
                            
                              
                            
                            ⁢
                            π
                          
                          )
                        
                      
                      ·
                      
                        cos
                        ⁡
                        
                          (
                          
                            
                              
                                
                                  
                                    φ
                                    0
                                  
                                  -
                                
                              
                            
                            
                              
                                
                                  
                                    
                                      φ
                                      0
                                      -
                                    
                                    +
                                    
                                      φ
                                      0
                                      +
                                    
                                  
                                  2
                                
                              
                            
                          
                          )
                        
                      
                      ·
                      
                        [
                        
                          
                            
                              
                                
                                  cos
                                  ⁢
                                  
                                    (
                                    
                                      
                                        
                                          ω
                                          m
                                        
                                        ⁢
                                        t
                                      
                                      -
                                      
                                        
                                          ω
                                          m
                                        
                                        ⁢
                                        
                                          T
                                          1
                                        
                                      
                                      -
                                      
                                        
                                          
                                            φ
                                            0
                                            -
                                          
                                          -
                                          
                                            φ
                                            0
                                            +
                                          
                                        
                                        2
                                      
                                    
                                    )
                                  
                                
                                -
                              
                            
                          
                          
                            
                              
                                cos
                                ⁡
                                
                                  (
                                  
                                    
                                      
                                        ω
                                        m
                                      
                                      ⁢
                                      t
                                    
                                    -
                                    
                                      
                                        
                                          φ
                                          0
                                          -
                                        
                                        -
                                        
                                          φ
                                          0
                                          +
                                        
                                      
                                      2
                                    
                                  
                                  )
                                
                              
                            
                          
                        
                        ]
                      
                    
                  
                  +
                  
                    2
                    ⁢
                    
                      
                        sin
                        ⁡
                        
                          (
                          
                            Δ
                            ⁢
                            
                              
                            
                            ⁢
                            n
                            ⁢
                            
                              
                            
                            ⁢
                            π
                          
                          )
                        
                      
                      ·
                      
                        sin
                        ⁡
                        
                          (
                          
                            
                              φ
                              0
                            
                            -
                            
                              
                                
                                  φ
                                  0
                                  -
                                
                                +
                                
                                  φ
                                  0
                                  +
                                
                              
                              2
                            
                          
                          )
                        
                      
                      ·
                      
                        [
                        
                          
                            
                              
                                
                                  cos
                                  ⁢
                                  
                                    (
                                    
                                      
                                        
                                          ω
                                          m
                                        
                                        ⁢
                                        t
                                      
                                      -
                                      
                                        
                                          ω
                                          m
                                        
                                        ⁢
                                        
                                          T
                                          1
                                        
                                      
                                      -
                                      
                                        
                                          
                                            φ
                                            0
                                            -
                                          
                                          -
                                          
                                            φ
                                            0
                                            +
                                          
                                        
                                        2
                                      
                                    
                                    )
                                  
                                
                                +
                              
                            
                          
                          
                            
                              
                                cos
                                ⁡
                                
                                  (
                                  
                                    
                                      
                                        ω
                                        m
                                      
                                      ⁢
                                      t
                                    
                                    -
                                    
                                      
                                        
                                          φ
                                          0
                                          -
                                        
                                        -
                                        
                                          φ
                                          0
                                          +
                                        
                                      
                                      2
                                    
                                  
                                  )
                                
                              
                            
                          
                        
                        ]
                      
                    
                  
                
              
            
          
        
      
    
  




Since ωmT1=(1−Δ)·π=π−Δπ, then




  
    
      
        [
        
          
            
              
                
                  cos
                  ⁢
                  
                    (
                    
                      
                        
                          ω
                          m
                        
                        ⁢
                        t
                      
                      +
                      
                        
                          ω
                          0
                          -
                        
                        ⁢
                        
                          T
                          1
                        
                      
                      +
                      
                        n
                        ⁢
                        
                          
                        
                        ⁢
                        π
                        ⁢
                        
                          
                        
                        ⁢
                        Δ
                      
                      +
                      
                        φ
                        0
                      
                      -
                      
                        φ
                        0
                        -
                      
                    
                    )
                  
                
                +
              
            
          
          
            
              
                cos
                ⁡
                
                  (
                  
                    
                      
                        -
                        
                          ω
                          m
                        
                      
                      ⁢
                      t
                    
                    +
                    
                      
                        ω
                        0
                        +
                      
                      ⁢
                      
                        T
                        1
                      
                    
                    +
                    
                      n
                      ⁢
                      
                        
                      
                      ⁢
                      π
                      ⁢
                      
                        
                      
                      ⁢
                      Δ
                    
                    +
                    
                      φ
                      0
                    
                    -
                    
                      φ
                      0
                      +
                    
                  
                  )
                
              
            
          
        
        ]
      
      -
      
        [
        
          
            
              
                
                  cos
                  ⁢
                  
                    (
                    
                      
                        
                          ω
                          m
                        
                        ⁢
                        t
                      
                      +
                      
                        n
                        ⁢
                        
                          
                        
                        ⁢
                        π
                        ⁢
                        
                          
                        
                        ⁢
                        Δ
                      
                      +
                      
                        φ
                        0
                      
                      -
                      
                        φ
                        0
                        -
                      
                    
                    )
                  
                
                +
              
            
          
          
            
              
                cos
                ⁡
                
                  (
                  
                    
                      
                        -
                        
                          ω
                          m
                        
                      
                      ⁢
                      t
                    
                    +
                    
                      n
                      ⁢
                      
                        
                      
                      ⁢
                      π
                      ⁢
                      
                        
                      
                      ⁢
                      Δ
                    
                    +
                    
                      φ
                      0
                    
                    -
                    
                      φ
                      0
                      +
                    
                  
                  )
                
              
            
          
        
        ]
      
    
    =
    
      
        
          2
          ⁢
          
            
              cos
              ⁡
              
                (
                
                  Δ
                  ⁢
                  
                    
                  
                  ⁢
                  n
                  ⁢
                  
                    
                  
                  ⁢
                  π
                
                )
              
            
            ·
            
              cos
              ⁡
              
                (
                
                  
                    φ
                    0
                  
                  -
                  
                    
                      
                        φ
                        0
                        -
                      
                      +
                      
                        φ
                        0
                        +
                      
                    
                    2
                  
                
                )
              
            
            ·
            
              [
              
                
                  
                    
                      
                        cos
                        ⁢
                        
                          (
                          
                            
                              
                                ω
                                m
                              
                              ⁢
                              t
                            
                            -
                            
                              
                                
                                  φ
                                  0
                                  -
                                
                                -
                                
                                  φ
                                  0
                                  +
                                
                              
                              2
                            
                            +
                            
                              Δ
                              ⁢
                              
                                
                              
                              ⁢
                              π
                            
                            -
                            π
                          
                          )
                        
                      
                      -
                    
                  
                
                
                  
                    
                      cos
                      ⁡
                      
                        (
                        
                          
                            
                              ω
                              m
                            
                            ⁢
                            t
                          
                          -
                          
                            
                              
                                φ
                                0
                                -
                              
                              -
                              
                                φ
                                0
                                +
                              
                            
                            2
                          
                        
                        )
                      
                    
                  
                
              
              ]
            
          
        
        +
        
          2
          ⁢
          
            
              sin
              ⁡
              
                (
                
                  Δ
                  ⁢
                  
                    
                  
                  ⁢
                  n
                  ⁢
                  
                    
                  
                  ⁢
                  π
                
                )
              
            
            ·
            
              sin
              ⁡
              
                (
                
                  
                    φ
                    0
                  
                  -
                  
                    
                      
                        φ
                        0
                        -
                      
                      +
                      
                        φ
                        0
                        +
                      
                    
                    2
                  
                
                )
              
            
            ·
            
              [
              
                
                  
                    
                      
                        cos
                        ⁢
                        
                          (
                          
                            
                              
                                ω
                                m
                              
                              ⁢
                              t
                            
                            -
                            
                              
                                
                                  φ
                                  0
                                  -
                                
                                -
                                
                                  φ
                                  0
                                  +
                                
                              
                              2
                            
                            +
                            
                              Δ
                              ⁢
                              
                                
                              
                              ⁢
                              π
                            
                            -
                            π
                          
                          )
                        
                      
                      +
                    
                  
                
                
                  
                    
                      cos
                      ⁡
                      
                        (
                        
                          
                            
                              ω
                              m
                            
                            ⁢
                            t
                          
                          -
                          
                            
                              
                                φ
                                0
                                -
                              
                              -
                              
                                φ
                                0
                                +
                              
                            
                            2
                          
                        
                        )
                      
                    
                  
                
              
              ]
            
          
        
      
      =
      
        
          
            2
            ⁢
            
              
                cos
                ⁡
                
                  (
                  
                    Δ
                    ⁢
                    
                      
                    
                    ⁢
                    n
                    ⁢
                    
                      
                    
                    ⁢
                    π
                  
                  )
                
              
              ·
              
                cos
                ⁡
                
                  (
                  
                    
                      φ
                      0
                    
                    -
                    
                      
                        
                          φ
                          0
                          -
                        
                        +
                        
                          φ
                          0
                          +
                        
                      
                      2
                    
                  
                  )
                
              
              ·
              
                [
                
                  
                    
                      
                        
                          
                            -
                            cos
                          
                          ⁢
                          
                            (
                            
                              
                                
                                  ω
                                  m
                                
                                ⁢
                                t
                              
                              -
                              
                                
                                  
                                    φ
                                    0
                                    -
                                  
                                  -
                                  
                                    φ
                                    0
                                    +
                                  
                                
                                2
                              
                              +
                              
                                Δ
                                ⁢
                                
                                  
                                
                                ⁢
                                π
                              
                            
                            )
                          
                        
                        -
                      
                    
                  
                  
                    
                      
                        cos
                        ⁡
                        
                          (
                          
                            
                              
                                ω
                                m
                              
                              ⁢
                              t
                            
                            -
                            
                              
                                
                                  φ
                                  0
                                  -
                                
                                -
                                
                                  φ
                                  0
                                  +
                                
                              
                              2
                            
                          
                          )
                        
                      
                    
                  
                
                ]
              
            
          
          +
          
            2
            ⁢
            
              
                sin
                ⁡
                
                  (
                  
                    Δ
                    ⁢
                    
                      
                    
                    ⁢
                    n
                    ⁢
                    
                      
                    
                    ⁢
                    π
                  
                  )
                
              
              ·
              
                sin
                ⁡
                
                  (
                  
                    
                      φ
                      0
                    
                    -
                    
                      
                        
                          φ
                          0
                          -
                        
                        +
                        
                          φ
                          0
                          +
                        
                      
                      2
                    
                  
                  )
                
              
              ·
              
                [
                
                  
                    
                      
                        
                          
                            -
                            cos
                          
                          ⁢
                          
                            (
                            
                              
                                
                                  ω
                                  m
                                
                                ⁢
                                t
                              
                              -
                              
                                
                                  
                                    φ
                                    0
                                    -
                                  
                                  -
                                  
                                    φ
                                    
                                      0
                                      ⁢
                                      
                                        
                                      
                                    
                                    +
                                  
                                
                                2
                              
                              +
                              
                                Δ
                                ⁢
                                
                                  
                                
                                ⁢
                                π
                              
                            
                            )
                          
                        
                        +
                      
                    
                  
                  
                    
                      
                        cos
                        ⁡
                        
                          (
                          
                            
                              
                                ω
                                m
                              
                              ⁢
                              t
                            
                            -
                            
                              
                                
                                  φ
                                  0
                                  -
                                
                                -
                                
                                  φ
                                  0
                                  +
                                
                              
                              2
                            
                          
                          )
                        
                      
                    
                  
                
                ]
              
            
          
        
        =
        
          
            
              
                -
                4
              
              ⁢
              
                
                  cos
                  ⁡
                  
                    (
                    
                      Δ
                      ⁢
                      
                        
                      
                      ⁢
                      n
                      ⁢
                      
                        
                      
                      ⁢
                      π
                    
                    )
                  
                
                ·
                
                  cos
                  ⁡
                  
                    (
                    
                      
                        φ
                        0
                      
                      -
                      
                        
                          
                            φ
                            0
                            -
                          
                          +
                          
                            φ
                            0
                            +
                          
                        
                        2
                      
                    
                    )
                  
                
                ·
                
                  cos
                  ⁡
                  
                    (
                    
                      
                        Δ
                        ⁢
                        
                          
                        
                        ⁢
                        π
                      
                      2
                    
                    )
                  
                
                ·
                
                  cos
                  ⁡
                  
                    (
                    
                      
                        
                          ω
                          m
                        
                        ⁢
                        t
                      
                      -
                      
                        
                          
                            φ
                            0
                            -
                          
                          -
                          
                            φ
                            0
                            +
                          
                        
                        2
                      
                      +
                      
                        
                          Δ
                          ⁢
                          
                            
                          
                          ⁢
                          π
                        
                        2
                      
                    
                    )
                  
                
              
            
            +
            
              4
              ⁢
              
                
                  sin
                  ⁡
                  
                    (
                    
                      Δ
                      ⁢
                      
                        
                      
                      ⁢
                      n
                      ⁢
                      
                        
                      
                      ⁢
                      π
                    
                    )
                  
                
                ·
                
                  sin
                  ⁡
                  
                    (
                    
                      
                        φ
                        0
                      
                      -
                      
                        
                          
                            φ
                            0
                            -
                          
                          +
                          
                            φ
                            0
                            +
                          
                        
                        2
                      
                    
                    )
                  
                
                ·
                
                  sin
                  ⁡
                  
                    (
                    
                      
                        Δ
                        ⁢
                        
                          
                        
                        ⁢
                        π
                      
                      2
                    
                    )
                  
                
                ·
                
                  sin
                  ⁡
                  
                    (
                    
                      
                        
                          ω
                          m
                        
                        ⁢
                        t
                      
                      -
                      
                        
                          
                            φ
                            0
                            -
                          
                          -
                          
                            φ
                            0
                            +
                          
                        
                        2
                      
                      +
                      
                        
                          Δ
                          ⁢
                          
                            
                          
                          ⁢
                          π
                        
                        2
                      
                    
                    )
                  
                
              
            
          
          =
          
            
              -
              4
            
            ⁢
            
              
                cos
                ⁡
                
                  (
                  
                    Δ
                    ⁢
                    
                      
                    
                    ⁢
                    n
                    ⁢
                    
                      
                    
                    ⁢
                    π
                  
                  )
                
              
              ·
              
                cos
                ⁡
                
                  (
                  
                    
                      φ
                      0
                    
                    -
                    
                      
                        
                          φ
                          0
                          -
                        
                        +
                        
                          φ
                          0
                          +
                        
                      
                      2
                    
                  
                  )
                
              
              ·
              
                cos
                ⁡
                
                  (
                  
                    
                      Δ
                      ⁢
                      
                        
                      
                      ⁢
                      π
                    
                    2
                  
                  )
                
              
              ·
              
                cos
                ⁡
                
                  (
                  
                    
                      
                        ω
                        m
                      
                      ⁢
                      t
                    
                    +
                    
                      
                        Δ
                        ⁢
                        
                          
                        
                        ⁢
                        π
                      
                      2
                    
                    -
                    
                      
                        
                          φ
                          0
                          -
                        
                        -
                        
                          φ
                          0
                          +
                        
                      
                      2
                    
                    +
                    
                      δ
                      ⁢
                      
                        
                      
                      ⁢
                      φ
                    
                  
                  )
                
              
            
          
        
      
    
  




  
    
      
        δ
        ·
        
          [
          
            
              
                
                  
                    cos
                    ⁢
                    
                      (
                      
                        
                          
                            ω
                            m
                          
                          ⁢
                          t
                        
                        +
                        
                          
                            ω
                            0
                            -
                          
                          ⁢
                          
                            T
                            1
                          
                        
                        +
                        
                          n
                          ⁢
                          
                            
                          
                          ⁢
                          π
                          ⁢
                          
                            
                          
                          ⁢
                          Δ
                        
                        +
                        
                          φ
                          0
                        
                        -
                        
                          φ
                          0
                          -
                        
                      
                      )
                    
                  
                  -
                
              
            
            
              
                
                  cos
                  ⁡
                  
                    (
                    
                      
                        
                          -
                          
                            ω
                            m
                          
                        
                        ⁢
                        t
                      
                      +
                      
                        
                          ω
                          0
                          +
                        
                        ⁢
                        
                          T
                          1
                        
                      
                      +
                      
                        n
                        ⁢
                        
                          
                        
                        ⁢
                        π
                        ⁢
                        
                          
                        
                        ⁢
                        Δ
                      
                      +
                      
                        φ
                        0
                      
                      -
                      
                        φ
                        0
                        +
                      
                    
                    )
                  
                
              
            
          
          ]
        
      
      -
      
        δ
        ·
        
          [
          
            
              cos
              ⁡
              
                (
                
                  
                    
                      ω
                      m
                    
                    ⁢
                    t
                  
                  +
                  
                    n
                    ⁢
                    
                      
                    
                    ⁢
                    π
                    ⁢
                    
                      
                    
                    ⁢
                    Δ
                  
                  +
                  
                    φ
                    0
                  
                  -
                  
                    φ
                    0
                    -
                  
                
                )
              
            
            -
            
              cos
              ⁡
              
                (
                
                  
                    
                      -
                      
                        ω
                        m
                      
                    
                    ⁢
                    t
                  
                  +
                  
                    n
                    ⁢
                    
                      
                    
                    ⁢
                    π
                    ⁢
                    
                      
                    
                    ⁢
                    Δ
                  
                  +
                  
                    φ
                    0
                  
                  -
                  
                    φ
                    0
                    +
                  
                
                )
              
            
          
          ]
        
      
    
    =
    
      
        
          
            -
            2
          
          ⁢
          
            δ
            ·
            
              sin
              ⁡
              
                (
                
                  
                    
                      ω
                      0
                    
                    ⁢
                    
                      T
                      1
                    
                  
                  +
                  
                    n
                    ⁢
                    
                      
                    
                    ⁢
                    π
                    ⁢
                    
                      
                    
                    ⁢
                    Δ
                  
                  +
                  
                    φ
                    0
                  
                  -
                  
                    
                      
                        φ
                        0
                        -
                      
                      +
                      
                        φ
                        0
                        +
                      
                    
                    2
                  
                
                )
              
            
            ·
            
              sin
              ⁡
              
                (
                
                  
                    
                      ω
                      m
                    
                    ⁢
                    t
                  
                  -
                  
                    
                      ω
                      m
                    
                    ⁢
                    
                      T
                      1
                    
                  
                  -
                  
                    
                      
                        φ
                        0
                        -
                      
                      -
                      
                        φ
                        0
                        +
                      
                    
                    2
                  
                
                )
              
            
          
        
        +
        
          2
          ⁢
          
            δ
            ·
            
              sin
              ⁡
              
                (
                
                  
                    n
                    ⁢
                    
                      
                    
                    ⁢
                    π
                    ⁢
                    
                      
                    
                    ⁢
                    Δ
                  
                  +
                  
                    φ
                    0
                  
                  -
                  
                    
                      
                        φ
                        0
                        -
                      
                      +
                      
                        φ
                        0
                        +
                      
                    
                    2
                  
                
                )
              
            
            ·
            
              sin
              ⁡
              
                (
                
                  
                    
                      ω
                      m
                    
                    ⁢
                    t
                  
                  -
                  
                    
                      
                        φ
                        0
                        -
                      
                      -
                      
                        φ
                        0
                        +
                      
                    
                    2
                  
                
                )
              
            
          
        
      
      =
      
        
          
            
              -
              2
            
            ⁢
            
              δ
              ·
              
                sin
                ⁡
                
                  (
                  
                    
                      2
                      ⁢
                      n
                      ⁢
                      
                        
                      
                      ⁢
                      π
                    
                    -
                    
                      n
                      ⁢
                      
                        
                      
                      ⁢
                      π
                      ⁢
                      
                        
                      
                      ⁢
                      Δ
                    
                    +
                    
                      φ
                      0
                    
                    -
                    
                      
                        
                          φ
                          0
                          -
                        
                        +
                        
                          φ
                          0
                          +
                        
                      
                      2
                    
                  
                  )
                
              
              ·
              
                sin
                ⁡
                
                  (
                  
                    
                      
                        ω
                        m
                      
                      ⁢
                      
                        
                      
                      ⁢
                      t
                    
                    -
                    
                      
                        ω
                        m
                      
                      ⁢
                      
                        T
                        1
                      
                    
                    -
                    
                      
                        
                          φ
                          0
                          -
                        
                        -
                        
                          ω
                          0
                          +
                        
                      
                      2
                    
                  
                  )
                
              
            
          
          +
          
            2
            ⁢
            
              δ
              ·
              
                sin
                ⁡
                
                  (
                  
                    
                      n
                      ⁢
                      
                        
                      
                      ⁢
                      π
                      ⁢
                      
                        
                      
                      ⁢
                      Δ
                    
                    +
                    
                      φ
                      0
                    
                    -
                    
                      
                        
                          φ
                          0
                          +
                        
                        +
                        
                          φ
                          0
                          +
                        
                      
                      2
                    
                  
                  )
                
              
              ·
              
                sin
                ⁡
                
                  (
                  
                    
                      
                        ω
                        m
                      
                      ⁢
                      t
                    
                    -
                    
                      
                        
                          φ
                          0
                          -
                        
                        -
                        
                          φ
                          0
                          +
                        
                      
                      2
                    
                  
                  )
                
              
            
          
        
        =
        
          
            
              
                -
                2
              
              ⁢
              
                δ
                ·
                
                  sin
                  ⁡
                  
                    (
                    
                      
                        
                          -
                          n
                        
                        ⁢
                        
                          
                        
                        ⁢
                        π
                        ⁢
                        
                          
                        
                        ⁢
                        Δ
                      
                      +
                      
                        φ
                        0
                      
                      -
                      
                        
                          
                            φ
                            0
                            -
                          
                          +
                          
                            φ
                            0
                            +
                          
                        
                        2
                      
                    
                    )
                  
                
                ·
                
                  sin
                  ⁡
                  
                    (
                    
                      
                        
                          ω
                          m
                        
                        ⁢
                        t
                      
                      -
                      
                        
                          ω
                          m
                        
                        ⁢
                        
                          T
                          1
                        
                      
                      -
                      
                        
                          
                            φ
                            0
                            -
                          
                          +
                          
                            φ
                            0
                            +
                          
                        
                        2
                      
                    
                    )
                  
                
              
            
            +
            
              2
              ⁢
              
                δ
                ·
                
                  sin
                  ⁡
                  
                    (
                    
                      
                        n
                        ⁢
                        
                          
                        
                        ⁢
                        π
                        ⁢
                        
                          
                        
                        ⁢
                        Δ
                      
                      +
                      
                        φ
                        0
                      
                      -
                      
                        
                          
                            φ
                            0
                            -
                          
                          +
                          
                            φ
                            0
                            +
                          
                        
                        2
                      
                    
                    )
                  
                
                ·
                
                  sin
                  ⁡
                  
                    (
                    
                      
                        
                          ω
                          m
                        
                        ⁢
                        t
                      
                      -
                      
                        
                          
                            φ
                            0
                            -
                          
                          -
                          
                            φ
                            0
                            +
                          
                        
                        2
                      
                    
                    )
                  
                
              
            
          
          =
          
            
              
                
                  -
                  2
                
                ⁢
                
                  δ
                  ·
                  
                    sin
                    ⁡
                    
                      (
                      
                        
                          
                            
                              
                                
                                  ω
                                  m
                                
                                ⁢
                                t
                              
                              -
                              
                                
                                  ω
                                  m
                                
                                ⁢
                                
                                  T
                                  1
                                
                              
                              -
                            
                          
                        
                        
                          
                            
                              
                                
                                  φ
                                  0
                                
                                -
                                
                                  φ
                                  0
                                  +
                                
                              
                              2
                            
                          
                        
                      
                      )
                    
                  
                  ·
                  
                    [
                    
                      
                        
                          
                            
                              
                                -
                                
                                  sin
                                  ⁡
                                  
                                    (
                                    
                                      n
                                      ⁢
                                      
                                        
                                      
                                      ⁢
                                      π
                                      ⁢
                                      
                                        
                                      
                                      ⁢
                                      Δ
                                    
                                    )
                                  
                                
                              
                              ·
                              
                                cos
                                ⁡
                                
                                  (
                                  
                                    
                                      φ
                                      0
                                    
                                    -
                                    
                                      
                                        
                                          φ
                                          0
                                          -
                                        
                                        +
                                        
                                          φ
                                          0
                                          +
                                        
                                      
                                      2
                                    
                                  
                                  )
                                
                              
                            
                            +
                          
                        
                      
                      
                        
                          
                            
                              cos
                              ⁡
                              
                                (
                                
                                  n
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  π
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  Δ
                                
                                )
                              
                            
                            ·
                            
                              sin
                              ⁡
                              
                                (
                                
                                  
                                    φ
                                    0
                                  
                                  -
                                  
                                    
                                      
                                        φ
                                        0
                                        -
                                      
                                      +
                                      
                                        φ
                                        0
                                        +
                                      
                                    
                                    2
                                  
                                
                                )
                              
                            
                          
                        
                      
                    
                    ]
                  
                
              
              +
              
                2
                ⁢
                
                  δ
                  ·
                  
                    sin
                    ⁡
                    
                      (
                      
                        
                          
                            ω
                            m
                          
                          ⁢
                          t
                        
                        -
                        
                          
                            
                              φ
                              0
                              -
                            
                            -
                            
                              φ
                              0
                              +
                            
                          
                          2
                        
                      
                      )
                    
                  
                  ·
                  
                    [
                    
                      
                        
                          
                            
                              
                                sin
                                ⁡
                                
                                  (
                                  
                                    n
                                    ⁢
                                    
                                      
                                    
                                    ⁢
                                    π
                                    ⁢
                                    
                                      
                                    
                                    ⁢
                                    Δ
                                  
                                  )
                                
                              
                              ·
                              
                                cos
                                ⁡
                                
                                  (
                                  
                                    
                                      φ
                                      0
                                    
                                    -
                                    
                                      
                                        
                                          φ
                                          0
                                          -
                                        
                                        +
                                        
                                          φ
                                          0
                                          +
                                        
                                      
                                      2
                                    
                                  
                                  )
                                
                              
                            
                            +
                          
                        
                      
                      
                        
                          
                            
                              cos
                              ⁡
                              
                                (
                                
                                  n
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  π
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  Δ
                                
                                )
                              
                            
                            ·
                            
                              sin
                              ⁡
                              
                                (
                                
                                  
                                    φ
                                    0
                                  
                                  -
                                  
                                    
                                      
                                        φ
                                        0
                                        -
                                      
                                      +
                                      
                                        φ
                                        0
                                        +
                                      
                                    
                                    2
                                  
                                
                                )
                              
                            
                          
                        
                      
                    
                    ]
                  
                
              
            
            =
            
              2
              ⁢
              
                δ
                ·
                
                  sin
                  ⁡
                  
                    (
                    
                      n
                      ⁢
                      
                        
                      
                      ⁢
                      π
                      ⁢
                      
                        
                      
                      ⁢
                      Δ
                    
                    )
                  
                
                ·
                
                  cos
                  ⁡
                  
                    (
                    
                      
                        φ
                        0
                      
                      -
                      
                        
                          
                            φ
                            0
                            -
                          
                          +
                          
                            φ
                            0
                            +
                          
                        
                        2
                      
                    
                    )
                  
                
                ·
                
                   
                  
                    
                      [
                      
                        
                          
                            
                              
                                sin
                                ⁢
                                
                                  (
                                  
                                    
                                      
                                        ω
                                        m
                                      
                                      ⁢
                                      
                                        
                                      
                                      ⁢
                                      t
                                    
                                    -
                                    
                                      
                                        
                                          φ
                                          0
                                          -
                                        
                                        +
                                        
                                          φ
                                          0
                                          +
                                        
                                      
                                      2
                                    
                                  
                                  )
                                
                              
                              +
                            
                          
                        
                        
                          
                            
                              sin
                              ⁡
                              
                                (
                                
                                  
                                    
                                      ω
                                      m
                                    
                                    ⁢
                                    
                                      
                                    
                                    ⁢
                                    t
                                  
                                  -
                                  
                                    
                                      ω
                                      m
                                    
                                    ⁢
                                    
                                      
                                    
                                    ⁢
                                    
                                      T
                                      1
                                    
                                  
                                  -
                                  
                                    
                                      
                                        φ
                                        0
                                        -
                                      
                                      -
                                      
                                        φ
                                        0
                                        +
                                      
                                    
                                    2
                                  
                                
                                )
                              
                            
                          
                        
                      
                      ]
                    
                    +
                    
                      2
                      ⁢
                      
                        δ
                        ·
                        
                          cos
                          ⁡
                          
                            (
                            
                              n
                              ⁢
                              
                                
                              
                              ⁢
                              π
                              ⁢
                              
                                
                              
                              ⁢
                              Δ
                            
                            )
                          
                        
                        ·
                        
                          sin
                          ⁡
                          
                            (
                            
                              
                                φ
                                0
                              
                              -
                              
                                
                                  
                                    φ
                                    0
                                    -
                                  
                                  +
                                  
                                    φ
                                    0
                                    +
                                  
                                
                                2
                              
                            
                            )
                          
                        
                        ·
                        
                           
                          
                            
                              [
                              
                                
                                  
                                    
                                      
                                        sin
                                        ⁢
                                        
                                          (
                                          
                                            
                                              
                                                ω
                                                m
                                              
                                              ⁢
                                              t
                                            
                                            -
                                            
                                              
                                                
                                                  φ
                                                  0
                                                  -
                                                
                                                -
                                                
                                                  φ
                                                  0
                                                  +
                                                
                                              
                                              2
                                            
                                          
                                          )
                                        
                                      
                                      -
                                    
                                  
                                
                                
                                  
                                    
                                      sin
                                      ⁡
                                      
                                        (
                                        
                                          
                                            
                                              ω
                                              m
                                            
                                            ⁢
                                            t
                                          
                                          -
                                          
                                            
                                              ω
                                              m
                                            
                                            ⁢
                                            
                                              T
                                              1
                                            
                                          
                                          -
                                          
                                            
                                              
                                                φ
                                                0
                                                -
                                              
                                              -
                                              
                                                φ
                                                0
                                                +
                                              
                                            
                                            2
                                          
                                        
                                        )
                                      
                                    
                                  
                                
                              
                              ]
                            
                            =
                            
                              2
                              ⁢
                              
                                δ
                                ·
                                
                                  sin
                                  ⁡
                                  
                                    (
                                    
                                      n
                                      ⁢
                                      
                                        
                                      
                                      ⁢
                                      π
                                      ⁢
                                      
                                        
                                      
                                      ⁢
                                      Δ
                                    
                                    )
                                  
                                
                                ·
                                
                                  cos
                                  ⁡
                                  
                                    (
                                    
                                      
                                        φ
                                        0
                                      
                                      -
                                      
                                        
                                          
                                            φ
                                            0
                                            -
                                          
                                          +
                                          
                                            φ
                                            0
                                            +
                                          
                                        
                                        2
                                      
                                    
                                    )
                                  
                                
                                ·
                                
                                   
                                  
                                    
                                      [
                                      
                                        
                                          
                                            
                                              
                                                sin
                                                ⁢
                                                
                                                  (
                                                  
                                                    
                                                      
                                                        ω
                                                        m
                                                      
                                                      ⁢
                                                      t
                                                    
                                                    -
                                                    
                                                      
                                                        
                                                          φ
                                                          0
                                                          -
                                                        
                                                        -
                                                        
                                                          φ
                                                          0
                                                          +
                                                        
                                                      
                                                      2
                                                    
                                                  
                                                  )
                                                
                                              
                                              +
                                            
                                          
                                        
                                        
                                          
                                            
                                              sin
                                              ⁡
                                              
                                                (
                                                
                                                  
                                                    
                                                      ω
                                                      m
                                                    
                                                    ⁢
                                                    t
                                                  
                                                  -
                                                  π
                                                  +
                                                  
                                                    Δ
                                                    ⁢
                                                    
                                                      
                                                    
                                                    ⁢
                                                    π
                                                  
                                                  -
                                                  
                                                    
                                                      
                                                        φ
                                                        0
                                                        -
                                                      
                                                      -
                                                      
                                                        φ
                                                        0
                                                        +
                                                      
                                                    
                                                    2
                                                  
                                                
                                                )
                                              
                                            
                                          
                                        
                                      
                                      ]
                                    
                                    +
                                    
                                      2
                                      ⁢
                                      
                                        δ
                                        ·
                                        
                                          cos
                                          ⁡
                                          
                                            (
                                            
                                              n
                                              ⁢
                                              
                                                
                                              
                                              ⁢
                                              π
                                              ⁢
                                              
                                                
                                              
                                              ⁢
                                              Δ
                                            
                                            )
                                          
                                        
                                        ·
                                        
                                          sin
                                          ⁡
                                          
                                            (
                                            
                                              
                                                
                                                  
                                                    
                                                      φ
                                                      0
                                                    
                                                    -
                                                  
                                                
                                              
                                              
                                                
                                                  
                                                    
                                                      
                                                        φ
                                                        0
                                                        -
                                                      
                                                      +
                                                      
                                                        φ
                                                        0
                                                        +
                                                      
                                                    
                                                    2
                                                  
                                                
                                              
                                            
                                            )
                                          
                                        
                                        ·
                                        
                                           
                                          
                                            
                                              [
                                              
                                                
                                                  
                                                    
                                                      
                                                        sin
                                                        ⁢
                                                        
                                                          (
                                                          
                                                            
                                                              
                                                                ω
                                                                m
                                                              
                                                              ⁢
                                                              t
                                                            
                                                            -
                                                            
                                                              
                                                                
                                                                  φ
                                                                  0
                                                                  -
                                                                
                                                                -
                                                                
                                                                  φ
                                                                  0
                                                                  +
                                                                
                                                              
                                                              2
                                                            
                                                          
                                                          )
                                                        
                                                      
                                                      -
                                                    
                                                  
                                                
                                                
                                                  
                                                    
                                                      sin
                                                      ⁡
                                                      
                                                        (
                                                        
                                                          
                                                            
                                                              
                                                                
                                                                  
                                                                    ω
                                                                    m
                                                                  
                                                                  ⁢
                                                                  t
                                                                
                                                                -
                                                                π
                                                                +
                                                              
                                                            
                                                          
                                                          
                                                            
                                                              
                                                                
                                                                  Δ
                                                                  ⁢
                                                                  
                                                                    
                                                                  
                                                                  ⁢
                                                                  π
                                                                
                                                                -
                                                                
                                                                  
                                                                    
                                                                      φ
                                                                      0
                                                                      -
                                                                    
                                                                    -
                                                                    
                                                                      φ
                                                                      0
                                                                      +
                                                                    
                                                                  
                                                                  2
                                                                
                                                              
                                                            
                                                          
                                                        
                                                        )
                                                      
                                                    
                                                  
                                                
                                              
                                              ]
                                            
                                            =
                                            
                                              2
                                              ⁢
                                              
                                                δ
                                                ·
                                                
                                                  sin
                                                  ⁡
                                                  
                                                    (
                                                    
                                                      n
                                                      ⁢
                                                      
                                                        
                                                      
                                                      ⁢
                                                      π
                                                      ⁢
                                                      
                                                        
                                                      
                                                      ⁢
                                                      Δ
                                                    
                                                    )
                                                  
                                                
                                                ·
                                                
                                                  cos
                                                  ⁡
                                                  
                                                    (
                                                    
                                                      
                                                        φ
                                                        0
                                                      
                                                      -
                                                      
                                                        
                                                          
                                                            φ
                                                            0
                                                            -
                                                          
                                                          +
                                                          
                                                            φ
                                                            0
                                                            +
                                                          
                                                        
                                                        2
                                                      
                                                    
                                                    )
                                                  
                                                
                                                ·
                                                
                                                   
                                                  
                                                     
                                                    
                                                      
                                                        [
                                                        
                                                          
                                                            
                                                              
                                                                
                                                                  sin
                                                                  ⁢
                                                                  
                                                                    (
                                                                    
                                                                      
                                                                        
                                                                          ω
                                                                          m
                                                                        
                                                                        ⁢
                                                                        t
                                                                      
                                                                      -
                                                                      
                                                                        
                                                                          
                                                                            φ
                                                                            0
                                                                            -
                                                                          
                                                                          -
                                                                          
                                                                            φ
                                                                            0
                                                                            +
                                                                          
                                                                        
                                                                        2
                                                                      
                                                                    
                                                                    )
                                                                  
                                                                
                                                                -
                                                              
                                                            
                                                          
                                                          
                                                            
                                                              
                                                                sin
                                                                ⁡
                                                                
                                                                  (
                                                                  
                                                                    
                                                                      
                                                                        ω
                                                                        m
                                                                      
                                                                      ⁢
                                                                      t
                                                                    
                                                                    +
                                                                    
                                                                      Δ
                                                                      ⁢
                                                                      
                                                                        
                                                                      
                                                                      ⁢
                                                                      π
                                                                    
                                                                    -
                                                                    
                                                                      
                                                                        
                                                                          φ
                                                                          0
                                                                          -
                                                                        
                                                                        -
                                                                        
                                                                          φ
                                                                          0
                                                                          +
                                                                        
                                                                      
                                                                      2
                                                                    
                                                                  
                                                                  )
                                                                
                                                              
                                                            
                                                          
                                                        
                                                        ]
                                                      
                                                      +
                                                      
                                                        2
                                                        ⁢
                                                        
                                                          δ
                                                          ·
                                                          
                                                            cos
                                                            ⁡
                                                            
                                                              (
                                                              
                                                                n
                                                                ⁢
                                                                
                                                                  
                                                                
                                                                ⁢
                                                                π
                                                                ⁢
                                                                
                                                                  
                                                                
                                                                ⁢
                                                                Δ
                                                              
                                                              )
                                                            
                                                          
                                                          ·
                                                          
                                                            sin
                                                            ⁡
                                                            
                                                              (
                                                              
                                                                
                                                                  φ
                                                                  0
                                                                
                                                                -
                                                                
                                                                  
                                                                    
                                                                      φ
                                                                      0
                                                                      -
                                                                    
                                                                    +
                                                                    
                                                                      φ
                                                                      0
                                                                      +
                                                                    
                                                                  
                                                                  2
                                                                
                                                              
                                                              )
                                                            
                                                          
                                                          ·
                                                          
                                                             
                                                            
                                                              
                                                                [
                                                                
                                                                  
                                                                    
                                                                      
                                                                        
                                                                          sin
                                                                          ⁢
                                                                          
                                                                            (
                                                                            
                                                                              
                                                                                
                                                                                  ω
                                                                                  m
                                                                                
                                                                                ⁢
                                                                                t
                                                                              
                                                                              -
                                                                              
                                                                                
                                                                                  
                                                                                    φ
                                                                                    0
                                                                                    -
                                                                                  
                                                                                  -
                                                                                  
                                                                                    φ
                                                                                    0
                                                                                    +
                                                                                  
                                                                                
                                                                                2
                                                                              
                                                                            
                                                                            )
                                                                          
                                                                        
                                                                        +
                                                                      
                                                                    
                                                                  
                                                                  
                                                                    
                                                                      
                                                                        sin
                                                                        ⁡
                                                                        
                                                                          (
                                                                          
                                                                            
                                                                              
                                                                                ω
                                                                                m
                                                                              
                                                                              ⁢
                                                                              t
                                                                            
                                                                            +
                                                                            
                                                                              Δ
                                                                              ⁢
                                                                              
                                                                                
                                                                              
                                                                              ⁢
                                                                              π
                                                                            
                                                                            -
                                                                            
                                                                              
                                                                                
                                                                                  φ
                                                                                  0
                                                                                  -
                                                                                
                                                                                -
                                                                                
                                                                                  φ
                                                                                  0
                                                                                  +
                                                                                
                                                                              
                                                                              2
                                                                            
                                                                          
                                                                          )
                                                                        
                                                                      
                                                                    
                                                                  
                                                                
                                                                ]
                                                              
                                                              =
                                                              
                                                                
                                                                  
                                                                    -
                                                                    4
                                                                  
                                                                  ⁢
                                                                  
                                                                    δ
                                                                    ·
                                                                    
                                                                      sin
                                                                      ⁡
                                                                      
                                                                        (
                                                                        
                                                                          n
                                                                          ⁢
                                                                          
                                                                            
                                                                          
                                                                          ⁢
                                                                          π
                                                                          ⁢
                                                                          
                                                                            
                                                                          
                                                                          ⁢
                                                                          Δ
                                                                        
                                                                        )
                                                                      
                                                                    
                                                                    ·
                                                                    
                                                                      cos
                                                                      ⁡
                                                                      
                                                                        (
                                                                        
                                                                          
                                                                            φ
                                                                            0
                                                                          
                                                                          -
                                                                          
                                                                            
                                                                              
                                                                                φ
                                                                                0
                                                                                -
                                                                              
                                                                              +
                                                                              
                                                                                φ
                                                                                0
                                                                                +
                                                                              
                                                                            
                                                                            2
                                                                          
                                                                        
                                                                        )
                                                                      
                                                                    
                                                                    ·
                                                                    
                                                                      sin
                                                                      ⁡
                                                                      
                                                                        (
                                                                        
                                                                          
                                                                            Δ
                                                                            ⁢
                                                                            
                                                                              
                                                                            
                                                                            ⁢
                                                                            π
                                                                          
                                                                          2
                                                                        
                                                                        )
                                                                      
                                                                    
                                                                    ·
                                                                    
                                                                      cos
                                                                      ⁡
                                                                      
                                                                        (
                                                                        
                                                                          
                                                                            
                                                                              ω
                                                                              m
                                                                            
                                                                            ⁢
                                                                            t
                                                                          
                                                                          -
                                                                          
                                                                            
                                                                              
                                                                                φ
                                                                                0
                                                                                -
                                                                              
                                                                              -
                                                                              
                                                                                φ
                                                                                0
                                                                                +
                                                                              
                                                                            
                                                                            2
                                                                          
                                                                          +
                                                                          
                                                                            
                                                                              Δ
                                                                              ⁢
                                                                              
                                                                                
                                                                              
                                                                              ⁢
                                                                              π
                                                                            
                                                                            2
                                                                          
                                                                        
                                                                        )
                                                                      
                                                                    
                                                                  
                                                                
                                                                +
                                                                
                                                                  4
                                                                  ⁢
                                                                  
                                                                    δ
                                                                    ·
                                                                    
                                                                      cos
                                                                      ⁡
                                                                      
                                                                        (
                                                                        
                                                                          n
                                                                          ⁢
                                                                          
                                                                            
                                                                          
                                                                          ⁢
                                                                          π
                                                                          ⁢
                                                                          
                                                                            
                                                                          
                                                                          ⁢
                                                                          Δ
                                                                        
                                                                        )
                                                                      
                                                                    
                                                                    ·
                                                                    
                                                                      sin
                                                                      ⁡
                                                                      
                                                                        (
                                                                        
                                                                          
                                                                            
                                                                              
                                                                                
                                                                                  φ
                                                                                  0
                                                                                
                                                                                -
                                                                              
                                                                            
                                                                          
                                                                          
                                                                            
                                                                              
                                                                                
                                                                                  
                                                                                    φ
                                                                                    0
                                                                                    -
                                                                                  
                                                                                  +
                                                                                  
                                                                                    φ
                                                                                    0
                                                                                    +
                                                                                  
                                                                                
                                                                                2
                                                                              
                                                                            
                                                                          
                                                                        
                                                                        )
                                                                      
                                                                    
                                                                    ·
                                                                    
                                                                      cos
                                                                      ⁡
                                                                      
                                                                        (
                                                                        
                                                                          
                                                                            Δ
                                                                            ⁢
                                                                            
                                                                              
                                                                            
                                                                            ⁢
                                                                            π
                                                                          
                                                                          2
                                                                        
                                                                        )
                                                                      
                                                                    
                                                                    ·
                                                                    
                                                                      sin
                                                                      ⁡
                                                                      
                                                                        (
                                                                        
                                                                          
                                                                            
                                                                              ω
                                                                              m
                                                                            
                                                                            ⁢
                                                                            t
                                                                          
                                                                          -
                                                                          
                                                                            
                                                                              
                                                                                φ
                                                                                0
                                                                                -
                                                                              
                                                                              -
                                                                              
                                                                                φ
                                                                                0
                                                                                +
                                                                              
                                                                            
                                                                            2
                                                                          
                                                                          +
                                                                          
                                                                            
                                                                              Δ
                                                                              ⁢
                                                                              
                                                                                
                                                                              
                                                                              ⁢
                                                                              π
                                                                            
                                                                            2
                                                                          
                                                                        
                                                                        )
                                                                      
                                                                    
                                                                  
                                                                
                                                              
                                                            
                                                          
                                                        
                                                      
                                                    
                                                  
                                                
                                              
                                            
                                          
                                        
                                      
                                    
                                  
                                
                              
                            
                          
                        
                      
                    
                  
                
              
            
          
        
      
    
  



Therefore,





  
    
      
        (
        
          1
          +
          δ
        
        )
      
      ·
      
        [
        
          
            cos
            ⁡
            
              (
              
                
                  
                    ω
                    m
                  
                  ⁢
                  t
                
                +
                
                  
                    ω
                    0
                    -
                  
                  ⁢
                  
                    T
                    1
                  
                
                +
                
                  n
                  ⁢
                  
                    
                  
                  ⁢
                  π
                  ⁢
                  
                    
                  
                  ⁢
                  Δ
                
                +
                
                  φ
                  0
                
                -
                
                  φ
                  0
                  -
                
              
              )
            
          
          -
          
            cos
            ⁡
            
              (
              
                
                  
                    ω
                    m
                  
                  ⁢
                  t
                
                +
                
                  n
                  ⁢
                  
                    
                  
                  ⁢
                  π
                  ⁢
                  
                    
                  
                  ⁢
                  Δ
                
                +
                
                  φ
                  0
                
                -
                
                  φ
                  0
                  -
                
              
              )
            
          
        
        ]
      
    
    +
    
      
        (
        
          1
          -
          δ
        
        )
      
      ·
      
        [
        
          
            
              
                
                  cos
                  ⁢
                  
                    (
                    
                      
                        
                          -
                          
                            ω
                            m
                          
                        
                        ⁢
                        t
                      
                      +
                      
                        
                          ω
                          0
                          +
                        
                        ⁢
                        
                          T
                          1
                        
                      
                      +
                      
                        n
                        ⁢
                        
                          
                        
                        ⁢
                        π
                        ⁢
                        
                          
                        
                        ⁢
                        Δ
                      
                      +
                      
                        φ
                        0
                      
                      -
                      
                        φ
                        0
                        +
                      
                    
                    )
                  
                
                -
              
            
          
          
            
              
                cos
                ⁡
                
                  (
                  
                    
                      
                        -
                        
                          ω
                          m
                        
                      
                      ⁢
                      t
                    
                    +
                    
                      n
                      ⁢
                      
                        
                      
                      ⁢
                      π
                      ⁢
                      
                        
                      
                      ⁢
                      Δ
                    
                    +
                    
                      φ
                      0
                    
                    -
                    
                      φ
                      0
                      +
                    
                  
                  )
                
              
            
          
        
        ]
      
    
  
  =
  
    
      
        
          -
          4
        
        ⁢
        
          
            cos
            ⁡
            
              (
              
                Δ
                ⁢
                
                  
                
                ⁢
                n
                ⁢
                
                  
                
                ⁢
                π
              
              )
            
          
          ·
          
            cos
            ⁡
            
              (
              
                
                  φ
                  0
                
                -
                
                  
                    
                      φ
                      0
                      -
                    
                    +
                    
                      φ
                      0
                      +
                    
                  
                  2
                
              
              )
            
          
          ·
          
            cos
            ⁡
            
              (
              
                
                  Δ
                  ⁢
                  
                    
                  
                  ⁢
                  π
                
                2
              
              )
            
          
          ·
          
            cos
            ⁡
            
              (
              
                
                  
                    ω
                    
                      m
                      ⁢
                      
                        
                      
                    
                  
                  ⁢
                  t
                
                -
                
                  
                    
                      φ
                      0
                      -
                    
                    -
                    
                      φ
                      0
                      +
                    
                  
                  2
                
                +
                
                  
                    Δ
                    ⁢
                    
                      
                    
                    ⁢
                    π
                  
                  2
                
              
              )
            
          
        
      
      +
      
        4
        ⁢
        
          
            sin
            ⁡
            
              (
              
                Δ
                ⁢
                
                  
                
                ⁢
                n
                ⁢
                
                  
                
                ⁢
                π
              
              )
            
          
          ·
          
            sin
            ⁡
            
              (
              
                
                  φ
                  0
                
                -
                
                  
                    
                      φ
                      0
                      -
                    
                    +
                    
                      φ
                      0
                      +
                    
                  
                  2
                
              
              )
            
          
          ·
          
            sin
            ⁡
            
              (
              
                
                  Δ
                  ⁢
                  
                    
                  
                  ⁢
                  π
                
                2
              
              )
            
          
          ·
          
            sin
            ⁡
            
              (
              
                
                  
                    ω
                    m
                  
                  ⁢
                  t
                
                -
                
                  
                    
                      φ
                      0
                      -
                    
                    -
                    
                      φ
                      0
                      +
                    
                  
                  2
                
                +
                
                  
                    Δ
                    ⁢
                    
                      
                    
                    ⁢
                    π
                  
                  2
                
              
              )
            
          
        
      
      -
      
        4
        ⁢
        
          δ
          ·
          
            sin
            ⁡
            
              (
              
                n
                ⁢
                
                  
                
                ⁢
                π
                ⁢
                
                  
                
                ⁢
                Δ
              
              )
            
          
          ·
          
            cos
            ⁡
            
              (
              
                
                  φ
                  0
                
                -
                
                  
                    
                      φ
                      0
                      -
                    
                    +
                    
                      φ
                      0
                      +
                    
                  
                  2
                
              
              )
            
          
          ·
          
            sin
            ⁡
            
              (
              
                
                  Δ
                  ⁢
                  
                    
                  
                  ⁢
                  π
                
                2
              
              )
            
          
          ·
          
            cos
            ⁡
            
              (
              
                
                  
                    ω
                    m
                  
                  ⁢
                  t
                
                -
                
                  
                    
                      φ
                      0
                      -
                    
                    -
                    
                      φ
                      0
                      +
                    
                  
                  2
                
                +
                
                  
                    Δ
                    ⁢
                    
                      
                    
                    ⁢
                    t
                  
                  2
                
              
              )
            
          
        
      
      +
      
        4
        ⁢
        
          δ
          ·
          
            cos
            ⁡
            
              (
              
                n
                ⁢
                
                  
                
                ⁢
                π
                ⁢
                
                  
                
                ⁢
                Δ
              
              )
            
          
          ·
          
            sin
            ⁡
            
              (
              
                
                  φ
                  0
                
                -
                
                  
                    
                      φ
                      0
                      -
                    
                    +
                    
                      φ
                      0
                      +
                    
                  
                  2
                
              
              )
            
          
          ·
          
            cos
            ⁡
            
              (
              
                
                  Δ
                  ⁢
                  
                    
                  
                  ⁢
                  π
                
                
                  2
                  ⁢
                  
                    
                  
                
              
              )
            
          
          ·
          
            cos
            ⁡
            
              (
              
                
                  Δ
                  ⁢
                  
                    
                  
                  ⁢
                  π
                
                2
              
              )
            
          
          ·
          
            sin
            ⁡
            
              (
              
                
                  
                    ω
                    m
                  
                  ⁢
                  t
                
                -
                
                  
                    
                      φ
                      0
                      -
                    
                    -
                    
                      φ
                      0
                      +
                    
                  
                  2
                
                +
                
                  
                    Δ
                    ⁢
                    
                      
                    
                    ⁢
                    π
                  
                  2
                
              
              )
            
          
        
      
    
    =
    
      
        
          
            -
            4
          
          ⁢
          
            
              cos
              ⁡
              
                (
                
                  
                    φ
                    0
                  
                  -
                  
                    
                      
                        φ
                        0
                        -
                      
                      +
                      
                        φ
                        0
                        +
                      
                    
                    2
                  
                
                )
              
            
            ·
            
              [
              
                
                  
                    cos
                    ⁡
                    
                      (
                      
                        Δ
                        ⁢
                        
                          
                        
                        ⁢
                        n
                        ⁢
                        
                          
                        
                        ⁢
                        π
                      
                      )
                    
                  
                  ·
                  
                    cos
                    ⁡
                    
                      (
                      
                        
                          Δ
                          ⁢
                          
                            
                          
                          ⁢
                          π
                        
                        2
                      
                      )
                    
                  
                
                +
                
                  δ
                  ·
                  
                    sin
                    ⁡
                    
                      (
                      
                        n
                        ⁢
                        
                          
                        
                        ⁢
                        π
                        ⁢
                        
                          
                        
                        ⁢
                        Δ
                      
                      )
                    
                  
                  ·
                  
                    sin
                    ⁡
                    
                      (
                      
                        
                          Δ
                          ⁢
                          
                            
                          
                          ⁢
                          π
                        
                        2
                      
                      )
                    
                  
                
              
              ]
            
            ·
            
              cos
              ⁡
              
                (
                
                  
                    
                      ω
                      m
                    
                    ⁢
                    t
                  
                  -
                  
                    
                      
                        φ
                        0
                        -
                      
                      -
                      
                        φ
                        0
                        +
                      
                    
                    2
                  
                  +
                  
                    
                      Δ
                      ⁢
                      
                        
                      
                      ⁢
                      π
                    
                    2
                  
                
                )
              
            
          
        
        +
        
          4
          ⁢
          
            
              sin
              ⁡
              
                (
                
                  
                    φ
                    0
                  
                  -
                  
                    
                      
                        φ
                        0
                        -
                      
                      +
                      
                        φ
                        0
                        +
                      
                    
                    2
                  
                
                )
              
            
            ·
            
              [
              
                
                  
                    sin
                    ⁡
                    
                      (
                      
                        Δ
                        ⁢
                        
                          
                        
                        ⁢
                        n
                        ⁢
                        
                          
                        
                        ⁢
                        π
                      
                      )
                    
                  
                  ·
                  
                    sin
                    ⁡
                    
                      (
                      
                        
                          Δ
                          ⁢
                          
                            
                          
                          ⁢
                          π
                        
                        2
                      
                      )
                    
                  
                
                +
                
                  δ
                  ·
                  
                    cos
                    ⁡
                    
                      (
                      
                        n
                        ⁢
                        
                          
                        
                        ⁢
                        π
                        ⁢
                        
                          
                        
                        ⁢
                        Δ
                      
                      )
                    
                  
                  ·
                  
                    cos
                    ⁡
                    
                      (
                      
                        
                          Δ
                          ⁢
                          
                            
                          
                          ⁢
                          π
                        
                        2
                      
                      )
                    
                  
                
              
              ]
            
            ·
            
              sin
              ⁡
              
                (
                
                  
                    
                      ω
                      m
                    
                    ⁢
                    t
                  
                  -
                  
                    
                      
                        φ
                        0
                        -
                      
                      -
                      
                        φ
                        0
                        +
                      
                    
                    2
                  
                  +
                  
                    
                      Δ
                      ⁢
                      
                        
                      
                      ⁢
                      π
                    
                    2
                  
                
                )
              
            
          
        
      
      ≈
      
        
          -
          4
        
        ⁢
        
          
            cos
            ⁡
            
              (
              
                
                  φ
                  0
                
                -
                
                  
                    
                      φ
                      0
                      -
                    
                    +
                    
                      φ
                      0
                      +
                    
                  
                  2
                
              
              )
            
          
          ·
          
            cos
            ⁡
            
              (
              
                Δ
                ⁢
                
                  
                
                ⁢
                n
                ⁢
                
                  
                
                ⁢
                π
              
              )
            
          
          ·
          
            cos
            ⁡
            
              (
              
                
                  Δ
                  ⁢
                  
                    
                  
                  ⁢
                  π
                
                2
              
              )
            
          
          ·
          
            
              cos
              ⁡
              
                (
                
                  
                    
                      ω
                      m
                    
                    ⁢
                    
                      
                    
                    ⁢
                    t
                  
                  +
                  
                    
                      Δ
                      ⁢
                      
                        
                      
                      ⁢
                      π
                    
                    2
                  
                  -
                  
                    
                      
                        φ
                        0
                        -
                      
                      -
                      
                        φ
                        0
                        +
                      
                    
                    2
                  
                  +
                  
                    δ
                    ⁢
                    
                      
                    
                    ⁢
                    φ
                  
                
                )
              
            
            .

## Claims

1. A line locator, comprising:
a plurality of coil detectors, each of the plurality of coil detectors oriented to measure a component of an electromagnetic field;
circuitry coupled to provide quadrature signals indicating a complex electromagnetic field strength;
a position locator for indicating a position of the line locator;
a processor coupled to receive the complex electromagnetic field strength and the position, and to calculate values related to a target line, wherein
the processor is configured to execute instructions for:
modeling an expected complex field strength using a phase error function, and using an estimation of a current attenuation at two nearest side tones to update a fit factor, the side tones defined by a carrier frequency and a modulation frequency; and
determining parameters related to the target line from a best model.

2. The line locator of claim 1, wherein the parameters related to the target line include at least one estimated value chosen from a list consisting of a centerline position, a depth of the target line, and a current carried in the target line.

3. The line locator of claim 1, wherein the best model includes a selected value of a cost function.

4. The line locator of claim 1, further including estimating a confidence interval for a plurality of selected values associated with the best model.

5. The line locator of claim 1, wherein the plurality of coil detectors includes a first coil to determine a first electromagnetic field strength and a second coil for determining a second electromagnetic field strength, the first coil being orthogonal to the second coil.

6. The line locator of claim 5, wherein the plurality of coil detectors includes a third coil for determining a third electromagnetic field strength, wherein the third coil is oriented orthogonally from the first coil.

7. The line locator of claim 1, wherein the position locator can indicate the position of the locator along a plurality of positions along a path determined by inertial tracking.

8. The line locator of claim 1, wherein the position locator can detect pitch, roll, and yaw angles of the locator by inertial tracking and the processor can correct the complex electromagnetic field strengths for aberrations that occur while measuring components of the complex electromagnetic field strength.

9. The line locator of claim 1, wherein modeling an expected complex field strength includes optimizing each of a plurality of models.

10. A method of locating a target line, comprising:
measuring an electromagnetic field using a plurality of coil detectors;
modeling, by a processor, a phase error based on the presence of lines neighboring the target line, wherein modeling comprises using a phase error function including a plurality of parameters from the target line and from the neighboring lines, and an estimation of a current attenuation at two nearest side tones to update a fit factor, the side tones defined by a carrier frequency and a modulation frequency; and
utilizing the phase error in an enhanced electromagnetic field model to locate the target line.

11. The method of claim 10 wherein updating the fit factor comprises using a ratio of a higher frequency to a lower frequency of the two nearest side tones in combination with a slope of a current gain at an integer multiple of the carrier frequency.

12. The method of claim 10, further comprising displaying a plurality of parameters related to the target line.

13. The method of claim 12, wherein the plurality of parameters related to the target line correspond to a best model.

14. The method of claim 13, wherein the plurality of parameters includes an indication of a number of neighboring lines utilized in the best model.

15. The method of claim 13, wherein the plurality of parameters includes an indication of a relative strength of different directions of the measured electromagnetic field.

16. The method of claim 10, wherein measuring an electromagnetic field comprises measuring a set of complex electromagnetic field strengths at a plurality of positions along a path that crosses the target line.

17. The method of claim 16, wherein modeling the phase error comprises modeling an expected complex field strength at each of the plurality of positions along the path that crosses the target line, the model including a hypothesized number of bleed over lines; and
forming a set of hypothesized values corresponding to a set of individual models for the target line for each bleed over line from the hypothesized number of bleed over lines.

18. The method of claim 17, further including estimating a confidence interval for the hypothesized values associated with a best model.

19. The method of claim 17, wherein modeling an expected complex field strengths at each of the plurality of positions includes optimizing each individual model from the set of individual models.

20. The method of claim 19, wherein optimizing each individual model comprises applying a Levenberg-Marquardt algorithm.

21. The method of claim 17, wherein modeling the phase error comprises determining which of the set of individual models is a best model, and determining a plurality of parameters related to the target line from the best model.

22. The method of claim 21, wherein the plurality of parameters related to the target line include at least one estimated value chosen from a list consisting of at least one of a centerline position, depth of the target line, and a current carried in the target line.

23. The method of claim 21, wherein determining which of the set of individual models is the best model includes:
applying a cost function to each of the set of individual models; and
choosing the best model.

24. The method of claim 23, wherein the cost function includes a penalty for a number of neighboring lines used.

25. The method of claim 23, wherein the cost function includes a mean square error fit to the set of complex electromagnetic field strengths.

26. The method of claim 23, further including rejecting false positive locates due to excessive noise and interference.

27. The method of claim 23, further including optimizing the cost function.

28. The method of claim 27, wherein optimizing the cost function comprises applying a global grid search algorithm.

29. The method of claim 27, wherein optimizing the cost function includes applying a minimum norm algorithm.

30. The method of claim 16, wherein measuring a set of complex electromagnetic field strengths includes traversing a handheld locate receiver over the target line by walking.

31. The method of claim 16, wherein measuring a set of complex electromagnetic field strengths includes traversing a handheld locate receiver over the target line by using a pull cart.

32. The method of claim 16, wherein measuring a set of complex electromagnetic field strengths includes determining a first electromagnetic field strength from a first coil and determining a second electromagnetic field strength from a second coil, the first coil being orthogonal to the second coil.

33. The method of claim 32, wherein measuring a set of complex electromagnetic field strengths includes determining a third electromagnetic field strength from a third coil.

34. The method of claim 32, wherein the first coil and the second coil are oriented in a “V” configuration, with one coil tilted above horizontal and the other tilted below horizontal, the first coil and the second coil aligned in a direction transverse to a line direction.

35. The method of claim 32, wherein modeling an expected complex field strength includes employing a first field model to estimate the first electromagnetic field strength by adjusting a plurality of hypothesized values and employing a second field model to estimate the second electromagnetic field strength by adjusting the plurality of hypothesized values.

36. The method of claim 35, further including detecting pitch, roll, and yaw angles by inertial tracking and correcting the complex electromagnetic field strengths for aberrations that occur while measuring the set of complex electromagnetic field strengths.

37. The method of claim 35, wherein a best field model to be utilized for determining the first electromagnetic field strength and the second electromagnetic field strength is chosen for each of the plurality of positions depending on which of the first electromagnetic field strength or second electromagnetic field strength is stronger, thereby avoiding null fields sensed by one of the first coil or the second coil by replacing that complex field measurement with values from an alternative model.

38. The method of claim 37, wherein a dedicated digital phase locked loop operates on the signal from each of the first coil and the second coil, thereby enhancing performance when null fields are sensed by one of the first coil and the second coil and are at a maximum on the other of the first coil and the second coil.

39. The method of claim 16, wherein measuring a set of complex electromagnetic field strengths comprises measuring electromagnetic field strengths with three orthogonal coils, and modeling an expected complex field strength includes utilizing a full three-dimensional field model that adjusts a plurality of hypothesized values to estimate the complex electromagnetic field strengths.

40. The method of claim 16, wherein the plurality of coils comprises a first coil, a top coil, and two additional coils, the two additional coils oriented orthogonally to each other and to the first coil and the top coil, and wherein modeling a phase error includes utilizing a two dimensional field model that relates a plurality of hypothesized values to generate an expected complex field strength at two of the three orthogonal coils, and modeling a phase error includes utilizing a third of the three orthogonal coils to validate that the path that crosses the target line is roughly perpendicular to a length of the target line.

41. The method of claim 16, further including demodulating an in-phase signal and a quadrature signal from the measured electromagnetic field to obtain an amplitude and a phase of the complex electromagnetic field strengths.

42. The method of claim 16, wherein the plurality of positions along the path are determined by inertial tracking.

43. The method of claim 16, wherein the plurality of positions along the path are determined by stereo imaging.

44. The method of claim 16, wherein the plurality of positions along the path are determined by laser or ultrasonic range finding.

45. The method of claim 16, wherein the plurality of positions along the path are determined with a measuring wheel.

46. The method of claim 16, wherein the plurality of positions along the path are determined using remote sensing.

47. The method of claim 46, wherein the remote sensing includes detection of tick marks on a tape measure.

48. The method of claim 46, wherein the remote sensing includes user input that a predetermined measurement position has been reached.

49. The method of claim 10, wherein the plurality of coil detectors comprises a first coil and a second coil, wherein:
the first coil comprises a horizontal coil arranged to detect a horizontal component of an electromagnetic field; and
the second coil comprises a vertical coil arranged to detect a vertical component of the electromagnetic field.