# Patent

## Title
Enhanced precise location

## Abstract

A method for determining the location of underground cables and pipes is disclosed. In some embodiments, the method includes measuring a set of electromagnetic field magnitudes and phases at a plurality of positions while traversing a target line parallelly using 3D electromagnetic coil sensors, the 3D electromagnetic coil sensors being orthogonally oriented to the target line, modeling a set of expected complex electromagnetic field magnitudes of a single underground conductor at each of the positions to form a set of values corresponding to a set of individual models for the target line, determining which of the set of individuals models is a best model, determining confidence information at each of the positions based on a comparison between the measured set of complex electromagnetic magnitudes and phases and the best model, and determining parameters at each of positions related to the target line from the best model.

## Background
BENEFIT OF PRIORITY
This application in a continuation-in-part of application Ser. No. 12/122,136, filed May 16, 2008 now U.S. Pat. No. 8,209,136, which claims the benefit of Provisional Application No. 60/930,753, filed May 18, 2007, both of which are incorporated herein by reference in their entirety. This application further claims the benefit of Provisional Application No. 60/993,741, filed Sep. 13, 2007, which is also which is incorporated herein by reference in its entirety.


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 method for determining the location of underground cables and pipes may include measuring a set of complex electromagnetic field magnitudes and phases at a plurality of positions while traversing a target line in a substantially parallel direction using a plurality of 3-dimensional electromagnetic coil sensors, the plurality of 3-dimensional electromagnetic coil sensors being oriented substantially orthogonal to the target line, modeling a set of expected complex electromagnetic field magnitudes of a single underground conductor at each of the plurality of positions to form a set of hypothesized values corresponding to a set of individual models for the target line, determining which of the set of individuals models is a best model, determining confidence information at each of the plurality of positions based on a comparison between the measured set of complex electromagnetic magnitudes and phases and the best model, and determining parameters at each of the plurality of positions related to the target line from the best model.
Consistent with some embodiments of the present invention, a line locator may include a plurality of 3-dimensional electromagnetic coil detectors configured to measure a complex electromagnetic field magnitude and phase generated at least by a target line, the plurality of 3-dimensional electromagnetic coil detectors being oriented substantially orthogonal to the target line, circuitry coupled to receive signals from the plurality of coil detectors and provide Quadrature signals indicating a measured complex electromagnetic field magnitude and phase, a position locator for indicating a position of the line locator, a processor coupled to receive the complex electromagnetic field magnitude and phase and the position and calculate values related to the target line, and a display coupled to the processor, the display indicating to a user the values related to the target line, wherein the processor includes software for modeling a set of expected complex electromagnetic field magnitudes of a single underground conductor at each of a plurality of positions determined by the position locator to form a set of hypothesized values corresponding to a set of individual models for the target line, determining which of the set of individuals models is a best model, determining target line location confidence information at each of the plurality of positions based on a comparison between the measured set of complex electromagnetic field complex electromagnetic field magnitudes and phases and the best model, and determining parameters at each of the plurality of positions related to the target line from the best model.
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.
FIG. 7 illustrates the aboveground use of a cart-based line locator consistent with some embodiments of the present invention to locate the position of an underground targeted conductor during a vector mode locate process.



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 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 1. 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
                    ⁢
                    
                      
                    
                    ⁢
                    ξ
                    ⁢
                    
                      
                    
                    ⁢
                    
                      e
                      
                        
                          -
                          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 (4π 10−7),
σ=conductivity of the ground, and
f=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
        
        =
        
          
            I
            z
          
          
            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.
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
                      ⁢
                      
                        
                      
                      ⁢
                      πΔ
                    
                    )
                  
                
              
              
                n
                ⁢
                
                  
                
                ⁢
                πΔ
              
            
            ·
            
              cos
              ⁡
              
                (
                
                  
                    
                      ω
                      0
                    
                    ⁢
                    t
                  
                  +
                  
                    n
                    ⁢
                    
                      
                    
                    ⁢
                    πΔ
                  
                
                )
              
            
          
          +
          
            
              
                
                  
                    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
                      ⁢
                      
                        
                      
                      ⁢
                      πΔ
                    
                    )
                  
                
              
              
                n
                ⁢
                
                  
                
                ⁢
                πΔ
              
            
            ·
            
              cos
              ⁡
              
                (
                
                  
                    
                      ω
                      0
                    
                    ⁢
                    t
                  
                  +
                  
                    n
                    ⁢
                    
                      
                    
                    ⁢
                    πΔ
                  
                  +
                  
                    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
                        ⁢
                        
                          
                        
                        ⁢
                        πΔ
                      
                      )
                    
                  
                
                
                  n
                  ⁢
                  
                    
                  
                  ⁢
                  πΔ
                
              
              ·
              
                cos
                ⁡
                
                  (
                  
                    
                      
                        ω
                        0
                      
                      ⁢
                      t
                    
                    +
                    
                      n
                      ⁢
                      
                        
                      
                      ⁢
                      πΔ
                    
                    +
                    
                      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
                      ⁢
                      
                        
                      
                      ⁢
                      πΔ
                    
                    )
                  
                
                
                  n
                  ⁢
                  
                    
                  
                  ⁢
                  πΔ
                
              
              ·
              
                
                  ∑
                  
                    k
                    =
                    0
                  
                  M
                
                ⁢
                
                  
                    A
                    k
                  
                  ·
                  
                    cos
                    ⁡
                    
                      (
                      
                        
                          
                            ω
                            0
                          
                          ⁢
                          t
                        
                        +
                        
                          n
                          ⁢
                          
                            
                          
                          ⁢
                          πΔ
                        
                        +
                        
                          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
                          +
                        
                      
                      )
                    
                  
                
              
            
          
        
        ⁢
        
          

        
        ⁢
        
          where
          ,
        
      
    
    
      
        (
        25
        )
      
    
  
  
    
      
        
          
            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
                      ⁢
                      
                        
                      
                      ⁢
                      πΔ
                    
                    )
                  
                
                
                  n
                  ⁢
                  
                    
                  
                  ⁢
                  πΔ
                
              
              ·
              A
              ·
              
                cos
                ⁡
                
                  (
                  
                    
                      
                        ω
                        0
                      
                      ⁢
                      t
                    
                    +
                    
                      n
                      ⁢
                      
                        
                      
                      ⁢
                      πΔ
                    
                    +
                    
                      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
                      +
                    
                  
                  )
                
              
            
          
        
        ⁢
        
          

        
        ⁢
        
          where
          ,
        
      
    
    
      
        (
        29
        )
      
    
  
  
    
      
        
          
            
              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
                      ⁢
                      
                        
                      
                      ⁢
                      πΔ
                    
                    +
                    
                      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
                      ⁢
                      
                        
                      
                      ⁢
                      πΔ
                    
                    +
                    
                      φ
                      0
                    
                    -
                    
                      φ
                      0
                      +
                    
                  
                  )
                
              
            
          
          =
          
            
              
                
                  
                    n
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Δ
                      /
                      π
                    
                  
                  
                    1
                    -
                    
                      
                        (
                        
                          2
                          ⁢
                          n
                          ⁢
                          
                            
                          
                          ⁢
                          Δ
                        
                        )
                      
                      2
                    
                  
                
                ·
                
                  A
                  -
                
                ·
                
                  [
                  
                    
                      cos
                      ⁡
                      
                        (
                        
                          
                            
                              ω
                              m
                            
                            ⁢
                            t
                          
                          +
                          
                            
                              ω
                              0
                              -
                            
                            ⁢
                            
                              T
                              1
                            
                          
                          +
                          
                            n
                            ⁢
                            
                              
                            
                            ⁢
                            πΔ
                          
                          +
                          
                            φ
                            0
                          
                          -
                          
                            φ
                            0
                            -
                          
                        
                        )
                      
                    
                    -
                    
                      cos
                      ⁡
                      
                        (
                        
                          
                            
                              ω
                              m
                            
                            ⁢
                            t
                          
                          +
                          
                            n
                            ⁢
                            
                              
                            
                            ⁢
                            πΔ
                          
                          +
                          
                            φ
                            0
                          
                          -
                          
                            φ
                            0
                            -
                          
                        
                        )
                      
                    
                  
                  ]
                
              
              +
              
                
                  
                    n
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Δ
                      /
                      π
                    
                  
                  
                    1
                    -
                    
                      
                        (
                        
                          2
                          ⁢
                          n
                          ⁢
                          
                            
                          
                          ⁢
                          Δ
                        
                        )
                      
                      2
                    
                  
                
                ·
                
                  A
                  +
                
                ·
                
                  [
                  
                    
                      cos
                      ⁡
                      
                        (
                        
                          
                            
                              -
                              
                                ω
                                m
                              
                            
                            ⁢
                            t
                          
                          +
                          
                            
                              ω
                              0
                              +
                            
                            ⁢
                            
                              T
                              1
                            
                          
                          +
                          
                            n
                            ⁢
                            
                              
                            
                            ⁢
                            πΔ
                          
                          +
                          
                            φ
                            0
                          
                          -
                          
                            φ
                            0
                            +
                          
                        
                        )
                      
                    
                    -
                    
                      cos
                      ⁡
                      
                        (
                        
                          
                            
                              -
                              
                                ω
                                m
                              
                            
                            ⁢
                            t
                          
                          +
                          
                            n
                            ⁢
                            
                              
                            
                            ⁢
                            πΔ
                          
                          +
                          
                            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
                                              ⁢
                                              
                                                
                                              
                                              ⁢
                                              πΔ
                                            
                                            +
                                            
                                              φ
                                              0
                                            
                                            -
                                            
                                              φ
                                              0
                                              -
                                            
                                          
                                          )
                                        
                                      
                                      -
                                    
                                  
                                
                                
                                  
                                    
                                      cos
                                      ⁡
                                      
                                        (
                                        
                                          
                                            
                                              ω
                                              m
                                            
                                            ⁢
                                            t
                                          
                                          +
                                          
                                            n
                                            ⁢
                                            
                                              
                                            
                                            ⁢
                                            πΔ
                                          
                                          +
                                          
                                            φ
                                            0
                                          
                                          -
                                          
                                            φ
                                            0
                                            -
                                          
                                        
                                        )
                                      
                                    
                                  
                                
                              
                              ]
                            
                          
                          +
                        
                      
                    
                    
                      
                        
                          
                            (
                            
                              1
                              -
                              δ
                            
                            )
                          
                          ·
                          
                            [
                            
                              
                                
                                  
                                    
                                      cos
                                      ⁡
                                      
                                        (
                                        
                                          
                                            
                                              -
                                              
                                                ω
                                                m
                                              
                                            
                                            ⁢
                                            t
                                          
                                          +
                                          
                                            
                                              ω
                                              0
                                              +
                                            
                                            ⁢
                                            
                                              T
                                              1
                                            
                                          
                                          +
                                          
                                            n
                                            ⁢
                                            
                                              
                                            
                                            ⁢
                                            πΔ
                                          
                                          +
                                          
                                            φ
                                            0
                                          
                                          -
                                          
                                            φ
                                            0
                                            +
                                          
                                        
                                        )
                                      
                                    
                                    -
                                  
                                
                              
                              
                                
                                  
                                    cos
                                    ⁡
                                    
                                      (
                                      
                                        
                                          
                                            -
                                            
                                              ω
                                              m
                                            
                                          
                                          ⁢
                                          t
                                        
                                        +
                                        
                                          n
                                          ⁢
                                          
                                            
                                          
                                          ⁢
                                          πΔ
                                        
                                        +
                                        
                                          φ
                                          0
                                        
                                        -
                                        
                                          φ
                                          0
                                          +
                                        
                                      
                                      )
                                    
                                  
                                
                              
                            
                            ]
                          
                        
                      
                    
                  
                  }
                
              
              ≈
              
                
                  -
                  4
                
                ⁢
                
                  
                    cos
                    ⁡
                    
                      (
                      
                        
                          φ
                          0
                        
                        -
                        
                          
                            
                              φ
                              0
                              -
                            
                            +
                            
                              φ
                              0
                              +
                            
                          
                          2
                        
                      
                      )
                    
                  
                  ·
                  
                    cos
                    ⁡
                    
                      (
                      
                        Δ
                        ⁢
                        
                          
                        
                        ⁢
                        n
                        ⁢
                        
                          
                        
                        ⁢
                        π
                      
                      )
                    
                  
                  ·
                  
                    cos
                    ⁡
                    
                      (
                      
                        Δπ
                        2
                      
                      )
                    
                  
                  ·
                  
                    cos
                    ⁡
                    
                      (
                      
                        
                          
                            ω
                            m
                          
                          ⁢
                          t
                        
                        +
                        
                          Δπ
                          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
                              ⁡
                              
                                (
                                
                                  Δπ
                                  2
                                
                                )
                              
                            
                            ·
                            
                              tan
                              ⁡
                              
                                (
                                
                                  Δ
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  n
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  π
                                
                                )
                              
                            
                          
                        
                        ]
                      
                    
                    )
                  
                
                ⁢
                
                  

                
                ≈
                
                  
                    [
                    
                      δ
                      +
                      
                        
                          tan
                          ⁡
                          
                            (
                            
                              Δπ
                              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 re-modulated 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
                      ⁢
                      
                        
                      
                      ⁢
                      πΔ
                    
                    )
                  
                
                
                  n
                  ⁢
                  
                    
                  
                  ⁢
                  πΔ
                
              
              ⁢
              
                {
                
                  
                    
                      
                        
                          
                            [
                            
                              
                                
                                  
                                    h
                                    n
                                    I
                                  
                                  ⁡
                                  
                                    (
                                    
                                      x
                                      j
                                    
                                    )
                                  
                                
                                ·
                                
                                  cos
                                  ⁡
                                  
                                    (
                                    Δφ
                                    )
                                  
                                
                              
                              +
                              
                                
                                  
                                    h
                                    n
                                    Q
                                  
                                  ⁡
                                  
                                    (
                                    
                                      x
                                      j
                                    
                                    )
                                  
                                
                                ·
                                
                                  sin
                                  ⁡
                                  
                                    (
                                    Δφ
                                    )
                                  
                                
                              
                            
                            ]
                          
                          ·
                          
                            cos
                            ⁡
                            
                              (
                              
                                
                                  
                                    ω
                                    0
                                  
                                  ⁢
                                  t
                                
                                +
                                
                                  n
                                  ⁢
                                  
                                    
                                  
                                  ⁢
                                  πΔ
                                
                                +
                                Δφ
                              
                              )
                            
                          
                        
                        -
                      
                    
                  
                  
                    
                      
                        
                          [
                          
                            
                              
                                
                                  h
                                  n
                                  Q
                                
                                ⁡
                                
                                  (
                                  
                                    x
                                    j
                                  
                                  )
                                
                              
                              ·
                              
                                cos
                                ⁡
                                
                                  (
                                  Δφ
                                  )
                                
                              
                            
                            -
                            
                              
                                
                                  
                                    h
                                    n
                                    I
                                  
                                  ⁡
                                  
                                    (
                                    
                                      x
                                      j
                                    
                                    )
                                  
                                
                                ·
                                sin
                              
                              ⁢
                              
                                (
                                Δφ
                                )
                              
                            
                          
                          ]
                        
                        ·
                        
                          sin
                          ⁡
                          
                            (
                            
                              
                                
                                  ω
                                  0
                                
                                ⁢
                                t
                              
                              +
                              
                                n
                                ⁢
                                
                                  
                                
                                ⁢
                                πΔ
                              
                              +
                              Δφ
                            
                            )
                          
                        
                      
                    
                  
                
                }
              
            
          
          =
          
            
              
                sin
                ⁡
                
                  (
                  
                    n
                    ⁢
                    
                      
                    
                    ⁢
                    πΔ
                  
                  )
                
              
              
                n
                ⁢
                
                  
                
                ⁢
                πΔ
              
            
            ⁢
            
              {
              
                
                  
                    
                      
                        h
                        n
                        I
                      
                      ⁡
                      
                        (
                        
                          x
                          j
                        
                        )
                      
                    
                    mod
                  
                  ·
                  
                    cos
                    ⁡
                    
                      (
                      
                        
                          
                            ω
                            0
                          
                          ⁢
                          t
                        
                        +
                        
                          n
                          ⁢
                          
                            
                          
                          ⁢
                          πΔ
                        
                        +
                        Δφ
                      
                      )
                    
                  
                
                -
                
                  
                    
                      
                        h
                        n
                        Q
                      
                      ⁡
                      
                        (
                        
                          x
                          j
                        
                        )
                      
                    
                    mod
                  
                  ·
                  
                    sin
                    ⁡
                    
                      (
                      
                        
                          
                            ω
                            0
                          
                          ⁢
                          t
                        
                        +
                        
                          n
                          ⁢
                          
                            
                          
                          ⁢
                          πΔ
                        
                        +
                        Δφ
                      
                      )
                    
                  
                
              
              }
            
            ⁢
            
              
            
            ⁢
            where
          
        
        ,
        
          
        
        ⁢
        
          
            
              
                h
                n
                I
              
              ⁡
              
                (
                
                  x
                  j
                
                )
              
            
            mod
          
          ⁢
          
            
          
          ⁢
          and
          ⁢
          
            
          
          ⁢
          
            
              
                h
                n
                Q
              
              ⁡
              
                (
                
                  x
                  j
                
                )
              
            
            mod
          
          ⁢
          
            
          
          ⁢
          are
          ⁢
          
            
          
          ⁢
          defined
          ⁢
          
            
          
          ⁢
          as
          ⁢
          
            :
          
        
      
    
    
      
        (
        41
        )
      
    
  
  
    
      
        
          
            
              
                h
                n
                I
              
              ⁡
              
                (
                
                  x
                  j
                
                )
              
            
            mod
          
          =
          
            
              
                sin
                ⁡
                
                  (
                  
                    n
                    ⁢
                    
                      
                    
                    ⁢
                    πΔ
                  
                  )
                
              
              
                n
                ⁢
                
                  
                
                ⁢
                πΔ
              
            
            ·
            
              [
              
                
                  
                    
                      h
                      n
                      I
                    
                    ⁡
                    
                      (
                      
                        x
                        j
                      
                      )
                    
                  
                  ·
                  
                    cos
                    ⁡
                    
                      (
                      Δφ
                      )
                    
                  
                
                +
                
                  
                    
                      h
                      n
                      Q
                    
                    ⁡
                    
                      (
                      
                        x
                        j
                      
                      )
                    
                  
                  ·
                  
                    sin
                    ⁡
                    
                      (
                      Δφ
                      )
                    
                  
                
              
              ]
            
          
        
        ⁢
        
          

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

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




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
                    ⁢
                    
                      
                    
                    ⁢
                    πΔ
                  
                  )
                
              
              
                n
                ⁢
                
                  
                
                ⁢
                πΔ
              
            
            ·
            
              (
              
                
                  
                    
                      cos
                      ⁡
                      
                        (
                        Δφ
                        )
                      
                    
                  
                  
                    
                      sin
                      ⁡
                      
                        (
                        Δφ
                        )
                      
                    
                  
                
                
                  
                    
                      -
                      
                        sin
                        ⁡
                        
                          (
                          Δφ
                          )
                        
                      
                    
                  
                  
                    
                      cos
                      ⁡
                      
                        (
                        Δφ
                        )
                      
                    
                  
                
              
              )
            
            ·
            
              
                [
                
                  
                    
                      
                        
                          h
                          t
                          I
                        
                        ⁡
                        
                          (
                          
                            x
                            j
                          
                          )
                        
                      
                    
                  
                  
                    
                      
                        
                          h
                          t
                          Q
                        
                        ⁡
                        
                          (
                          
                            x
                            j
                          
                          )
                        
                      
                    
                  
                
                ]
              
              ⁢
              
                

              
              [
              
                
                  
                    
                      
                        h
                        t
                        I
                      
                      ⁡
                      
                        (
                        
                          x
                          j
                        
                        )
                      
                    
                  
                
                
                  
                    
                      
                        h
                        t
                        Q
                      
                      ⁡
                      
                        (
                        
                          x
                          j
                        
                        )
                      
                    
                  
                
              
              ]
            
          
          =
          
            
              
                sin
                ⁡
                
                  (
                  
                    n
                    ⁢
                    
                      
                    
                    ⁢
                    πΔ
                  
                  )
                
              
              
                n
                ⁢
                
                  
                
                ⁢
                πΔ
              
            
            ·
            
              (
              
                
                  
                    
                      cos
                      ⁡
                      
                        (
                        Δφ
                        )
                      
                    
                  
                  
                    
                      -
                      
                        sin
                        ⁡
                        
                          (
                          Δφ
                          )
                        
                      
                    
                  
                
                
                  
                    
                      sin
                      ⁡
                      
                        (
                        Δφ
                        )
                      
                    
                  
                  
                    
                      cos
                      ⁡
                      
                        (
                        Δφ
                        )
                      
                    
                  
                
              
              )
            
            ·
            
              
                [
                
                  
                    
                      
                        
                          
                            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, hvQ(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 detectable 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
                
                  i
                  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, φ1, dI1, 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
                                  
                                  )
                                
                              
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                          ]
                        
                        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
                      2
                    
                    ⁡
                    
                      (
                      a
                      )
                    
                  
                
                
                  ∂
                  
                    a
                    k
                  
                
              
              ·
              
                
                  ∂
                  
                    S
                    ⁡
                    
                      (
                      a
                      )
                    
                  
                
                
                  ∂
                  
                    a
                    l
                  
                
              
            
            ]
          
        
      
    
    
      
        (
        52
        )
      
    
  




often approximated with only the first derivatives of the model as





  
    
      
        
          α
          kI
        
        =
        
          
            ∑
            
              j
              =
              1
            
            N
          
          ⁢
          
            
          
          ⁢
          
            (
            
              
                
                  
                    
                      
                        
                          ∂
                          
                            
                              
                                h
                                n
                                I
                              
                              ⁡
                              
                                (
                                
                                  x
                                  j
                                
                                )
                              
                            
                            mod
                          
                        
                        
                          ∂
                          
                            a
                            k
                          
                        
                      
                      ·
                      
                        
                          ∂
                          
                            
                              
                                h
                                n
                                I
                              
                              ⁡
                              
                                (
                                
                                  x
                                  j
                                
                                )
                              
                            
                            mod
                          
                        
                        
                          ∂
                          
                            a
                            
                              
                                
                              
                              I
                            
                          
                        
                      
                    
                    +
                    
                      
                        
                          ∂
                          
                            
                              
                                h
                                n
                                Q
                              
                              ⁡
                              
                                (
                                
                                  x
                                  j
                                
                                )
                              
                            
                            mod
                          
                        
                        
                          ∂
                          
                            a
                            k
                          
                        
                      
                      ·
                      
                        
                          ∂
                          
                            
                              
                                h
                                n
                                Q
                              
                              ⁡
                              
                                (
                                
                                  x
                                  j
                                
                                )
                              
                            
                            mod
                          
                        
                        
                          ∂
                          
                            a
                            
                              
                                
                              
                              I
                            
                          
                        
                      
                    
                    +
                  
                
              
              
                
                  
                    
                      
                        
                          ∂
                          
                            
                              
                                h
                                v
                                I
                              
                              ⁡
                              
                                (
                                
                                  x
                                  j
                                
                                )
                              
                            
                            mod
                          
                        
                        
                          ∂
                          
                            a
                            k
                          
                        
                      
                      ·
                      
                        
                          ∂
                          
                            
                              
                                h
                                v
                                I
                              
                              ⁡
                              
                                (
                                
                                  x
                                  j
                                
                                )
                              
                            
                            mod
                          
                        
                        
                          ∂
                          
                            a
                            
                              
                                
                              
                              I
                            
                          
                        
                      
                    
                    +
                    
                      
                        
                          ∂
                          
                            
                              
                                h
                                v
                                Q
                              
                              ⁡
                              
                                (
                                
                                  x
                                  j
                                
                                )
                              
                            
                            mod
                          
                        
                        
                          ∂
                          
                            a
                            k
                          
                        
                      
                      ·
                      
                        
                          ∂
                          
                            
                              
                                h
                                v
                                Q
                              
                              ⁡
                              
                                (
                                
                                  x
                                  j
                                
                                )
                              
                            
                            mod
                          
                        
                        
                          ∂
                          
                            a
                            
                              
                                
                              
                              I
                            
                          
                        
                      
                    
                    +
                  
                
              
              
                
                  
                    
                      
                        
                          ∂
                          
                            
                              
                                h
                                t
                                I
                              
                              ⁡
                              
                                (
                                
                                  x
                                  j
                                
                                )
                              
                            
                            mod
                          
                        
                        
                          ∂
                          
                            a
                            k
                          
                        
                      
                      ·
                      
                        
                          ∂
                          
                            
                              
                                h
                                t
                                I
                              
                              ⁡
                              
                                (
                                
                                  x
                                  j
                                
                                )
                              
                            
                            mod
                          
                        
                        
                          ∂
                          
                            a
                            
                              
                                
                              
                              I
                            
                          
                        
                      
                    
                    +
                    
                      
                        
                          ∂
                          
                            
                              
                                h
                                t
                                Q
                              
                              ⁡
                              
                                (
                                
                                  x
                                  j
                                
                                )
                              
                            
                            mod
                          
                        
                        
                          ∂
                          
                            a
                            k
                          
                        
                      
                      ·
                      
                        
                          ∂
                          
                            
                              
                                h
                                t
                                Q
                              
                              ⁡
                              
                                (
                                
                                  x
                                  j
                                
                                )
                              
                            
                            mod
                          
                        
                        
                          ∂
                          
                            a
                            
                              
                                
                              
                              I
                            
                          
                        
                      
                    
                  
                
              
            
            )
          
        
      
    
    
      
        (
        53
        )
      
    
  




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−ä)·(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
            
            ]
          
          hpII
        
        
          
            Re
            ⁡
            
              [
              
                h
                n
              
              ]
            
          
          hpII
        
      
      )
    
  
  ;




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
                                  
                                  ⁡
                                  
                                    (
                                    
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                                      j
                                    
                                    )
                                  
                                
                                meas
                              
                              -
                              
                                
                                  
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                                    n
                                    Q
                                  
                                  ⁡
                                  
                                    (
                                    
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                            ]
                          
                          2
                        
                        +
                      
                    
                  
                  
                    
                      
                        
                          
                            [
                            
                              
                                
                                  
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                                    v
                                    I
                                  
                                  ⁡
                                  
                                    (
                                    
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                                meas
                              
                              -
                              
                                
                                  
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                                    v
                                    I
                                  
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                                    (
                                    
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                                      j
                                    
                                    )
                                  
                                
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                            ]
                          
                          2
                        
                        +
                        
                          
                            [
                            
                              
                                
                                  
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                                    v
                                    Q
                                  
                                  ⁡
                                  
                                    (
                                    
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                                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
                        
                      
                    
                  
                
                }
              
            
          
        
      
    
  




  
    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
                i
                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
          max
        
      
      ≥
      
        
          (
          
            
              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
        max
      
    
    ≤
    
      
        (
        
          
            f
            0
          
          
            
              f
              0
            
            /
            2
          
        
        )
      
      
        N
        ⁡
        
          (
          
            f
            0
          
          )
        
      
    
  
  =
  
    
      2
      
        N
        ⁡
        
          (
          
            f
            0
          
          )
        
      
    
    .
  




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

Imax·100N(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
                max
              
              ·
              
                100
                
                  N
                  ⁡
                  
                    (
                    
                      f
                      0
                    
                    )
                  
                
              
            
          
          ,
          
            
              
                I
                max
              
              ·
              
                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.
Vector Mode Locate
FIG. 7 illustrates the aboveground use of a cart-based line locator 700 consistent with some embodiments of the present invention to locate the position of an underground targeted conductor 704 during a vector mode locate process. While a vector mode locate process is described in the context of a cart-based locate system, as shown in FIG. 7, embodiments of a line locator consistent with the present invention may be integrated in hand-held devices, mounted on vehicles, or integrated in any device that can be moved relative to target line 704.
As shown in FIG. 7, cart-based locate receiver 700 may include cart frame 702, display 718, one or more wheels 714, EM detector coils 712, and locator receiver electronics 716. In some embodiments, locator receiver electronics 716 may include at least some of the components included in the locator receiver shown in FIG. 2. In some embodiments, EM detector coils 712 may be internally integrated within locator receiver electronics 716. Further, in some embodiments locator receiver electronics 716 may include systems for determining the location of cart-based location receiver including, for example, GPS, real-time kinematic (RTK) GPS, odometers, gyroscopes, internal sensors, accelerometers, digital level sensors, compasses, and/or precise survey grade location equipment (e.g., laser rangefinders, ultrasonic rangefinders, and/or on-board reflectors capable of integration with a laser theodelite total station system). In certain embodiments, the systems for determining the location of cart-based location receiver may also provide confidence information related to the accuracy of the estimated location of the cart. The location of the cart and any confidence information may be provided to a geographic information system (GIS) mapping platform, which in some embodiments, may be displayed to a user via display 718.
A target line 704 energized by an electric current from a transmitter emits an EM field 708. EM field 708 may induce a current in nearby bleedover conductors such as line 706 because of resistive, inductive, and/or capacitive bleedover. The sum of the fields generated from target line 708 and/or bleedover line 710 can be detected at the surface by EM detector coils 712 of cart-based line locator 700. As shown in FIG. 7, cart-based line locator 700 may include a pair of EM detector coils 712 mounted on each side of the cart frame 702, and each EM detector coil 712 may include three EM coils configured to provide 3-dimensional electromagnetic field measurement information. In some embodiments, 3-dimensional magnetometers may be used in lieu of and/or in addition to EM coils. In some embodiments, EM detector coils 712 may be mounted on cart frame 702 within the wells of wheels 714, providing the coils some protection as the cart traverses the locate path. In general, cart-based line locator 700 may include any number of detector coils positioned in other configurations.
To estimate the location of target line 704, several complex electromagnetic field measurements taken by EM detector coils 712 can be performed as cart-based line locator 700 is traversed over target line 704 in a substantially parallel direction (i.e., substantially parallel with target line 704). These measurements, along with location based measurements taken by locator receiver electronics 716, can be compared with a model of the electromagnetic field generated by a hypothetical target line as a function of position to estimate the depth of target line 704 and the horizontal centerline coordinate. In addition, an estimate of current in target line 704 and the soil conductivity may be extracted from the results. This information may then be provided to a GIS mapping platform, which in some embodiments, may be displayed to a user via display 718.
The model consistent with some embodiments of the present invention used in vector mode assumes a single hypothetical target line locally long enough that the induced electromagnetic field is cylindrical along the line. The results of the electromagnetic phase and field strength amplitude measurements taken by EM detector coils 712 as well as location measurements from cart location systems may be provided to a model optimization module that takes into account that the complex electromagnetic field measurements were measured while traversing the target line in a substantially parallel direction. In this manner, the vector mode line detection process used by cart-based line locator 700 is similar to the previously discussed line detection methods that utilize a “walkover” technique for electromagnetic field measurements and location optimization modeling, the difference being that in vector mode, measurements are taken as the EM detector coils 712 are moved along the target line 704 and with a pair of horizontally separated EM detector coils 712, and a single conductor line is used in generating the hypothetical line model used in model optimization.
Using the previously discussed model optimization techniques, a model may be selected that best represents the measured data. A cost function can be formed using the mean square error of the predicted field (after optimization) compared to the measured data. The results of this comparison may be used to determine confidence information that describes the accuracy of the target line 704 parameters. This target line location measurement confidence information may be exported to the GIS system along with estimated target line 704 location parameters and displayed to a user via display 718. Similarly, cart location confidence information provided by cart location determination systems may also be displayed to a user. In some embodiments, cart location confidence information may be used with the target line location measurement confidence information to provide an overall confidence measurement for the line locate.
In some embodiments, the estimated location and depth of target line 704 may be determined using the above described techniques within a cart-based coordinate frame. Utilizing systems for determining the location of cart-based location receiver 700 included within locator receiver electronics 716 (e.g., gyroscopes, accelerometers, internal sensors, and/or digital level sensors), the position of the cart 700 within the cart-based coordinate frame may be determined. Further, systems for determining the location of cart-based location receiver 700 included within locator receiver electronics 716 (e.g., RTK GPS and/or precise survey grade location equipment) may be used to determine the position of the cart 700 within an earth coordinate frame. In some embodiments, the estimated location and depth of target line 704 determined in reference to the cart-based coordinate frame may be translated into an estimated location and depth of target line 704 within an earth coordinate frame. In this manner, variations in the position of the cart (e.g., pitch, yaw, and/or roll) may be accounted for in determining the estimated parameters of target line 704 within the earth based coordinate frame, which may then be exported to a GIS mapping platform and displayed to a user via display 718.
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 ak and 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
                  ⁢
                  
                    
                  
                  ⁢
                  πΔ
                
                )
              
            
          
          +
          
            
              
                I
                tx
              
              
                
                  ω
                  1
                
                -
                
                  ω
                  0
                
              
            
            ·
            
              sin
              ⁡
              
                (
                
                  2
                  ⁢
                  n
                  ⁢
                  
                    
                  
                  ⁢
                  πΔ
                
                )
              
            
          
          +
        
      
    
  
  
    
      
        
        ⁢
        
          
            
              
                I
                tx
              
              
                
                  ω
                  1
                
                -
                
                  ω
                  0
                
              
            
            ·
            
              sin
              ⁡
              
                (
                
                  2
                  ⁢
                  n
                  ⁢
                  
                    
                  
                  ⁢
                  πΔ
                
                )
              
            
          
          +
          
            
              
                I
                tx
              
              
                
                  ω
                  2
                
                +
                
                  ω
                  0
                
              
            
            ·
            
              sin
              ⁡
              
                (
                
                  2
                  ⁢
                  n
                  ⁢
                  
                    
                  
                  ⁢
                  πΔ
                
                )
              
            
          
          +
        
      
    
  
  
    
      
        
        ⁢
        
          
            
              I
              tx
            
            
              
                ω
                2
              
              -
              
                ω
                0
              
            
          
          ·
          
            [
            
              -
              
                sin
                ⁡
                
                  (
                  
                    2
                    ⁢
                    n
                    ⁢
                    
                      
                    
                    ⁢
                    πΔ
                  
                  )
                
              
            
            ]
          
        
      
    
  
  
    
      
        ≈
        
        ⁢
        
          
            
              
                I
                tx
              
              
                
                  ω
                  0
                
                ·
                Δ
              
            
            ·
            
              sin
              ⁡
              
                (
                
                  2
                  ⁢
                  n
                  ⁢
                  
                    
                  
                  ⁢
                  πΔ
                
                )
              
            
          
          +
          
            
              
                I
                tx
              
              
                
                  -
                  
                    ω
                    0
                  
                
                ·
                Δ
              
            
            ·
            
              [
              
                -
                
                  sin
                  ⁡
                  
                    (
                    
                      2
                      ⁢
                      n
                      ⁢
                      
                        
                      
                      ⁢
                      πΔ
                    
                    )
                  
                
              
              ]
            
          
        
      
    
  
  
    
      
        =
        
        ⁢
        
          
            
              2
              ⁢
              
                I
                tx
              
            
            
              
                ω
                0
              
              ·
              Δ
            
          
          ·
          
            sin
            ⁡
            
              (
              
                2
                ⁢
                n
                ⁢
                
                  
                
                ⁢
                πΔ
              
              )
            
          
        
      
    
  




leading to




  
    
      
        a
        
          2
          ⁢
          n
        
      
      ≈
      
        
          2
          ⁢
          
            I
            tx
          
          ⁢
          
            sin
            ⁡
            
              (
              
                2
                ⁢
                n
                ⁢
                
                  
                
                ⁢
                πΔ
              
              )
            
          
        
        
          
            
              (
              
                
                  T
                  1
                
                +
                
                  T
                  2
                
              
              )
            
            ·
            
              ω
              0
            
          
          ⁢
          Δ
        
      
    
    ⁢
    
      

    
    ⁢
    
      
    
    =
    
      
        
          I
          tx
        
        ⁢
        
          
            sin
            ⁡
            
              (
              
                n
                ⁢
                
                  
                
                ⁢
                πΔ
              
              )
            
          
          ·
          
            cos
            ⁡
            
              (
              
                n
                ⁢
                
                  
                
                ⁢
                πΔ
              
              )
            
          
        
      
      
        n
        ⁢
        
          
        
        ⁢
        πΔ
      
    
  




  
    
      
        (
        
          
            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
                  tx
                
                ·
                
                  cos
                  ⁡
                  
                    (
                    
                      
                        w
                        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
                            ⁢
                            
                              
                            
                            ⁢
                            πΔ
                          
                          )
                        
                      
                    
                    ]
                  
                
                +
                
                  

                
                ⁢
                
                  
                
                ⁢
                
                  
                    
                      I
                      tx
                    
                    
                      
                        ω
                        1
                      
                      -
                      
                        ω
                        0
                      
                    
                  
                  ·
                  
                    [
                    
                      
                        cos
                        ⁡
                        
                          (
                          
                            2
                            ⁢
                            n
                            ⁢
                            
                              
                            
                            ⁢
                            πΔ
                          
                          )
                        
                      
                      -
                      1
                    
                    ]
                  
                
                +
                
                  

                
                ⁢
                
                  
                
                ⁢
                
                  
                    
                      I
                      tx
                    
                    
                      
                        ω
                        2
                      
                      +
                      
                        ω
                        0
                      
                    
                  
                  ·
                  
                    [
                    
                      
                        cos
                        ⁡
                        
                          (
                          
                            2
                            ⁢
                            n
                            ⁢
                            
                              
                            
                            ⁢
                            πΔ
                          
                          )
                        
                      
                      -
                      1
                    
                    ]
                  
                
                +
                
                  

                
                ⁢
                
                  
                
                ⁢
                
                  
                    
                      I
                      tx
                    
                    
                      
                        ω
                        2
                      
                      -
                      
                        ω
                        0
                      
                    
                  
                  ·
                  
                    [
                    
                      1
                      -
                      
                        cos
                        ⁡
                        
                          (
                          
                            2
                            ⁢
                            n
                            ⁢
                            
                              
                            
                            ⁢
                            πΔ
                          
                          )
                        
                      
                    
                    ]
                  
                
              
              ⁢
              
                

              
              ⁢
              
                
              
              ≈
              
                
                  
                    
                      I
                      tx
                    
                    
                      
                        ω
                        0
                      
                      ·
                      Δ
                    
                  
                  ·
                  
                    [
                    
                      
                        cos
                        ⁡
                        
                          (
                          
                            2
                            ⁢
                            b
                            ⁢
                            
                              
                            
                            ⁢
                            πΔ
                          
                          )
                        
                      
                      -
                      1
                    
                    ]
                  
                
                +
                
                  

                
                ⁢
                
                  
                
                ⁢
                
                  
                    
                      I
                      tx
                    
                    
                      
                        -
                        
                          ω
                          0
                        
                      
                      ·
                      Δ
                    
                  
                  ·
                  
                    [
                    
                      1
                      -
                      
                        cos
                        ⁡
                        
                          (
                          
                            2
                            ⁢
                            n
                            ⁢
                            
                              
                            
                            ⁢
                            πΔ
                          
                          )
                        
                      
                    
                    ]
                  
                
              
            
            ⁢
            
              

            
            ⁢
            
              
            
            =
            
              
                
                  
                    2
                    ⁢
                    
                      I
                      tx
                    
                  
                  
                    
                      ω
                      0
                    
                    ·
                    Δ
                  
                
                ·
                
                  [
                  
                    
                      cos
                      ⁡
                      
                        (
                        
                          2
                          ⁢
                          n
                          ⁢
                          
                            
                          
                          ⁢
                          πΔ
                        
                        )
                      
                    
                    -
                    1
                  
                  ]
                
              
              ⁢
              
                

              
              ⁢
              
                
              
              =
              
                
                  
                    -
                    4
                  
                  ⁢
                  
                    I
                    tx
                  
                  ⁢
                  
                    
                      sin
                      2
                    
                    ⁡
                    
                      (
                      
                        n
                        ⁢
                        
                          
                        
                        ⁢
                        πΔ
                      
                      )
                    
                  
                
                
                  
                    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
                    ⁢
                    
                      
                    
                    ⁢
                    πΔ
                  
                  )
                
              
            
          
          
            n
            ⁢
            
              
            
            ⁢
            πΔ
          
        
        ·
        
          

        
        ⁢
        
          
        
        ⁢
        
          cos
          ⁡
          
            (
            
              
                ω
                0
              
              ⁢
              t
            
            )
          
        
      
      +
      
        
          
            
              -
              
                I
                tx
              
            
            ⁢
            
              
                sin
                2
              
              ⁡
              
                (
                
                  n
                  ⁢
                  
                    
                  
                  ⁢
                  πΔ
                
                )
              
            
          
          
            n
            ⁢
            
              
            
            ⁢
            πΔ
          
        
        ·
        
          sin
          ⁡
          
            (
            
              
                ω
                0
              
              ⁢
              t
            
            )
          
        
      
    
  
  ⁢
  
    

  
  ⁢
  
    
  
  =
  
    
      
        
          
            I
            tx
          
          ⁢
          
            sin
            ⁡
            
              (
              
                n
                ⁢
                
                  
                
                ⁢
                πΔ
              
              )
            
          
        
        
          n
          ⁢
          
            
          
          ⁢
          πΔ
        
      
      ·
      
        
          cos
          ⁡
          
            (
            
              
                
                  ω
                  0
                
                ⁢
                t
              
              +
              
                n
                ⁢
                
                  
                
                ⁢
                πΔ
              
            
            )
          
        
        .
        
          

        
        ⁢
        
          (
          
            
              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
                        2
                      
                      -
                      
                        ω
                        0
                        +
                      
                    
                  
                  ·
                  
                    sin
                    ⁡
                    
                      (
                      
                        
                          ω
                          0
                          +
                        
                        ⁢
                        
                          T
                          1
                        
                      
                      )
                    
                  
                
              
              ⁢
              
                

              
              ⁢
              
                
              
              ≈
              
                
                  
                    
                      2
                      ⁢
                      
                        nl
                        tx
                      
                    
                    
                      
                        (
                        
                          
                            2
                            ⁢
                            n
                            ⁢
                            
                              
                            
                            ⁢
                            Δ
                          
                          -
                          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
                  
                  ·
                  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
                ·
                
                  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
                      ⁢
                      
                        nI
                        tx
                      
                    
                    
                      
                        (
                        
                          
                            2
                            ⁢
                            n
                            ⁢
                            
                              
                            
                            ⁢
                            Δ
                          
                          -
                          1
                        
                        )
                      
                      ⁢
                      
                        ω
                        0
                      
                    
                  
                  ·
                  
                    [
                    
                      
                        cos
                        ⁡
                        
                          (
                          
                            
                              ω
                              0
                              +
                            
                            ⁢
                            
                              T
                              1
                            
                          
                          )
                        
                      
                      -
                      1
                    
                    ]
                  
                
                +
                
                  

                
                ⁢
                
                  
                
                ⁢
                
                  
                    
                      
                        -
                        2
                      
                      ⁢
                      
                        nI
                        tx
                      
                    
                    
                      
                        (
                        
                          
                            2
                            ⁢
                            n
                            ⁢
                            
                              
                            
                            ⁢
                            Δ
                          
                          +
                          1
                        
                        )
                      
                      ⁢
                      
                        ω
                        0
                      
                    
                  
                  ·
                  
                    [
                    
                      1
                      -
                      
                        cos
                        ⁡
                        
                          (
                          
                            
                              ω
                              0
                              +
                            
                            ⁢
                            
                              T
                              1
                            
                          
                          )
                        
                      
                    
                    ]
                  
                
              
            
            ⁢
            
              

            
            ⁢
            
              
            
            =
            
              
                
                  2
                  ⁢
                  
                    
                      nI
                      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
            -
            
              
                (
                
                  2
                  ⁢
                  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
                        2
                      
                      -
                      
                        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
                      ⁢
                      
                        nI
                        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
                  ⁢
                  
                    
                      nI
                      tx
                    
                    ·
                    4
                  
                  ⁢
                  n
                  ⁢
                  
                    
                  
                  ⁢
                  Δ
                
                
                  
                    ω
                    0
                  
                  ·
                  
                    (
                    
                      1
                      -
                      
                        
                          (
                          
                            2
                            ⁢
                            n
                            ⁢
                            
                              
                            
                            ⁢
                            Δ
                          
                          )
                        
                        2
                      
                    
                    )
                  
                
              
              ·
              
                [
                
                  1
                  -
                  
                    cos
                    ⁡
                    
                      (
                      
                        
                          ω
                          0
                          -
                        
                        ⁢
                        
                          T
                          1
                        
                      
                      )
                    
                  
                
                ]
              
            
          
        
      
    
  




leads to





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

          
          ⁢
          
            
              cos
              ⁡
              
                (
                
                  
                    
                      -
                      
                        ω
                        m
                      
                    
                    ⁢
                    t
                  
                  +
                  
                    
                      
                        φ
                        0
                      
                      -
                      
                        φ
                        0
                        +
                      
                    
                    2
                  
                
                )
              
            
            -
            
              cos
              ⁡
              
                (
                
                  
                    
                      -
                      
                        ω
                        m
                      
                    
                    ⁢
                    t
                  
                  +
                  
                    
                      θ
                      0
                      +
                    
                    ⁢
                    
                      T
                      1
                    
                  
                  +
                  
                    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
                    
                    +
                    
                      n
                      ⁢
                      
                        
                      
                      ⁢
                      πΔ
                    
                    +
                    
                      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
                  ⁢
                  
                    
                  
                  ⁢
                  πΔ
                
                +
                
                  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
              ⁡
              
                (
                
                  
                    2
                    ⁢
                    n
                    ⁢
                    
                      
                    
                    ⁢
                    π
                  
                  -
                  
                    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
                ⁡
                
                  (
                  
                    
                      2
                      ⁢
                      n
                      ⁢
                      
                        
                      
                      ⁢
                      πΔ
                      ⁢
                      
                        
                      
                      ⁢
                      n
                      ⁢
                      
                        
                      
                      ⁢
                      π
                    
                    -
                    
                      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
                      ⁢
                      
                        
                      
                      ⁢
                      πΔ
                    
                    +
                    
                      φ
                      0
                    
                    -
                    
                      
                        
                          φ
                          0
                          -
                        
                        +
                        
                          φ
                          0
                          +
                        
                      
                      2
                    
                  
                  )
                
              
              ·
              
                cos
                ⁡
                
                  (
                  
                    
                      
                        ω
                        m
                      
                      ⁢
                      t
                    
                    -
                    
                      
                        
                          φ
                          0
                          -
                        
                        +
                        
                          φ
                          0
                          +
                        
                      
                      2
                    
                  
                  )
                
              
            
          
        
        =
        
          
            
              2
              ⁢
              
                
                  cos
                  ⁡
                  
                    (
                    
                      
                        -
                        Δnπ
                      
                      +
                      
                        φ
                        0
                      
                      -
                      
                        
                          
                            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
                                ⁢
                                
                                  
                                
                                ⁢
                                πΔ
                              
                              +
                              
                                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
                      
                      +
                      Δπ
                      -
                      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
                      
                      +
                      Δπ
                      -
                      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
                        
                        +
                        Δπ
                      
                      )
                    
                  
                  -
                  
                    cos
                    ⁢
                    
                      (
                      
                        
                          
                            ω
                            m
                          
                          ⁢
                          t
                        
                        -
                        
                          
                            
                              φ
                              0
                              -
                            
                            -
                            
                              φ
                              0
                              +
                            
                          
                          2
                        
                      
                      )
                    
                  
                
                ]
              
            
          
        
        =
        
          
        
        ⁢
        
          
            
              -
              4
            
            ⁢
            
              
                cos
                ⁡
                
                  (
                  
                    Δ
                    ⁢
                    
                      
                    
                    ⁢
                    n
                    ⁢
                    
                      
                    
                    ⁢
                    π
                  
                  )
                
              
              ·
              
                cos
                ⁡
                
                  (
                  
                    φ
                    -
                    
                      
                        
                          φ
                          0
                          -
                        
                        +
                        
                          φ
                          0
                          +
                        
                      
                      2
                    
                  
                  )
                
              
              ·
              
                cos
                ⁡
                
                  (
                  
                    Δπ
                    2
                  
                  )
                
              
              ·
              sin
            
            ⁢
            
              (
              
                
                  
                    ω
                    m
                  
                  ⁢
                  t
                
                -
                
                  
                    
                      φ
                      0
                      -
                    
                    -
                    
                      φ
                      0
                      +
                    
                  
                  2
                
                +
                
                  Δπ
                  2
                
              
              )
            
          
          +
          
            4
            ⁢
            sin
            ⁢
            
              
                (
                
                  Δ
                  ⁢
                  
                    
                  
                  ⁢
                  n
                  ⁢
                  
                    
                  
                  ⁢
                  π
                
                )
              
              ·
              
                sin
                ⁡
                
                  (
                  
                    
                      φ
                      0
                    
                    -
                    
                      
                        
                          φ
                          0
                          -
                        
                        +
                        
                          φ
                          0
                          +
                        
                      
                      2
                    
                  
                  )
                
              
            
          
        
      
    
  
  ⁣
  
    
      
        ·
        
          sin
          ⁡
          
            (
            
              Δπ
              2
            
            )
          
        
        ·
        sin
      
      ⁢
      
        (
        
          
            ω
            
              m
              ⁢
              
                
              
              ⁢
              t
            
          
          -
          
            
              
                φ
                0
                -
              
              -
              
                φ
                0
                +
              
            
            2
          
          +
          
            Δπ
            2
          
        
        )
      
    
    ≈
    
      
    
    ⁢
    
      
        -
        4
      
      ⁢
      
        
          cos
          ⁡
          
            (
            
              Δ
              ⁢
              
                
              
              ⁢
              n
              ⁢
              
                
              
              ⁢
              π
            
            )
          
        
        ·
        
          cos
          ⁡
          
            (
            
              
                φ
                0
              
              -
              
                
                  
                    φ
                    0
                    -
                  
                  +
                  
                    φ
                    0
                    +
                  
                
                2
              
            
            )
          
        
        ·
        
          cos
          ⁡
          
            (
            
              Δπ
              2
            
            )
          
        
        ·
        
          cos
          ⁡
          
            (
            
              
                
                  ω
                  m
                
                ⁢
                t
              
              +
              
                Δπ
                2
              
              -
              
                
                  
                    φ
                    0
                    -
                  
                  -
                  
                    φ
                    0
                    +
                  
                
                2
              
              +
              δφ
            
            )
          
        
      
    
  








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

## Claims

1. A method for determining a location of underground cables and pipes, comprising:
measuring a set of complex electromagnetic field magnitudes and phases at a plurality of positions while traversing a target line in a substantially parallel direction using a plurality of 3-dimensional electromagnetic coil sensors, the plurality of 3-dimensional electromagnetic coil sensors being mounted on opposite sides of a cart frame within wheel wells of a cart-based location receiver such that there is a horizontal separation between the plurality of 3-dimensional electromagnetic coil sensors, and such that the plurality of 3-dimensional electromagnetic coil sensors are oriented substantially orthogonal to the target line;
modeling a set of expected complex electromagnetic field magnitudes of a single underground conductor at each of the plurality of positions to form a set of hypothesized values corresponding to a set of individual models for the target line, the modeling including 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;
determining which of the set of individuals models is a best model;
determining confidence information at each of the plurality of positions based on a comparison between the measured set of complex electromagnetic magnitudes and phases and the best model; and
determining parameters at each of the plurality of positions related to the target line from the best model.

2. The method 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, depth of the target line, and a current carried in the target line.

3. The method of claim 1, wherein the set of complex electromagnetic field magnitudes includes electromagnetic field magnitudes and phases.

4. The method of claim 1, wherein modeling the set of expected complex field magnitudes includes optimizing each of the set of individual models.

5. The method of claim 4, wherein optimizing includes applying a Levenburg-Marquardt algorithm.

6. The method of claim 1, further comprising:
determining a set of precise locations corresponding with the plurality of positions.

7. The method of claim 6, further comprising:
associating the set of precise locations with the determined parameters and determined confidence information; and
exporting the set of precise locations, the determined parameters, and the confidence information to a geographic information system (GIS) mapping environment.

8. The method of claim 1, wherein determining a set of precise locations includes determining the location of the plurality of positions within a first reference frame and translating the location of the plurality of positions within the first reference frame to a second reference frame.

9. The method of claim 1, further comprising:
generating a warning if the determined confidence information meets and/or exceeds a predetermined confidence value.

10. The method of claim 1, wherein measuring a set of complex electromagnetic field magnitudes while traversing a target line comprises traversing a path substantially parallel to the target line with the cart-based location receiver.

11. A line locator, comprising: a plurality of 3-dimensional electromagnetic coil detectors configured to measure a complex electromagnetic field magnitude and phase generated at least by a target line as the plurality of 3-dimensional electromagnetic coil detectors are moved along the target line, the plurality of 3-dimensional electromagnetic coil detectors being mounted on opposite sides of a cart frame within wheel wells of a cart-based location receiver such that there is a horizontal separation between the plurality of 3-dimensional electromagnetic coil detectors, and such that the plurality of 3-dimensional electromagnetic coil detectors are oriented substantially orthogonal to the target line;
circuitry coupled to receive signals from the plurality of 3-dimensional electromagnetic coil detectors and provide quadrature signals indicating a measured complex electromagnetic field magnitude and phase; a position locator for indicating a precise position of the line locator; a processor coupled to receive the complex electromagnetic field magnitude and phase and the position and calculate values related to the target line;
and a display coupled to the processor, the display indicating to a user the values related to the target line, wherein the processor includes software for performing the following:
modeling a set of expected complex electromagnetic field magnitudes of a single underground conductor at each of a plurality of positions determined by the position locator to form a set of hypothesized values corresponding to a set of individual models for the target line, the modeling including 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;
determining which of the set of individuals models is a best model;
determining target line location confidence information at each of the plurality of positions based on a comparison between the measured set of complex electromagnetic field magnitudes and phases and the best model;
and determining parameters at each of the plurality of positions related to the target line from the best model.

12. The line locator of claim 11, wherein the parameters related to the target line include at least one estimated value chosen from a list consisting of a centerline position, depth of the target line, and a current carried in the target line.

13. The line locator of claim 11, wherein the cart is capable of traversing a path substantially parallel to the target line.

14. The line locator of claim 13, wherein the cart includes at least one wheel.

15. The line locator of claim 14, wherein at least one of the plurality of 3-dimensional electromagnetic coil detectors is integrated within a well of the at least one wheel.

16. The line locator of claim 11, wherein the position locator further indicates locator position confidence information.

17. The line locator of claim 16, wherein the display further indicates to the user the locator position confidence information.

18. The line locator of claim 11, wherein the position locator includes an real-time kinematic GPS system.

19. The line locator of claim 11, wherein the position locator includes precise survey grade location equipment.

20. The line locator of claim 11, wherein the display further indicates to the user location information provided by the position locator.

21. The line locator of claim 11, wherein the display further indicates to the user the determined target line location confidence information.

22. The line locator of claim 11, wherein the position locator is configured to determine the position of the line locator within a line locator-based coordinate frame and within a earth-based coordinate frame.

23. The line locator of claim 22, wherein the position locator determines the precise position of the line locator by translating the determined position within the line locator-based coordinate frame into the earth-based coordinate frame.

24. A method for determining a location of underground cables and pipes, comprising:
measuring a set of complex electromagnetic field magnitudes and phases at a plurality of positions while traversing a target line in a substantially parallel direction using a plurality of 3-dimensional magnetometer sensors, the plurality of 3-dimensional magnetometer sensors being mounted on opposite sides of a cart frame within wheel wells of a cart-based location receiver such that there is a horizontal separation between the plurality of 3-dimensional magnetometer sensors, and such that the plurality of 3-dimensional magnetometer sensors are oriented substantially orthogonal to the target line;
modeling a set of expected complex electromagnetic field magnitudes of a single underground conductor at each of the plurality of positions to form a set of hypothesized values corresponding to a set of individual models for the target line, the modeling including 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;
determining which of the set of individuals models is a best model;
determining confidence information at each of the plurality of positions based on a comparison between the measured set of complex electromagnetic field magnitudes and phases and the best model; and
determining parameters at each of the plurality of positions related to the target line from the best model.

25. A line locator, comprising:
a plurality of 3-dimensional magnetometer detectors configured to measure a complex electromagnetic field magnitude and phase generated at least by a target line as the plurality of magnetometer detectors are moved along the target line, the plurality of 3-dimensional magnetometer detectors being mounted on opposite sides of a cart frame within wheel wells of a cart-based location receiver such that there is a horizontal separation between the plurality of 3-dimensional magnetometer detectors, and such that the plurality of 3-dimensional magnetometer detectors are oriented substantially orthogonal to the target line;
circuitry coupled to receive signals from the plurality of 3-dimensional magnetometer detectors and provide Quadrature signals indicating a measured complex electromagnetic field magnitude and phase;
a position locator for indicating a position of the line locator;
a processor coupled to receive the complex electromagnetic field magnitude and phase and the position and calculate values related to the target line; and
a display coupled to the processor, the display indicating to a user the values related to the target line, wherein the processor includes software for performing the following:
modeling a set of expected complex electromagnetic field magnitudes of a single underground conductor at each of a plurality of positions determined by the position locator to form a set of hypothesized values corresponding to a set of individual models for the target line, the modeling including 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;
determining which of the set of individuals models is a best model;
determining target line location confidence information at each of the plurality of positions based on a comparison between the measured set of complex electromagnetic field complex electromagnetic field magnitudes and phases and the best model; and
determining parameters at each of the plurality of positions related to the target line from the best model.