Abstract:
An apparatus and method for accurately determining the relative locations of sensors in a passive sonar or like monitoring system utilizes two non-parallel (but not necessarily orthogonal) calibrator signals to calculate the relative positions of sensors in a sensor array. Iteration, using a deformed line array constraint, permits reliable solution of the sensor position equations, even when the relative angle between the calibrator signals is initially unknown.

Description:
FIELD OF THE INVENTION 
     The present invention relates generally to the field of signal detecting and measuring systems, particularly those employing arrays of sensors and signal processing apparatus to post-process the signals received from such sensors. More specifically, the invention relates to a method of an apparatus for accurately determining the relative and/or absolute position of individual transducers in such signal-detection systems through the analysis of signals received from sources whose precise bearing is not known. 
     BACKGROUND OF THE INVENTION 
     Passive surveillance systems are widely used to detect underwater objects. These systems employ a plurality of individual signal sensors, sometimes as many as 1,000, placed at different locations surrounding the area to be monitored by the system. Because each sensor is situated at a different location, a single signal emanating from a single source will produce slightly different responses at each sensor. For example, sensors which are closer to the source will receive the signal earlier. Generally, the object of a passive surveillance system is to compare and contrast the signals received by individual sensors--looking particularly at the delays between the sensor-received signals--to determine the location of the signal source. 
     In an ideal environment, determination of the signal source location is a relatively straightforward process, and follows directly from well understood principles of analytic geometry and linear systems theory. Real systems, however, must contend with a variety of factors that substantially complicate the source location process. These factors include background noise (both environmental and system-generated), multi-path interference caused by reflection of the signal off the surface or bottom of the ocean, or simultaneous reception of multiple signals, to name a few. 
     To improve noise immunity and the ability to perform sophisticated signal-processing of the transducer-received signals, modern passive surveillance systems typically operate by (i) digitally sampling the signals at each transducer, (ii) converting each signal to the frequency domain and (iii) performing some type of statistical correlation analysis utilizing these frequency domain transducer signals. The type of correlation analysis performed varies depending upon the objective to be achieved. For example, U.S. Pat. No. 4,980,870, entitled ARRAY COMPENSATING BEAMFORMER, incorporated herein by reference, describes a passive surveillance system wherein the signals from individual transducers are mathematically combined in a manner that maximizes the reception of signals from a certain direction, and/or minimizes the reception of signals from other directions, so as to act like a highly-directional microphone pointed in the &#34;steering direction.&#34; U.S. Pat. No. 5,099,456, entitled PASSIVE LOCATING SYSTEM, incorporated herein by reference, uses a similar frequency-domain digital sampling apparatus, but instead analyzes the individual frequency-domain signals with an aim to ascertain the location of the common source from which these signals emanate. 
     In all passive surveillance systems, it is assumed that the location, or at least the relative location, of each sensor is known. This sensor location information is required to interpret the sensor-received signals in a meaningful way. Once the system knows the relative location of its sensors and the velocity of signal propagation in the particular medium, the system can anticipate the time it should take for a given signal to propagate (along a particular steering direction) from one sensor to another. By comparing the times at which the signal is actually received at various sensors to the predicted times-of-arrival, the system can determine the direction from which the signal emanates. 
     Inaccurate information regarding the sensor locations introduces serious errors into the process. Without accurate sensor position information, the system cannot predict the expected delays between sensors along given propagation paths and, as a result, will be unable to accurately compute the signal source location. If the array is instead used in a beam-forming application, the inaccuracy in sensor locations will be reflected as reduced directional specificity in the &#34;beam&#34; and/or reduced detectability of the signals monitored using the beam. 
     Unfortunately, measuring and maintaining exact sensor locations is not as simple as might appear, at least in the underwater environment. First, the long wavelengths involved often mandate sensor arrays of a mile or more. Clearly, therefore, a fixed, rigid array structure (wherein the relative sensor locations cannot shift) is impractical, or at least highly uneconomical. 
     Thus, there is a need for a method and apparatus for ascertaining the relative positions of sensors in a passive sonar system, preferably without need for high signal bandwidth in the sensors. 
     SUMMARY OF THE INVENTION 
     In light of the above, one object of the invention is an improved method and apparatus for determining the relative positions of a plurality of sensors in a passive monitoring system. 
     Another object of the invention is a method and apparatus for determining the locations of such sensors without need for expensive hardware. 
     Still another object of the invention is a method and apparatus for calibrating a passive monitoring system using calibrator signals whose bandwidth is not substantially greater than that of the signals to be monitored. 
     In accordance with one aspect of the invention, a passive monitoring system is calibrated by computing--preferably through correlation processing--the delays between sensors for each of two non-parallel calibration signals. These delays are used to formulate equations for the sensor positions (&#34;sensor position equations&#34;), where the &#34;unknown&#34; variables are the propagation angles of the calibration signals. Iterative solution, based upon assumptions about the deformation characteristics of the array, is used to find feasible values for the calibration signal angles. Once these angles are determined, the sensor position equations provide orthogonal coordinates, relative to a reference sensor, for each sensor in the array, thereby facilitating beam-forming and other phased array applications. 
     In accordance with other aspects of the invention, the calibrator signals may comprise sinusoidal signals or wideband signals, such as FM chirps or other broadband noise signals. Sinusoidal signals offer the advantage of not requiring high-bandwidth sensors for calibration purposes, and not requiring a synchronization between transmission of the calibration signal(s) and the sensors and other receiving apparatus which monitor the calibration signals. High-bandwidth calibration signals, on the other hand, offer the advantage of superior multi-path immunity. 
    
    
     BRIEF DESCRIPTION OF THE DRAWINGS AND APPENDICES 
     The present invention is described in the detailed description below, which description is intended to be read in conjunction with the following set of drawings and appendices, in which: 
     FIG. 1 depicts the overall physical organization of a passive sonar apparatus in accordance with the invention; 
     FIG. 2 depicts the process of acquiring input data from the sensors in an apparatus in accordance with the invention; 
     FIG. 3 depicts the geometric configuration from which the sensor position equations are derived; 
     FIG. 4 depicts the iteration process used to solve the sensor position equations; 
     APP. 1 contains the source code used in a present embodiment of the invention. 
    
    
     DETAILED DESCRIPTION OF THE INVENTION 
     Reference is now made to FIG. 1, which depicts the overall structure of a passive sonar apparatus in accordance with the present invention. As depicted, a passive sonar apparatus comprises an array 10 of sensors 1-n, and means 13 for communicating signals 12 received by said sensors to a computer 15, which processes said signals to, among other things, ascertain sensor locations and form beams. 
     The type of signals 12 which emanate from the sensors and the means 13 for communicating said signals to computer 15 can take many different forms. For example, sensors 10 may be analog transducers, preferably underwater piezoelectric microphones, in which case analog signals 12 are coupled by means 13, which may perform sampling and/or A/D conversion, to computer 15. Alternatively, sampling and/or A/D conversion may be performed at or proximate to the sensors themselves, in which case signals 12 will be of a sampled and/or digital form. It is also possible for means 13 to process--e.g., sample and compute FFT&#39;s--signals 12 before communicating said signals to computer 15, thus relieving the processing burden on computer 15. Those skilled in the art will recognize that numerous signal processing boards and chips are available to perform these functions. 
     Signals 12 may be communicated from sensors 1 to means 13 via any sort of energy pathway, and may be communicated by any means of transmission, including electrical, optical, acoustical, electro-magnetic, magnetic, electro-mechanical, mechanical, or a combination thereof. In a presently preferred embodiment, signals 12 are digitized at the Nyquist frequency, and communicated to the shore using fiber optic cables. 
     Reference is now made to FIG. 2, which depicts the input of calibration signals in accordance with the invention. As depicted, signals received by sensors 1-n are provided--by whatever means and in whatever form--to an FFT unit 22, which outputs a plurality of time-sampled FFT coefficient vectors z 1  -z n , where vector z k  contains the FFT coefficients derived from sensor k. (It is, of course, not required that input data be stored or organized as &#34;vectors&#34;; vector algebra formulations are used merely to facilitate concise description of the sequence of operations performed in accordance with the invention.) FFT unit 22 preferably performs a vernier filtering operation, so as to focus on the frequency of the calibrator signal of interest. 
     Delay Computation 
     A preliminary step in the source location process is to convert the narrow-band FFT measurements into relative range delays from sensor 1, which is used as a reference, to other sensors in the array. This delay computation is preferably performed twice, once for each calibrator signal. 
     Recall that z n  represents a vector, whose elements are a series of FFT coefficients corresponding to the signal received at sensor n at successive time intervals. A complex coherence vector v can be computed as follows: ##EQU1## where z n  &#39; represents the complex conjugate transpose of z n . (In theory, the summation operation is not actually necessary, but is preferred since its averaging effect tends to enhance the accuracy of the complex coherence calculation.) The phase delays, in radians, between sensor 1 and sensor n are computed as follows: 
     
         g=angle(v)rads 
    
     where g i  is the relative phase delay between sensors 1 and sensor i. Because the distance between sensor 1 and i may exceed a wavelength, it is preferable to &#34;unwrap&#34; the phase delays. Unwrapping is performed using an assumption that the phase difference between sensors does not exceed 180 degrees, or, if one or more of the sensors is defective, using a deformed line array assumption, as described below. The MATLAB™ software package provides a routine for performing the preferred unwrapping. The unwrapped phase delays are: 
     
         q=unwrap(g)rads 
    
     The phase delays q are converted to range delays as follows: 
     
         r=q(rads)×λ(meters/cycle)/2π(rads/cycle) 
    
     where λ is the wavelength of the calibrator sinusoid. As used below, the range delays from the first calibrator sinusoid are contained in the vector r 1 , while those from the second appear in the vector r 2 . 
     Orthogonal Mapping 
     Using range-delay vectors r 1  (i.e. the range delays, relative to sensor 1, from the first calibrator signal) and r 2  (i.e. the range delays for the second calibrator signal), one can formulate &#34;sensor position equations&#34; to determine the sensor positions, relative to sensor 1, in orthogonal x-y coordinates. 
     A vector d contains these sensor positions, where 
     
         d=x+jy 
    
     and d i  represents the x-y coordinates of sensor i relative to sensor 1. Referring to FIG. 3, and assuming a planar wavefront from each of the calibrators, the following 2n equations for x and y can be specified: 
     
         r.sub.1 =-[sin(Θ.sub.1)]x -[cos(Θ.sub.1)]y 
    
     
         r.sub.2 =-[sin(Θ.sub.2)]x -[cos(Θ.sub.2)]y 
    
     where Θ 1  is the angle from the y-axis to the direction of propagation of the first calibrator and Θ 2  is the angle to the second calibrator. Solving these equations for x and y, one obtains the following sensor position equations: 
     
         x=[(-r.sub.1 /cos(Θ.sub.1)tan(Θ.sub.2)) -r.sub.2 /sin(Θ.sub.2)]/[1-tan(Θ.sub.1)/tan(Θ.sub.2)] 
    
     
         y=-r.sub.1 /cos(Θ.sub.1)-xtan(Θ.sub.1) 
    
     If Θ 1  and Θ 2  are precisely known, the sensor position equations can be easily solved by well-known techniques to provide the relative x-y coordinates of the sensors. In actual underwater environments, however, Θ 1  and Θ 2  typically vary 0 to 15 degrees from the expected values, thus causing traditional solution techniques to diverge. Iteration provides a practical means for solving these equations. 
     Iterative Solution of Position Equations 
     Without accurate values for Θ 1  and Θ 2 , the x and y equations do not accurately predict the relative x-y locations of the sensors. Thus, iteration is used to find values for Θ 1  and Θ 2  which, when applied in the x and y equations, yield accurate sensor location results. The difficulty, even with iteration, is knowing whether a proposed set of Θ 1  /Θ 2  values yields accurate x-y sensor locations when the &#34;actual&#34; sensor locations are unknown. In accordance with the invention, use of a deformed line-array constraint allows the sensor position equations to be solved iteratively, at least for most cases of practical interest. 
     The deformed line-array constraint can be explained with reference to FIG. 1. It assumes that sensors 1-n are distributed along a line, which may be straight or (as depicted) deformed, with bends at the sensor locations. As illustrated, there is a segment S k ,k+1 between sensors k and (k+1). The deformed line-array constraint assumes that these individual segments are straight. With this assumption, it is apparent that the distance between adjacent sensors remains constant; these distances are simply the length of the segments connecting the sensors, since the individual segments are assumed to be straight. Since the individual segment lengths remain constant, so does the total length of all segments. It is this total length, which is known a priori, that is used to guide the iterative process using the deformed line-array assumption. 
     The deformed line-array constraint can be mathematically expressed as follows. Assuming some estimated Θ 1  and Θ 2 , one can use the previously described x and y equations to derive a vector of estimated sensor positions d=x+jy, where d depends upon the Θ 1  and Θ 2  estimates, and the measured range delays r 1  and r 2 . Using these estimated sensor positions d, total line length (assuming a deformed line array) can be computed as follows: ##EQU2## where d i  is the estimated position of sensor i. The difference between this estimated line length D, and the actual line length D, is used to guide the iteration process. Although this deformed line array assumption does not perfectly model the physical geometry of the system, it yields acceptable results, even in the face of one or more severe deformations. 
     Reference is now made to FIG. 4, which shows a flowchart of the iteration process. Iteration begins with initial calibrator angle Θ 1  /Θ 2 , estimates 32, and iterates to converge on a mathematically consistent value for Θ 1 . Field tests reveal that the initial estimates typically vary about 0-15 degrees from the actual values. A position estimation step 33 computes &#34;estimated&#34; sensor positions using the calibrator angle estimates and the measured range delays 30. Step 34 computes the total estimated length D of the array, using the deformed line-array assumption and the sensor position estimates computed in step 33. Step 35 compares the estimated length D to the actual line length D. Based on this comparison, step 36 makes a determination whether the sensor position estimates are accurate, in which case the iteration ends at step 37, or whether additional iteration is desirable, in which case step 38 modifies the estimate of calibrator angle Θ 1  and the iteration process returns to position estimation step 33. 
     Estimate modification step 38 operates by comparing the &#34;errors&#34; (i.e., the difference between the estimated and actual lengths, D--D) for the current iteration to that from the previous iteration to determine an appropriate modification for the Θ 1  estimate. Those skilled in the art will recognize that this Θ 1  update can be computed using a number of well-know techniques, such as adding or subtracting fixed angular increments according to the sign of the error, or updating according to the slope (or gradient) of the error function. Preferably, angular estimate updating is performed using knowledge about the nominal shape of the error function (or its transfer function), which permits convergence in approximately nine iterations. 
     Once the error is within acceptable bounds, final orthogonal positions of the sensors are computed, and the array is calibrated for operation. 
     Use of FM Chirps or Wideband Noise 
     If bandwidth and other constraints permit, one can employ FM chirp or other wideband noise type calibrator signals instead of sinusoids. With such signals, the delay computation step is performed by cross-correlating the sensor-1 waveform with respective waveforms at each other sensor to determine the relative time delays; no FFT processing or phase unwrapping is required. Since velocity is known, these time delays are easily converted into range delays, from which the invention further operates as described above. 
     Source Code 
     To ensure complete satisfaction of the disclosure obligations under 35 U.S.C. §112, attached APP. 1 contains a listing of the source code utilized in connection with a present embodiment of the invention. This source code is written in the MATLAB™ language, which permits compact expression of the required mathematical computations. (Of course, those skilled in the art will recognize that the invention could alternatively be implemented using a wide variety of available programming languages or, if desired, entirely in hardware.) 
     APP. 1 contains the following five sub-modules: 
     (1) pa2×1 a.m: CPL Measurements Start (pages 1-1 to 1-9). This is essentially a data input module for collecting and storing samples of the sinusoidal calibrator signals. 
     (2) pa×1b.m: CPL Measurements End (pages 2-1 to 2-8): This module, among other things, calculates the phase delays, and converts these into range delays for use in the mapping process. 
     (3) pa×2.m: CPL X-Y Map Iteration (pages 3-1 to 3-8): This module performs the iterative sensor position estimation. 
     (4) pa2×3.m: CPL Polar Mapping (pages 4-1 to 4-5): This module converts the computed relative x-y sensor positions to polar coordinates oriented to true North. 
     (5) pa×fm.m: FM Range Delay Measurement (pages 5-1 to 5-4): This is an alternative to modules (1)-(2); it computes range delays using FM chirp or other broadband noise type calibrator signals. ##SPC1##