Patent Publication Number: US-8116169-B2

Title: Active sonar system and active sonar method using noise reduction techniques and advanced signal processing techniques

Description:
FIELD OF THE INVENTION 
     This invention relates generally to active sonar systems and, more particularly, to active sonar systems and methods that provide reduced noise and also advanced processing techniques that result in improved detection of underwater objects, for example, underwater mines. 
     BACKGROUND OF THE INVENTION 
     It is known that an underwater vessel (i.e., a submarine) generates sound as it travels through the water. The sound is generated by a variety of sources, including, but not limited to, sound generated by a submarine propulsion system, sound generated by a submarine propeller, and sound generated by a submarine electrical power generator. It is known that submarine designers attempt to reduce these and other sound sources in order to make a submarine difficult to detect by passive acoustic means, therefore remaining as covert as possible. 
     Some water-born objects do not emit sound, for example, underwater mines. These objects cannot be detected by the sounds they make. 
     While a conventional passive sonar system merely listens for sounds made by a target of interest, a conventional active sonar system transmits a burst of acoustic energy, called a “ping,” which travels at the speed of sound through the water. Some of the acoustic energy reflects from objects in or on the water back toward the active sonar system. These reflections, referred to as “echoes,” are received by acoustic sensors at the active sonar system. After a sufficient amount of time has passed, the active sonar system transmits another ping and repeats the above process. 
     Both active and passive sonar systems must operate in an environment filled with acoustic noises generated by a variety of noise sources, including, but not limited to, ships, surface waves, wind, geologic noises, and biologic noises. 
     Detection electronics, which forms a part of the active sonar system, performs processing upon the received echoes to improve the likelihood that only echoes from targets of interest are identified and reported to a sonar system operator. However, as described above, the undersea acoustic environment is very noisy, and despite the application of sophisticated detection processing algorithms, the active sonar system may still falsely identify random bursts of noise as targets. These false detections are referred to as “false alarms.” If the consequences of reporting a false alarm are severe, then steps can be taken to further reduce a probability of the false alarms, but usually these steps also reduce the probability that a real target of interest will be detected. 
     A variety of approaches have been used in sonar systems to improve performance in the presence of the noisy ocean environment. For example, both active and passive sonar systems tend to do receive and/or transmit beamforming. Receive beamforming, for both passive and active sonar systems, tends to result in blocking out of directions from which noises may come. Transmit beamforming, for active sonar systems, tends to result in higher power in a transmit beam, and therefore, a stronger echo from an object in or on the water. 
     Another approach used in sonar systems to improve performance is a matched-filter technique, which will be understood to those of ordinary skill in the art to take a variety of forms in the time or frequency domains. 
     Another approach used in active sonar systems to improve performance is a “chaining algorithm” that attempts to identify echoes that appear in adjacent ping cycles at ranges consistent with a real target moving at a realistic speed. 
     It is known that sound can travel through the water in so-called “propagation paths,” which can be non-straight paths, particularly when the propagation paths extend over appreciable distances, e.g., miles. The propagation paths can be modeled with propagation models. Some propagation models assume that the sound travels in straight propagation paths. These models are often referred to as isovelocity models, since they presume that sound travels at the same sound speed at all water depths. Other propagation models do not assume that the sound travels in straight propagation paths. These models, which are sometimes referred to as “ray trace” models, can be used to more accurately predict the sound propagation paths and the resulting sound that arrives at a point in the ocean, for example, at a sonar system that receives passive sound from an underwater target. Other propagation models accomplish the equivalent function but are less computationally convenient. 
     As is also known, sound that travels underwater can often take more than one propagation path. For example, sound can take a “direct propagation path” from a sound source to a sound receiver, which path may curve but not intercept the surface or bottom of the ocean. The sounds can also travel upward from the sound source, on a so-called “surface reflected path,” reflecting (or scattering) from the surface of the water and traveling downward to the sound receiver. The sound can also travel downward from the sound source, on a so-called “bottom reflected path,” reflecting (or scattering) from the bottom of the water basin and traveling upward to the sound receiver. The sound can also take a variety of other propagation paths, having, for example, both a surface and a bottom reflection (or scattering) or more than one surface and bottom reflection (or scattering). 
     Through there exist a very large number of sound propagation paths between a sound source and a sound receiver, some of the propagation paths are dominant, i.e., sound received at a sound receiver will have an intensity largely from the dominant sound paths. In particular, because sound tends to lose intensity each time it reflects or scatters from the surface or the bottom, the propagation paths having the strongest sound intensity when received at a sound receiver tend to be the direct path, the surface reflected path, and the bottom reflected path. However, a surface to bottom reflected path and a bottom to surface reflected path can also be considered as well as paths with multiple boundary contacts. 
     Conventional active sonar systems tend to operate with direct sound paths between the active sonar system and the target of interest. However, conventional active sonar systems also experience (i.e., receive) sound reflecting from the ocean surface and from the ocean bottom. Active sonar systems must distinguish an echo from a target from a reflection from the oceans surface or from the ocean bottom. 
     It would be desirable to provide new approaches used in active sonar systems to improve performance in the presence of the noisy ocean environment. For example, it would be desirable to provide improved detection and localization of objects in the water. 
     SUMMARY OF THE INVENTION 
     The present invention provides an active sonar system and method having advanced noise reduction techniques and advanced processing techniques so as to better detect and localize an object in the water. 
     In accordance with one aspect of the present invention, a sonar system includes a noise reduction module coupled to receive a plurality of receive signals. The plurality of receive signals is representative of sound propagating in water from a respective plurality of different pointing directions. The plurality of receive signals is associated with a plurality of acoustic receive elements disposed in the water. The noise reduction module is configured to select a reference one of the plurality of receive signals, wherein the reference one of the plurality of receive signals comprises a combination of a noise signal portion and a target echo signal portion. The noise reduction processor is further configured to combine the plurality of receive signals resulting in a reduction of the noise signal portion to generate a corresponding noise reduced signal associated with the reference one of the plurality of receive signals. 
     In accordance with another aspect of the present invention, a method of sonar processing includes receiving a plurality of receive signals. The plurality of receive signals is representative of sound propagating in water from a respective plurality of different pointing directions. The plurality of receive signals is associated with a plurality of acoustic receive elements disposed in the water. The method also includes selecting a reference one of the plurality of receive signals, wherein the reference one of the plurality of receive signals comprises a combination of a noise signal portion and a target echo signal portion. The method also includes combining the plurality of receive signals resulting in a reduction of the noise signal portion to generate a corresponding noise reduced signal associated with the reference one of the plurality of receive signals. 
     In accordance with another aspect of the present invention, a computer-readable storage medium having computer readable code thereon for providing sonar processing includes instructions for receiving a plurality of receive signals with. The plurality of receive signals is representative of sound propagating in water from a respective plurality of different pointing directions. The plurality of receive signals is associated with a plurality of acoustic receive elements disposed in the water. The computer-readable storage medium also includes instructions for selecting a reference one of the plurality of receive signals, wherein the reference one of the plurality of receive signals comprises a combination of a noise signal portion and a target echo signal portion. The computer-readable storage medium also includes instructions for combining the plurality of receive signals resulting in a reduction of the noise signal portion to generate a corresponding noise reduced signal associated with the reference one of the plurality of receive signals. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       The foregoing features of the invention, as well as the invention itself may be more fully understood from the following detailed description of the drawings, in which: 
         FIG. 1  is a block diagram showing a sonar system in accordance with the present invention comprising a computer processor having a noise reduction module, a cross-term deleted Wigner representation (CDWR) detection statistic module, and a detection and localization module, wherein the computer processor is configured to couple to a receive array; 
         FIG. 2  is a block diagram of an exemplary receive array as a planar array to which the signal processor of  FIG. 1  can be coupled; 
         FIG. 2A  is a graph showing an exemplary vertical beampattern associated with the receive array of  FIG. 2 ; 
         FIG. 2B  is a graph showing an exemplary horizontal beampattern associated with the receive array of  FIG. 2 ; 
         FIG. 2C  is a graph showing another exemplary vertical beampattern associated with the receive array of  FIG. 2 ; 
         FIG. 2D  is a graph showing another exemplary horizontal beampattern associated with the receive array of  FIG. 2 ; 
         FIG. 3  is a graph showing another exemplary receive array as a line array to which the signal processor of  FIG. 1  can be coupled; 
         FIG. 3A  is a graph showing an exemplary vertical beampattern associated with the receive array of  FIG. 3 ; 
         FIG. 3B  is a graph showing an exemplary horizontal beampattern associated with the receive array of  FIG. 3 ; 
         FIG. 4  is a block diagram showing further details of the noise reduction processor of  FIG. 1 ; 
         FIG. 5  is a flow chart showing a processing method that can be performed by the noise reduction processor of  FIGS. 1 and 4 ; and 
         FIG. 6  is a flow chart showing a processing method that can be performed by the CDWR detection statistic module and the detection and localization module of  FIG. 1 . 
     
    
    
     DETAILED DESCRIPTION OF THE INVENTION 
     Referring to  FIG. 1 , an exemplary sonar system  10  includes a transmitter  12  configured to generate one or more transmit signals  12   a . A transmit array  14  disposed in the water, for example, in the ocean, is coupled to receive the one or more transmit signals  12   a  and configured to project sound into the water. The sound projected into the water can be a single frequency pulse, a multi-frequency pulse, a frequency modulated continuous wave (FMCW) sweep, a spread spectrum pulse (for example, a pseudo-random noise pulse), or any combination thereof. 
     In some arrangements, sound transmitted by the transmit array  14  is substantially omnidirectional in horizontal and vertical planes. In other arrangements, the sound transmitted by the transmit array  14  is somewhat directional in at least one of the horizontal or vertical planes, for example, limited to approximately one hundred eighty degrees. In still other arrangements, the sound transmitted by the transmit array  14  is contained in one or more directional beams, for example, beams having beamwidths of less than forty-five degrees in at least one of the horizontal or vertical planes. 
     The sonar system  10  also includes a receive array  16  disposed in the water and configured to receive sound propagating in the water. As will become apparent from discussion below in conjunction with  FIGS. 2-3B , the receive array can be in one of a variety of forms and can be used to generate a plurality of directional receive beampatterns, each pointing to receive sound from a different respective direction. The plurality of directional receive beampatterns provides a corresponding plurality of receive signals. 
     As will become apparent from discussion below, in some embodiments, the receive array  16  can have a plurality of receive elements, each of which can have a respective directional beampattern to provide the plurality of directional receive beampatterns and the corresponding plurality of receive signals. However, in other embodiments, each one of the plurality of receive element of the receive array  16  can have a substantially non-directional (i.e., omnidirectional) or widely directional beampattern, in which case, electronic beamforming techniques can be used to generate the plurality of directional receive beampatterns and the corresponding plurality of receive signals. 
     The receive array  16  itself is configured to generate a plurality of signals  16   a . A receiver  18  is coupled to receive the plurality of signals and configured to generate a plurality of receive signals  18   a . As described above, in some embodiments, the receiver  18  can include a beamformer configured to generate the plurality of receive signals  18   a . However, in other embodiments, the receiver  18  does not include a beamformer and the plurality of receive signals  18   a  is representative of the plurality of signals  16   a , but in amplified form. 
     The sonar system  10  can include a timer  20  coupled to the transmitter  12  and to the receiver  18 . The timer  20  is configured to provide timing signals  20   a ,  20   b  to result in the transmit signals  12   a  and the corresponding plurality of receive signals  18   a  in an active sonar timing pattern. 
     It should be understood that blocks identified as modules within a computer processor  22  can be hardware module, software modules, or any combination of hardware and software modules. 
     The computer processor  22  can include a noise reduction module  24  coupled to receive the plurality of receive signals  18   a . The plurality of receive signals  18   a  are representative of sound propagating in water from a respective plurality of different pointing directions, which are described more fully below in conjunction with  FIGS. 2-3B . The plurality of receive signals  18   a  is associated with a plurality of acoustic receive elements disposed in the water, which elements are within the receive array  16 . The noise reduction module  24  is configured to select a reference one of the plurality of receive signals  18   a . The reference one of the plurality of receive signals comprises a combination of a noise signal portion and a target echo signal portion associated with sound transmitted into the water by the transmit array  14  and returned as an echo from an object in the water. The noise reduction processor  24  is further configured to combine the plurality of receive signals resulting in a reduction of the noise signal portion to generate a corresponding noise reduced signal  24   a  associated with the reference one of the plurality of receive signals  18   a.    
     From discussion below in conjunction with  FIG. 6 , it will become apparent that, in some embodiments, the noise reduced signal  24   a  can include one or more noise reduced signals  24   a  corresponding to one or more respective selected ones of the plurality of receive signals  18   a . Thus the various function described below in conjunction with  FIGS. 1-4  can be applied either in series or in parallel to the more than one noise reduced signal  24   a . However, for clarity, only one noise reduced signal  24   a  and operations thereupon are discussed herein, with the exception of  FIG. 6 . 
     A cross-term deleted Wigner representation (CDWR) detection statistic module  26  can be coupled to receive the noise reduced signal  24   a . The CDWR function is described more fully below. Let it suffice here to say that the CDWR detection statistic module  26  can be configured to apply a CDWR function to the noise reduced signal  24   a . The CDWR function results in a representation of the noise reduced signal  24   a  in frequency and in time. 
     The CDWR detection statistic module  26  can also be configured to generate one or more detection statistic values  26   a  by computing products of cross-term deleted Wigner representations. The products are described more fully below. Let it suffice here to say that the CDWR detection statistic module  26  can apply an auto-CDWR function to the noise reduced signal, can apply a cross-CDWR function to a combination of the noise reduced signal  24   a  with the transmit signal  12   a , and can cross-correlate the auto-CDWR function and the cross-CDWR function. This results in the one or more detection statistic values  26   a.    
     In some embodiments, the computer processor  22  can also include a detection and localization module  28  coupled to receive the one or more detection statistic values  26   a . The detection and localization module  28  can be configured to determine a detection threshold, configured to compare the one or more detection statistic values to the detection threshold to determine a detection of an object in the water, e.g., a target, e.g., a mine, and configured to generate a target detection and localization signal  28   a  indicative of a direction (azimuth and elevation) of a target in the water, e.g., a mine, and a distance to the target. 
     Further details and operation of the noise reduction module  24  are described below in conjunction with  FIGS. 4 and 5 . Further details and operation of the CDWR detection statistic module  26  and the detection and localization module  28  are described below in conjunction with  FIG. 6 . 
     In some embodiments, the computer processor  22  can also include an other signal processing module  38  coupled to receive the plurality of receive signals  18   a  and configured to generate another one or more detection statistic values  38   a . The computer processor  22  can also include an other detection and localization module  40  coupled to receive the another one or more detection statistic values  38   a  and configured to generate another signal  40   a  indicative of the direction (azimuth and elevation) of the target in the water, e.g., a mine, and a distance to the target. The other signal processing module  38  can include one of a variety of conventional techniques, for example, a matched filter, and the other detection and localization module  40  can include a variety of conventional techniques, for example, noise-based thresholding. The other detection and localization module  40  can be configured to generate another target detection and localization signal  40   a.    
     The computer processor  22  can also include a combining module  30  coupled to receive the target detection and localization signals  28   a ,  40   a  and configured to generate a combined detection and localization signal  30   a . A computer  32  with a computer display  32   a  can be coupled to receive the combined detection and localization signal  30   a  and configured to display combined detection and localization signal information on the display  32   a . In some embodiments, the computer  32  can also include another computer processor  32   b  and a keyboard  32   c.    
     In some embodiments the computer processor  22  does not include the other signal processing module  38 , the other detection and localization module  40 , or the combining module  30 , in which case the computer  32  is coupled to receive the target detection and localization signal  28   a.    
     In some embodiments, the computer processor  22  can be coupled to other storage apparatus, for example, a disk storage  36  and other storage  34 , for example, random access memory. 
     It will be understood that the modules  24 ,  26 ,  28 ,  38 ,  40 ,  30  within the computer processor  22  can be software modules stored within associated program memory, which can be random access memory within the computer processor  22 , and the random access memory can be loaded with instructions that originate from the disk storage  36 . 
     Referring now to  FIG. 2 , in some embodiments, an array  50  having a plurality of acoustic elements (shown as small circles) can be the same as or similar to the receive array  16  of  FIG. 1 . The array  50  is shown to be a planar array but can be can be a planar array or a volumetric array, for example, an array of elements conformal to a non-planar hull of a water-borne vessel. 
     The elements of the array  50  are identified by particular nomenclature, wherein the letter “i” is representative of a row of the array, and the letter “j” is representative of a column of the array. Thus, an element labeled as (i,j) is at the ith row and the jth column. 
     The term “subarray” is used herein in two different but similar ways. In a first context, wherein physical array elements provide beampatterns without additional beamforming, the term subarray is used to indicate a selected plurality of elements of the array  50  (equivalent to beams) that are used to provide some of or all of the above-described plurality of receive signals, e.g., the plurality of receive signals  18   a  of  FIG. 1 . An exemplary subarray  50   a  within the receive array  50  includes five array elements, namely elements (i, j−1), (i,j), (i, j+1), (i+1,j), and (i−1,j). The darker element designated as (i,j) is representative of the above-described reference element selected from among the selected plurality of elements (i, j−1), (i,j), (i, j+1), (i+1,j), and (i−1,j). 
     In a second context, wherein physical array elements are used with beamforming as may be provided by the beamformer  18  of  FIG. 1 , to provide beams, the term subarray is used to indicate a selected plurality of beams associated with the array  50  that are used to provide some of or all of the above-described plurality of receive signals, e.g., the plurality of receive signals  18   a  of  FIG. 1 . Thus, in both contexts, the term subarray is used to describe a set of beams that may or may not map directly to an equal number of array elements. However, in examples set forth below, unless otherwise specified, the array elements are assumed to be associated with corresponding directional beams, and therefore, a subarray is assumed to be comprised of directional array elements. 
     As described above, it should be understood that the reference element (i,j) and the selected subarray  50   a  are processed by the noise reduction module  24  of  FIG. 1  to provide a noise reduced signal, e.g., the noise reduced signal  24   a  of  FIG. 1 , which is representative of a noise reduced version of a receive signal associated with the single reference element (i,j) (or a receive beam associated with the reference element (i,j). Other subarrays and other reference elements from within the receive array  50  can be processed in series or in parallel with the reference element (i,j) and the selected subarray  50   a  in order to provide noise reduced signals representative of noise reduced versions of receive signals associated with the other reference elements (or other beams). 
     While the exemplary subarray  50   a  is shown having five array elements, a subarray can have any number of array elements greater than one array element. 
     Referring now to  FIG. 2A , in which like elements of  FIG. 2  are shown having like reference designations, the elements (i+1,j), (i,j−1), (i,j), (i,j+1) and (i−1,j) of the subarray  50   a  provide vertical beampatterns  54 ,  56 ,  58 ,  60 ,  62  respectively, each directed to receive sound in the water from a different pointing direction. Vertical beampatterns  56 ,  58 ,  60  overlay each other in this view. 
     Referring now to  FIG. 2B , in which like elements of  FIG. 2  are shown having like reference designations, the elements (i, j−1), (i+1,j), (i,j), (i−1,j), and (i,j+1) of the subarray  50   a  provide horizontal beampatterns  66 ,  68 ,  70 ,  72 ,  74 , respectively, each directed to receive sound in the water from a different pointing direction. Horizontal beampatterns  68 ,  70 ,  72  overlay each other in this view. 
     While both vertical beampatterns  52  and horizontal beampatterns  64  are shown to have different pointing directions, in some embodiments, only the vertical beampatterns  52  or only the horizontal beampatterns  64  have different pointing directions. 
     It should be recognized that the beampatterns  52 ,  64  are shown to have narrow beam angles for clarity, however, in some embodiments, either the vertical beampatterns  52  or the horizontal beampatterns  64 , or both, have wide beam angles, for example, forty-five degrees, and the wide beam angles can overlap each other. 
     It should be understood that beampatterns  52 ,  64  of  FIGS. 2A and 2B  are indicative of array elements that have directional beampatterns without electronic beamforming and without spatial beamforming between elements of the array  50 . In these embodiments, the beampatterns  52 ,  64  each have phase centers at respective elements of the array  50 . 
     Referring now to  FIG. 2C , in which like elements of  FIG. 2  are shown having like reference designations, the elements (i+1,j), (i, j−1), (i,j), (i, j+1), and (i−1,j) of the subarray  50   a  provide vertical beampatterns  54   a ,  56   a ,  58   a ,  60   a ,  62   a , respectively, each directed to receive sound in the water from a different pointing direction. Vertical beampatterns  56   a ,  58   a ,  60   a  overlay each other in this view. 
     Referring now to  FIG. 2D , in which like elements of  FIG. 2  are shown having like reference designations, the elements (i, j−1), (i+1,j), (i,j), (i−1,j), and (i,j+1) of the subarray  50   a  provide horizontal beampatterns  66   a ,  68   a ,  70   a ,  72   a ,  74   a , respectively, each directed to receive sound in the water from a different pointing direction. Horizontal beampatterns  68   a ,  70   a ,  72   a  overlay each other in this view. 
     While both vertical beampatterns  52   a  and horizontal beampatterns  64   a  are shown to have different pointing directions, in some embodiments, only the vertical beampatterns  52   a  or only the horizontal beampatterns  64   a  have different pointing directions. 
     It should be recognized that the beampatterns  52   a ,  64   a  are shown to have narrow beam angles for clarity, however, in some embodiments, either the vertical beampatterns  52   a  or the horizontal beampatterns  64   a , or both, have wide beam angles, for example, forty-five degrees, and the wide beam angles can overlap each other. 
     It should be understood that beampatterns  52   a ,  64   a  of  FIGS. 2C and 2D  are indicative of array elements that have non-directional (i.e., omnidirectional) beampatterns, and the directional beampatterns  52   a ,  64   a  are provided by electronic beamforming (e.g.,  18  of  FIG. 1 ) for combining elements of the array  50  ( FIG. 2 ). In these embodiments, the beampatterns  52   a ,  64   a  each have phase centers at the selected reference element (i,j) of the array  50 . 
     Referring now to  FIG. 3 , in some embodiments, an array  100  having a plurality of acoustic elements (shown as small circles) can be the same as or similar to the receive array  16  of  FIG. 1 . The array  100  is shown to be a straight line array but can be can be a straight line array or a curved line array, for example, a line array of elements conformal to a non-planar hull of a water-borne vessel. 
     The elements of the array  100  are identified by particular nomenclature, wherein the letter “i” is representative of an element of the line array, and the letter “j,” here a “1,” is representative of a single column of the line array. Thus, an element labeled as (i,l) is at the ith element and the line array identified as “1.” 
     The term “subarray” is used herein to indicate a selected plurality of elements of the array  100  that are used to provide some of or all of the above-described plurality of receive signals, e.g., the plurality of receive signals  18   a  of  FIG. 1 . An exemplary subarray  100   a  within the receive array  100  includes three array elements, namely elements (i+1, 1), (i,1), (i−1, 1). The darker element designated as (i,1) is representative of the above-described reference element selected from among the selected plurality of elements (i+1, 1), (i,1), (i−1, 1). 
     As described above, it should be understood that the reference element (i,l) and the selected subarray  100   a  are processed by the noise reduction module  24  of  FIG. 1  to provide a noise reduced signal, e.g., the noise reduced signal  24   a  of  FIG. 1 , which is representative of a noise reduced version of a receive signal associated with the single reference element (i,1) (or a receive beam associated with the reference element (i,j). Other subarrays and other reference elements from within the receive array  100  can be processed in series or in parallel with the reference element (i,1) and the selected subarray  100   a  in order to provide noise reduced signals representative of noise reduced versions of receive signals associated with the other reference elements (or other beams). 
     While the exemplary subarray  100   a  is shown having three array elements, a subarray can have any number of array elements greater than one array element. 
     Referring now to  FIG. 3A , in which like elements of  FIG. 3  are shown having like reference designations, the elements (i+1,1), (i,1), and (i−1, 1) of the subarray  100   a  provide vertical beampatterns  102 ,  104 ,  106 , respectively, each directed to receive sound in the water from a different pointing direction. 
     Referring now to  FIG. 3B , in which like elements of  FIG. 3  are shown having like reference designations, the elements i+1,1), (i,1), and (i−1, 1) of the subarray  100   a  provide horizontal beampatterns  112 ,  114 ,  116 , respectively, each of which can be substantially omnidirectional. Horizontal beampatterns  112 ,  114 ,  116  overlay each other in this view. 
     It should be recognized that the vertical beampatterns  104 ,  106 ,  108  are shown to have narrow beam angles for clarity, however, in some embodiments, the vertical beampatterns  104 ,  106 ,  108  have wide beam angles, for example, forty-five degrees, and the wide beam angles can overlap each other. 
     It should be understood that beampatterns  102 ,  110  of  FIGS. 3A and 3B  are indicative of array elements that have non-directional (i.e., omnidirectional) beampatterns, and the directional beampatterns  102  are provided by electronic beamforming (e.g.,  18  of  FIG. 1 ) for combining elements of the array  100  ( FIG. 3 ). In these embodiments, the beampatterns  102 ,  110  each have phase centers at the selected reference element (i,j) of the array  100 . In other embodiments, it will be understood, that like the beams of  FIGS. 2A and 2B , the elements of the array  100  can have directional patterns and beampatterns similar to the beampatterns  102 ,  110  can be formed without beamforming, in which case the phase centers of the beampatterns will be at associated ones of the array elements. 
     Referring now to  FIG. 4 , in which like elements of  FIG. 1  are shown having like reference designations, the noise reduction processor  22  of  FIG. 1  is coupled to receive the plurality of receive signals  18   a.    
     The noise reduction processor  22  can include a subarray selection module  120  coupled to receive the plurality of receive signals  18   a  and configured to select a subarray associated with either all of the plurality of receive signals  18   a  or with a subset of the plurality of receive signals  18   a  (but still a plurality of receive signals  18   a , i.e., two or more). The subarray selection module is configured to generate a selected plurality of receive signals  120   a  associated with the selected subarray. 
     The noise reduction module  22  can also include a subarray data matrix formation module  122  coupled to receive the selected plurality of receive signals  120   a  and configured to assemble the selected plurality of receive signals  120   a  into a time bounded set of data samples (i.e., time samples), which can be arranged in a data (i.e., time sample) matrix  122   a . In examples described below in conjunction with  FIG. 5 , the array of time samples is associated with the subarray  50   a  of  FIG. 2 , e.g., five sets of time samples, each set ten samples long. However, as described above, a subarray can have any number of array elements greater than one. Furthermore, the number of time samples used for each element of the subarray (or each beam) can be greater than or less than ten time samples. 
     The time sample matrix  122   a  is received by a data matrix weighting module  124 . Operation of the data matrix weighting module is described more fully below in conjunction with  FIG. 4 . However, let it suffice here to say that the sets of time samples, e.g., five sets of time samples corresponding to the five element subarray  50   a  of  FIG. 2 , can be proportionately weighted according to their importance, i.e., the amount by which they correlate with the time samples associated with the selected reference element (i.j) of  FIG. 2 . The data weighting matrix module  124  can provide a weighted data matrix  124   a  to an eigen decomposition module  126 . 
     Operation the eigen decomposition module is also described more fully below in conjunction with  FIG. 5 . Let it suffice here to say that the eigen decomposition module is configured to perform an eigen decomposition of the weighted data matrix  124   a  into a combination of eigen vectors and associated eigen values equivalent to the weighted data matrix  124   a.    
     The eigen decomposition module  126  can provide the eigen decomposition  126   a  (i.e., eigen vectors and associate eigen values) to a signal subspace generate module  128 . Operation of the signal subspace generate module is also described more fully below in conjunction with  FIG. 5 . Let it suffice here to say that the signal subspace generation module  128  is configured to identify eigen vectors having eigen values above a threshold determined by the signal subspace generation module  128 . The identified eigen vectors and eigen values form a so-called “signal subspace.” The signal subspace  128   a  can be provided to a projection matrix module  130 . The projection matrix module  130  can also be coupled to receive the time sample matrix  122   a.    
     Operation of the projection matrix module  130  is also described more fully below in conjunction with  FIG. 5 . Let it suffice here to say here that the projection processing module  130  can compose the signal subspace into a projection matrix  130   a.    
     The noise reduction module  22  can also include an averaging module coupled to receive the projection matrix  130   a  and also to receive the time sample matrix  122   a . Operation of the averaging module is also described more fully below in conjunction with  FIG. 5 . However, let it suffice here o say that the averaging module  132  can be configured to average products of the projection matrix  130   a  and the time sample matrix  122   a  to generate the noise reduced signal  24   a  ( FIG. 1 ) associated with the reference one of the plurality of receive signals  18   a  ( FIG. 1 ) associated with the reference element, e.g., (i,j) of  FIG. 2 , of the subarray, e.g.,  50   a  of  FIG. 2 . 
     It should be appreciated that  FIGS. 5 and 6  show flowcharts corresponding to the below contemplated technique which would be implemented in computer system  10  ( FIG. 1 ). Rectangular elements (typified by element  154  in  FIG. 5 ), herein denoted “processing blocks,” represent computer software instructions or groups of instructions. Diamond shaped elements (typified by element  216  in  FIG. 6 ), herein denoted “decision blocks,” represent computer software instructions, or groups of instructions, which affect the execution of the computer software instructions represented by the processing blocks. 
     Alternatively, the processing and decision blocks represent steps performed by functionally equivalent circuits such as a digital signal processor circuit or an application specific integrated circuit (ASIC). The flow diagrams do not depict the syntax of any particular programming language. Rather, the flow diagrams illustrate the functional information one of ordinary skill in the art requires to fabricate circuits or to generate computer software to perform the processing required of the particular apparatus. It should be noted that many routine program elements, such as initialization of loops and variables and the use of temporary variables are not shown. It will be appreciated by those of ordinary skill in the art that unless otherwise indicated herein, the particular sequence of blocks described is illustrative only and can be varied without departing from the spirit of the invention. Thus, unless otherwise stated the blocks described below are unordered meaning that, when possible, the steps can be performed in any convenient or desirable order. 
     Referring now to  FIG. 5 , a process  150  can be representative of processing that can be performed by the noise reduction module  22  of  FIGS. 1 and 4 . The process begins at step  154 , where a subarray, for example the subarray  50   a  of  FIG. 2 , is selected and a reference element (or beam) from within the subarray is also selected. 
     Examples used below use a subarray having five elements (or beams), such as the subarray  50   a  of  FIG. 2 . However, in other embodiments, a subarray can have any number of elements greater than one element. Furthermore, the example below uses ten time samples for each of the five elements of the subarray. However, as discussed above, any number of time samples can be used. 
     Still further, the process  150  is described to process one subarray. However, as discussed below, each subarray of a larger array can be processed in the same way. It will become apparent from discussion in conjunction with  FIG. 6  that the process  150  can be repeated for other subarrays, for example, within the array  50  of  FIG. 2 . 
     As a subarray, it may be advantageous to select a reference element and four elements adjacent to the reference element (or, equivalently, a reference beam and four beams adjacent to the reference beam) because an underwater target, e.g., an underwater mine, is most likely seen in more than one of the five beams. Also, the reference beam would tend to have similar statistics corresponding to a target as compared to at least one of the four adjacent beams if the target is seen within the reference (i.e., center) beam. 
     At block  156 , time samples from the beams in the selected subarray from block  154  are collected and assembled into a data (i.e., time sample) matrix, also referred to herein as a “data space.” 
     Time samples of the subarray, i.e., the data space, can be represented as:
 
 X={x   i,j   ,x   i−1,j   ,x   i+1,j   ,x   i,j−1   ,x   i,j+1 },  (1)
 
where each small “x” is a set of time samples, for example, ten time samples, from an array element (or beam) identified by the “i,j” subscripts as in  FIG. 2 .
 
     Note that the first entry corresponds to the selected reference beam, (i,j), of  FIG. 2 . It should be appreciated that each small x can correspond to a 1×10 vector, and X can correspond to a 5×10 matrix, in which each row corresponds to a beam and each column corresponds to a time. This nomenclature will be presumed for continuations of this example below. 
     However, equivalently, each small x can correspond to a 10×1 vector, and X can correspond to a 10×5 matrix, in which each column corresponds to a beam and each row corresponds to a time. 
     The time samples, X, in matrix form, can correspond to the time sample matrix  122   a  of  FIG. 4 . 
     At block  158  a so-called “weighting matrix” can be formed in the following way: The symbol, W, is a diagonal weight matrix determined by the correlations between the center beam and the adjacent beams. The weighting matrix 
             W   =     [         α                                                                         w   1                                                                           w   2                                                                           w   3                                                                           w   4           ]           
can be calculated by the following equations:
 
     
       
         
           
             
               
                 
                   α 
                   ∈ 
                   
                     ( 
                     
                       0.4 
                       , 
                       0.5 
                     
                     ) 
                   
                 
               
               
                 
                   ( 
                   2 
                   ) 
                 
               
             
             
               
                 
                   β 
                   = 
                   
                     1.0 
                     - 
                     α 
                   
                 
               
               
                 
                   ( 
                   3 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       q 
                       k 
                     
                     = 
                     
                       cor 
                       ⁡ 
                       
                         ( 
                         
                           
                             x 
                             0 
                           
                           , 
                           
                             x 
                             k 
                           
                         
                         ) 
                       
                     
                   
                   , 
                   
                     k 
                     = 
                     1 
                   
                   , 
                   2 
                   , 
                   3 
                   , 
                   4. 
                 
               
               
                 
                   ( 
                   4 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       w 
                       k 
                     
                     = 
                     
                       
                         
                           q 
                           k 
                         
                         
                           
                             ∑ 
                             
                               i 
                               = 
                               1 
                             
                             4 
                           
                           ⁢ 
                           
                             q 
                             i 
                           
                         
                       
                       ⁢ 
                       β 
                     
                   
                   , 
                   
                     k 
                     = 
                     1 
                   
                   , 
                   2 
                   , 
                   3 
                   , 
                   4. 
                 
               
               
                 
                   ( 
                   5 
                   ) 
                 
               
             
           
         
       
     
     The value α corresponds to a weight to be applied to the reference beam. In some embodiments, the weight α is selected to be between 0.4 and 0.5. However, other weights less than 0.4 and grater than 0.5 can also be used. Note that the weighting coefficients W k  are associated with an amount by which time samples from each corresponding element (or beam) in the selected subarray correlate with time samples from the reference element (or beam). 
     At block  160 , the time sample matrix, X, from block  156  is combined with the weighting matrix, W, from block  158  to generate a weighted time sample matrix, Y, in the following way: 
     
       
         
           
             
               
                 
                   Y 
                   = 
                   
                     
                       [ 
                       
                         
                           
                             
                               α 
                               * 
                               
                                 x 
                                 1 
                               
                             
                           
                         
                         
                           
                             
                               
                                 w 
                                 1 
                               
                               * 
                               
                                 x 
                                 2 
                               
                             
                           
                         
                         
                           
                             
                               
                                 w 
                                 2 
                               
                               * 
                               
                                 x 
                                 3 
                               
                             
                           
                         
                         
                           
                             
                               
                                 w 
                                 3 
                               
                               * 
                               
                                 x 
                                 4 
                               
                             
                           
                         
                         
                           
                             
                               
                                 w 
                                 4 
                               
                               * 
                               
                                 x 
                                 5 
                               
                             
                           
                         
                       
                       ] 
                     
                     = 
                     
                       W 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         X 
                         H 
                       
                     
                   
                 
               
               
                 
                   ( 
                   6 
                   ) 
                 
               
             
           
         
       
         
         
           
             Where x 1 =x i,j   H , x 2 =x i−1j   H , x 3 =x i+1,j   H , x 4 =x i,j−1   H  and x 5 =x i,j+1   H , and the superscript H corresponds to a complex transpose. 
           
         
       
    
     The weighted time sample matrix can correspond to the weighted data matrix  124   a  of  FIG. 4 . 
     At block  162 , a covariance matrix is generated having the form: 
     
       
         
           
             
               
                 
                   C 
                   = 
                   
                     
                       E 
                       ⁡ 
                       
                         [ 
                         
                           YY 
                           H 
                         
                         ] 
                       
                     
                     = 
                     
                       [ 
                       
                         c 
                         ij 
                       
                       ] 
                     
                   
                 
               
               
                 
                   ( 
                   7 
                   ) 
                 
               
             
             
               
                 
                   
                     where 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       c 
                       ij 
                     
                   
                   = 
                   
                     
                       1 
                       N 
                     
                     ⁢ 
                     
                       
                         ∑ 
                         
                           i 
                           , 
                           j 
                         
                       
                       ⁢ 
                       
                         
                           y 
                           i 
                         
                         ⁢ 
                         
                           y 
                           j 
                           H 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   8 
                   ) 
                 
               
             
           
         
       
         
         
           
             and N is the number of time samples in each beam data, for example, ten. 
           
         
       
    
     At block  164 , the covariance matrix is decomposed using eigen analysis in the following way: 
     Theoretically, the covariance matrix can be represented by its spectral representation in terms of eigen value and eigen vectors. Namely 
                   C   =       ∑   k     ⁢       λ   k     ⁢     v   k     ⁢     v   k   H                 (   9   )               
Here, λ k  and v k  are the eigen value and eigenvector of the covariance matrix, respectively. To compute the eigen values and eigenvectors of the covariance matrix, we apply a so-called singular value decomposition (SVD) to the matrix. This results in:
 
                   C   =       U   ⁢           ⁢   Λ   ⁢           ⁢     V   H       =       ∑   k     ⁢       σ   k     ⁢     u   k     ⁢     v   k   H                   (   10   )               
where U={u 1 , . . . , h k } and V={v 1 , . . . , v k } are orthogonal matrices and Λ is a diagonal matrix with σ k  on the diagonal. Since the covariance is a symmetric nonnegative definite matrix, σ k =λ k  and v k  is a corresponding eigenvector. The SVD is an efficient computational tool to obtain the spectral representation described in Equation 9. With the spectral representation, we can determine the principle components of the weighted data space, Y.
 
     The decomposed matrix, C, can correspond to the eigen decomposition  126   a  of  FIG. 4 . 
     At block  166 , an eigen value threshold,  11 , is selected, namely, a predetermined threshold that can be used to select only those eigen vectors, v k , for which the corresponding eigen value, λ k , is sufficiently large. In some embodiments, the threshold is approximately 0.8. However, in other embodiments, the threshold is selected to have a value that will result in a selection of a few (e.g., three) of the eigen vectors. In still other embodiments, the threshold is selected to result in selection of some percentage of the eigen vectors, for example, fifty percent of the eigen vectors. 
     At block  168 , the predetermined threshold is applied to the eigen values to select associated eigen vectors in the following way: 
     If target signal energy is larger than noise energy, the principle components of the data space correspond to the signal subspace. To determine the principle components, we apply a thresholding technique to the eigen values. Specifically, for a given threshold η, the eigenvectors of the principle components are given as follows: 
     
       
         
           
             
               
                 
                   
                     
                       V 
                       p 
                     
                     = 
                     
                       { 
                       
                         
                           v 
                           k 
                         
                         ∈ 
                         
                           
                             V 
                             ⁢ 
                             
                               : 
                             
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               
                                 σ 
                                 k 
                               
                               
                                 σ 
                                 max 
                               
                             
                           
                           ≥ 
                           η 
                         
                       
                       } 
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   11 
                   ) 
                 
               
             
           
         
       
         
         
           
             where the symbol, ε, corresponds to “belongs to,” and where 
           
         
       
    
               σ   k       σ   max           
is a ratio of a k th  eigen value to a maximum eigen value. Thus, Vp includes only those eigen vectors having eigen values that are relatively large.
 
     At block  170 , the eigen vectors identified at block  168  are compiled into the signal subspace. It should be apparent that the vectors, Vp, can correspond to all of or a subset of all of the above eigen vectors, and therefore, the vectors, Vp, can be referred to herein as a “signal subspace.” The signal subspace, Vp, can correspond to the signal subspace  128   a  of  FIG. 4 . 
     At block  172 , the vectors, Vp, are arranged into a matrix to form a so-called “projection matrix.” 
     The projection matrix, T, is constructed by the eigenvectors from V p  that is given by
 
T=[v 1 , . . . , v s ]  (12)
 
where the subscript s is the number of dimension of the signal subspace. Since the noise is orthogonal to the signal subspace, to project the sub beam array onto the signal subspace can reduce the noise in the sub array.
 
     The projection matrix, T, can correspond to the projection matrix  128   a  of  FIG. 4 . 
     At block  174 , the projection matrix, T, can be combined with the original time sample matrix, X, (i.e., data space) of equation 1 in the following way to generate the noise reduced signal  24   a  of  FIGS. 1 and 4 . 
     In one embodiment, the noise reduced data from the center beam, {circumflex over (x)} i,j , can be computed by a weighted-average of the projection over the subarray. A noise reduced signal, U, can be computed as: 
     
       
         
           
             
               
                 
                   U 
                   = 
                   
                     
                       [ 
                       
                         
                           
                             
                               u 
                               1 
                             
                           
                         
                         
                           
                             
                               u 
                               2 
                             
                           
                         
                         
                           
                             ⋮ 
                           
                         
                         
                           
                             
                               u 
                               s 
                             
                           
                         
                       
                       ] 
                     
                     = 
                     
                       
                         T 
                         H 
                       
                       ⁢ 
                       
                         X 
                         H 
                       
                     
                   
                 
               
               
                 
                   ( 
                   13 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       
                         x 
                         ^ 
                       
                       
                         i 
                         , 
                         j 
                       
                       H 
                     
                     = 
                     
                       
                         
                           x 
                           ^ 
                         
                         1 
                       
                       = 
                       
                         
                           ∑ 
                           
                             i 
                             = 
                             1 
                           
                           s 
                         
                         ⁢ 
                         
                           
                             q 
                             i 
                           
                           * 
                           
                             u 
                             i 
                           
                         
                       
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   14 
                   ) 
                 
               
             
             
               
                 
                   
                     where 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       q 
                       i 
                     
                   
                   = 
                   
                     
                       λ 
                       i 
                     
                     
                       
                         ∑ 
                         
                           k 
                           = 
                           1 
                         
                         s 
                       
                       ⁢ 
                       
                         λ 
                         k 
                       
                     
                   
                 
               
               
                 
                     
                 
               
             
           
         
       
         
         
           
             and where the λ k , k=1, 2, . . . , s, are the eigen values of the eigenvectors in equation 12, 
             and where X is the subarray data given in equation 1. 
           
         
       
    
     It should be recognized that if the target is only seen in the reference element (or beam), the dimension of the signal subspace is one. In this case, the weighted-average of equation 14 has little significance since the projection becomes a one-dimensional signal. However, if the target is seen by more than one beam, the weighted-average operation results in a noise reduced signal associated with the reference beam, i.e., the noise reduced signal  24   a  of  FIGS. 1 and 4 . 
     Before describing functions of the CDWR detection statistic module  26  and detection and localization module  28  of  FIG. 1 , the CDWR function is first generally discussed below. 
     In time-frequency analysis, the Wigner Distribution (WD) is a useful analysis tool. The Wigner Distribution provides a signal representation in the time-frequency domain. For given signals s(t) and g(t) a cross-WD is defined as 
     
       
         
           
             
               
                 
                   
                     
                       WD 
                       
                         s 
                         , 
                         g 
                       
                     
                     ⁡ 
                     
                       ( 
                       
                         t 
                         , 
                         f 
                       
                       ) 
                     
                   
                   = 
                   
                     
                       ∫ 
                       
                         - 
                         ∞ 
                       
                       
                         + 
                         ∞ 
                       
                     
                     ⁢ 
                     
                       
                         s 
                         ⁡ 
                         
                           ( 
                           
                             t 
                             + 
                             
                               τ 
                               2 
                             
                           
                           ) 
                         
                       
                       ⁢ 
                       
                         
                           g 
                           * 
                         
                         ⁡ 
                         
                           ( 
                           
                             t 
                             - 
                             
                               τ 
                               2 
                             
                           
                           ) 
                         
                       
                       ⁢ 
                       
                         ⅇ 
                         
                           
                             - 
                             j2π 
                           
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           f 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           t 
                         
                       
                       ⁢ 
                       
                         ⅆ 
                         τ 
                       
                     
                   
                 
               
               
                 
                   ( 
                   15 
                   ) 
                 
               
             
           
         
       
     
     Equation 15 is called an auto-WD when s(t)=g(t). 
     The WD suffers from a problem of cross-term interference, which results in extra spectral content located between the two sets of signal spectrum in the time-frequency domain. This problem prevents the WD from being widely used in many applications. In order to overcome the problem, many techniques for reducing the undesirable cross-term interference have been identified. In particular, if the signals are Gaussian-shaped signals, the auto-terms and the cross-terms of the WD can be grouped into two terms separately; then the cross-terms can be easily removed from the WD. 
     According to Gabor expansion theory, an arbitrary shape signal s(t) can be represented by a set of basis functions as: 
                     s   ⁡     (   t   )       =       ∑     m   ,   n       ⁢       C     m   ,   n       ⁢       h     m   ,   n       ⁡     (   t   )                   (   16   )               where:  h   m,n ( t )= h ( t−Tm ) e   j2πnΩt   (17)
 
     The function h(t) is a synthesis function with unit energy; the C m,n  are complex coefficients; and the in and n are integers. The parameters T and Ω are the constant time and frequency sample intervals, respectively. If we use a Gaussian-shaped function as the synthesis function h(t), the basis functions h m,n (t) are all Gaussian-shaped functions because h m,n (t) are time-shift and frequency-modulated Gaussian-shaped functions. As a result, the arbitrary function s(t) is just a linear combination of Gaussian-shaped functions obtained from the Gabor expansion. If we take the WD of the Gabor expansion of the function s(t), we have the WD of s(t) given as: 
                       WD   s     ⁡     (     t   ,   f     )       =       2   ⁢       ∑     m   ,   n       ⁢              C     m   ,   n            2     ⁢     ⅇ     -     [           (     t   -   m     )     2     /     σ   2       +         (     2   ⁢   πσ     )     2     ⁢       (     f   -   n     )     2         ]               +     4   ⁢       ∑       k   ≠   m     ,     l   ≠   n         ⁢              C     m   ,   n       ⁢     C     k   ,   l     *            ×     ⅇ     -     [           (     t   -       (     m   +   k     )     /   2       )     2     /     σ   2       +         (     2   ⁢   πσ     )     2     ⁢       (     f   -       (     n   +   l     )     /   2       )     2         ]         ×       cos   [     2   ⁢     π   (         (     n   -   l     )     ⁢     (     t   -       m   +   k     2       )       +       (     m   -   k     )     ⁢     (     f   -       n   +   l     2       )         )       ]     .                     (   18   )               
The first term of equation 18 represents the auto-terms of the WD and the second term in equation 18 represents the cross-terms of the WD. Removal of the cross-term terms result in a cross-term deleted Wigner representation (CDWR) given as:
 
                       CDWR   s     ⁡     (     t   ,   f     )       =     2   ⁢       ∑     m   ,   n       ⁢              C     m   ,   n            2     ⁢     ⅇ     -     [           (     t   -   m     )     2     /     σ   2       +         (     2   ⁢   πσ     )     2     ⁢       (     f   -   n     )     2         ]                       (   19   )               
The CDWR has no cross-terms and correspond to the signal&#39;s contributions. Without the cross-terms, the CDWR is a powerful analysis tool for time-frequency analysis of a time variant signal.
 
     The CDWR can be optimized in ways described below. To compute the CDWR of a signal, one needs to represent the signal by a linear combination of Gaussian-shaped basis functions via the Gabor expansion as given in equation 16. In the Gabor expansion, the Gaussian-shaped basis functions can generated by a Gaussian-shaped synthesis function, which can be given by:
 
 h ( t )=(πσ 2 ) −0.25   e   −t     2     /2σ     2     (20)
         where σ 2  is a signal variance.       

     In equation 20, the variance, σ 2 , controls the resolutions in both the time and the frequency domains. A smaller value of σ 2  yields a better resolution in the time domain but a poorer resolution in the frequency domain, and vice-versa. It is desirable to optimize the CDWR to achieve the best resolutions in both the time domain and the frequency domain. Since the synthesis function only has one parameter, σ 2 , optimizing the synthesis function corresponds to finding an optimal value of the variance, σ 2 . In order to find the optimal value of σ 2 , an objective for the optimization is determined. In one arrangement, a function of the quadratic time-frequency moment can be used to measure the energy concentration of the WD. The function is given as: 
                           Σ   WD     =     ∫     ∫       [         (     t   -     t   s       )     2     +       (     f   -     f   s       )     2       ]     ⁢     WD   ⁡     (     t   ,   f     )       ⁢     ⅆ   t     ⁢     ⅆ   f                       =       1   2     ⁡     [       σ   2     +     1       (     2   ⁢   πσ     )     2         ]                     (   21   )               
where:
 
 t   s   =∫t|s ( t )| 2   dt  
 
 f   s   =∫f|S ( f )| 2   df   (22)
 
     The functions s(t) and S(f) are a Fourier transformation pair. Since a minimal energy concentration means a highest resolution in both the time and the frequency domains, the energy concentration measure can be used as the objective. Then, the goal is to find a value of σ 2  that minimizes the energy concentration measure defined in equation 21. To determine the optimal value, a derivative of the energy concentration with respect to the variance, σ 2 ; can be taken: 
                       ∂     Σ   WD         ∂     σ   2         =         1   2     ⁡     [     1   -       1     4   ⁢     π   2         ⁢       (     σ   2     )       -   2           ]       .             (   23   )               
Setting equation 23 to zero and solving for σ 2  results in an optimal value for σ 2 :
 
                     σ   opt   2     =     1     2   ⁢   π               (   24   )               
It will be appreciated that the optimal value of σ 2  is independent of the applied signal s(t).
 
     Referring now to  FIG. 6 , the noise reduced signal generated by the process  150  of  FIG. 5  can be used to detect a target. A process  200  begins at block  204 , where a detection statistic, η, is generated. The detection statistic can be generated in the following way: 
                       η   cdwr     =       ∫     -   ∞       +   ∞       ⁢       ∫     -   ∞       +   ∞       ⁢         CDWR     r   ,   s       ⁡     (     t   ,   f     )       ⁢       CDWR   s     ⁡     (     t   ,   f     )       ⁢     ⅆ   t     ⁢     ⅆ   f             ,           (   25   )               
where:
         the function CDWR s (t, f) is given in equation 19;   the function r(t) is a received signal,   the function s(t) is a transmitted signal; and   the complex coefficients C r     m,n    and C s     m,n    are the Gabor expansion coefficients of r(t) and s(t), respectively.       

     It should be recognized that equation 25 corresponds to a product of two CDWRs, a cross-CDWR of the noise reduced signal ( 24   a  of  FIGS. 1 and 4 ) with the transmitted signal ( 12   a  of  FIG. 1 ) and an auto-CDWR of the transmitted signal ( 12   a  of  FIG. 1 ) with itself, and a correlation thereof. 
     A discrete form of equation 25 is given below. A time window with a size of 2d can be used to compute the detection statistic, which makes the detection statistic time (i.e., range) dependent. The value d used below is a factor associated with a number of time steps (2d time steps). In some embodiments, fifty time steps can be used, and thus, 2d=50. However, in other embodiments, the number of time steps can be more than fifty or less then fifty. A time (i.e., range) dependent detection statistic can be calculated as follows: 
                       η   cdwr     ⁡     (   k   )       =       ∑     n   =   0     N     ⁢       ∑     m   =     k   -   d         k   +   d       ⁢         CDWR     r   ,   s       ⁡     (       m   ⁢           ⁢   Δ   ⁢           ⁢   t     ,     n   ⁢           ⁢   Δ   ⁢           ⁢   f       )       ⁢       CDWR   s     ⁡     (       m   ⁢           ⁢   Δ   ⁢           ⁢   t     ,     n   ⁢           ⁢   Δ   ⁢           ⁢   f       )       ⁢   Δ   ⁢           ⁢   t   ⁢           ⁢   Δ   ⁢           ⁢   f                 (   26   )               
where the indices k and in are discrete time variables and the index n is a discrete frequency variable. It will, therefore, be recognized that the discrete detection statistic, η(k), is comprised of a plurality of detection statistic values over time. In some embodiments, there are at least one thousand detection statistic values, equivalent to at least one thousand range increments.
 
     The variables Δt and Δf are the step sizes in both the time domain and the frequency domain respectively. The summation in the frequency domain is over all frequencies; that is NΔf=2π. The summation in the time domain is only over the time window having the size 2d. 
     At block  208  a detection statistic threshold is generated in the following way. First, a local window is identified within the plurality of detection statistic values in order to compute noise energy. In some embodiments, between fifty and one hundred detection statistic values are used to compute the noise energy, however, other quantities of detection statistic values can also be used. Then, the noise energy is used to generate the detection statistic threshold, Th, as a combination of a mean value m n  and a standard deviation σ n  of the noise energy:
 
 Th=m   n +ασ n   (27)
         where: α is a constant that, in some embodiments, can have a value between 1.5 to 3.0.       

     At block  210 , the detection statistic from equation 26 is compared with the detection statistic threshold from equation 27 in the following way: 
     For a given received signal r(t), a target detection is identified by the following hypothesis test:
         Hypothesis H 0 :r(t)=n(t), which represents no target detection   Hypothesis H 1 :r(t)=s(t)+n(t), which represents a target detection   where the function n(t) is white Gaussian noise.       

     
       
         
           
             
               
                 
                   { 
                   
                     
                       
                         
                           
                             
                               If 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 η 
                                 cdwr 
                               
                             
                             &gt;= 
                             
                               
                                 m 
                                 n 
                               
                               + 
                               
                                 ασ 
                                 n 
                               
                             
                           
                           , 
                           
                               
                           
                           ⁢ 
                           
                             
                               H 
                               1 
                             
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             is 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             true 
                           
                         
                       
                     
                     
                       
                         
                           
                             
                               If 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 η 
                                 cdwr 
                               
                             
                             &lt; 
                             
                               
                                 m 
                                 n 
                               
                               + 
                               
                                 ασ 
                                 n 
                               
                             
                           
                           , 
                           
                               
                           
                           ⁢ 
                           
                             
                               H 
                               o 
                             
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             is 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             true 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   28 
                   ) 
                 
               
             
           
         
       
         
         
           
             where, as described above, the constant α can have a value between 1.5 and 3.0. 
           
         
       
    
     If at block  210  a target is detected under hypothesis H 1 , then the process  150  continues to block  212 . 
     At block  212 , the target is thus detected in the reference beam under the H 1  hypothesis. 
     At block  214 , a direction and a range to the target can be computer. Azimuthal direction and elevation direction relative to the receive array can be computed merely by recognizing the pointing direction of the reference beam in which the target as detected. Distance to the target can be identified by the above-computed CDWR function, wherein a time delay to the target can be identified by the signal detected within the CDWR. 
     It should be recognized that the block within a boundary  202  can be performed by the CDWR detection statistic module  26  of  FIG. 1  and the blocks within a boundary  206  can be performed by the detection and localization module  28  of  FIG. 1 . 
     The above-described processing blocks of  FIGS. 5 and 6  produce a detection and localization associated with the reference beam, i.e., (i,j) of  FIG. 2 . Additional blocks of  FIG. 6  move to other reference beams that can be selected within the array  50  of  FIG. 2 . 
     At block  216 , it is determined whether the subarray and associated reference beam thus far processed are the last subarray and associated reference beam to be processed. If the subarray and associated reference beam thus far processed are the last subarray and associated reference beam to be processed, then process  150  continues to block  218 , where the first subarray and associated reference beam are again selected and the process returns to block  152  of  FIG. 5 . 
     If, at block  210 , the detection statistic is not greater than the detection statistic threshold, then at block  220 , a target is not detected and the process continues at block  216 . 
     If at block  216 , the subarray and associated reference beam thus far processed are not the last subarray and associated reference beam to be processed, then the process  200  continues to block  222 , where the next subarray and associated reference beam are selected and the process returns to block  156  of  FIG. 5 . 
     Having described preferred embodiments of the invention it will now become apparent to those of ordinary skill in the art that other embodiments incorporating these concepts may be used. Additionally, the software included as part of the invention may be embodied in a computer program product that includes a computer readable storage medium. For example, such a computer readable storage medium can include a readable memory device, such as a hard drive device, a CD-ROM, a DVD-ROM, or a computer diskette, having computer readable program code segments stored thereon. A computer readable transmission medium can include a communications link, either optical, wired, or wireless, having program code segments carried thereon as digital or analog signals. Accordingly, it is submitted that that the invention should not be limited to the described embodiments but rather should be limited only by the spirit and scope of the appended claims. All publications and references cited herein are expressly incorporated herein by reference in their entirety.