Abstract:
The invention relates to a method of comparing two or more series of signal analysis data samples, to produce a measure of their similarity. Corresponding data samples of each series are compared, and measures of the magnitude of the signal and the variation between corresponding data samples are calculated. The magnitude measure may simply be the average power of corresponding data samples. The produced measure of the similarity of the series is a test statistic having Snedecor&#39;s F distribution. When compared to typical cross correlation methods, a test statistic of the present invention provides a greater distinction between series displaying a high degree of similarity and series with a low degree of similarity. Accordingly, the present invention has applicability to the detection, classification and angular localisation of signals from acoustic or electromagnetic sources.

Description:
FIELD OF THE INVENTION 
       [0001]    The present invention relates to the field of digital signal processing. In particular, the present invention relates to assessing the similarity of two or more series of signal data samples, and applications of the present invention include (although are not necessarily limited to) the detection, classification and localisation of signals. 
       BACKGROUND OF THE INVENTION 
       [0002]    Sensors can be used to detect signal emissions—for example, acoustic or electromagnetic waves (e.g. radar). However, in a noisy environment, such as may be expected on a battlefield, the established methods of detection via amplitude discrepancies in a relatively steady background of noise have been found to result in a relatively high False Alarm Rate (FAR) as any intruding noise of high enough level will trigger the detector. 
         [0003]    Accordingly, Constant False Alarm Rate (CFAR) detectors have been developed which provide a specified (low) FAR. CFAR characteristics may be achieved by many different mechanisms. In addition to the current measurement they typically incorporate an estimate of the environmental state. The ‘environmental’ estimate may be used to set an adaptive threshold, or used to scale the current measurement which is then compared against a constant threshold. 
         [0004]    Once a signal is detected, the received waveform is typically classified by cross correlation with a template waveform stored in a library. Each template waveform corresponds to a particular class of signal source—for example, the class of weapon fired. 
         [0005]    Furthermore, if multiple sensors are used, the direction of the source can also be determined. This can be accomplished by a process of lag estimation—the time delay or lag between receiving the signal at each sensor is calculated and then used to estimate the angle of arrival of the signal. In the lag estimation process, sensors are typically partitioned into pairs, and lag estimates are calculated for each pair. Series of data samples taken from each sensor are cross correlated at different lags, and the lag with the highest correlation value is selected and used to determine the angle of arrival. 
         [0006]    There are a number of known cross correlation techniques which can be used for classification or lag estimation purposes. The standard cross correlation of two signals x 1 (n), x 2 (n) of N samples, where x 2  is shifted by l samples relative to x 1 , can be written as: 
         [0000]    
       
         
           
             
               
                 
                   
                     r 
                     12 
                   
                   = 
                   
                     
                       1 
                       N 
                     
                      
                     
                       
                         ∑ 
                         
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                           = 
                           0 
                         
                         
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                           - 
                           1 
                         
                       
                        
                       
                         
                           
                             x 
                             1 
                           
                            
                           
                             ( 
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                             ( 
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                   ( 
                   1 
                   ) 
                 
               
             
           
         
       
     
         [0007]    Standard cross correlation gives a positive figure for positive correlation, but a negative figure for negative correlation; however, the figures mean little in an absolute sense. Accordingly, normalised cross correlation is an improvement on standard cross correlation, giving a value that always lies between −1 and +1. +1 means 100% correlation, whereas −1 means 100% correlation in the opposing sense (e.g. signals in antiphase). The normalised cross correlation of two signals x 1 (n), x 2 (n) of N samples can be written as: 
         [0000]    
       
         
           
             
               
                 
                   
                     rho 
                     12 
                   
                   = 
                   
                     
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                       12 
                     
                     
                       
                         
                           1 
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         [0008]    In preference to standard or normalised cross correlation, the generalized correlation method is also used (see  The Generalized Correlation Method for Estimation for Time Delay,  Knapp and Carter, August 1976, IEE Transactions on Acoustics, Speech, and Signal Processing, Vol. ASSP-24, No. 4, 320). However, each of these techniques requires a preceding detection step. Furthermore each technique can sometimes result in either misclassification of a signal or an incorrect lag estimation—and accordingly, significant angular errors. 
       SUMMARY OF THE INVENTION 
       [0009]    In a first aspect of the present invention, there is provided a method of assessing the similarity of two or more series of signal data samples comprising:
       forming a test statistic which is inversely proportional to a variation measure, the variation measure indicating the variation between the series,   whereby the value of the test statistic therefore provides a measure of the similarity of the series.       
 
         [0012]    Of course, the formation of the test statistic may only be an intermediate step of the assessment process, and other calculations may be performed in addition to the formation of the test statistic when assessing the similarity of the series. 
         [0013]    The test statistic may be proportional to a magnitude measure, the magnitude measure indicating the magnitude of the signal. 
         [0014]    Preferably, the method of assessing the similarity of two or more series of signal data samples may further comprise:
       calculating, for each data sample, a sample variation measure indicating the variation between corresponding data samples of each series, and   calculating the variation measure from the sample variation measures.       
 
         [0017]    In some embodiments, the variation measure is 
         [0000]    
       
         
           
             
               
                 ∑ 
                 
                   n 
                   = 
                   0 
                 
                 
                   N 
                   - 
                   1 
                 
               
                
               
                 
                   σ 
                   ^ 
                 
                 n 
                 2 
               
             
             , 
           
         
       
       
         
           
             where N is the number of samples in each series, and 
             where {circumflex over (σ)} n  is the observed standard deviation of the set of the nth samples of each series, at a specific lag. 
           
         
       
     
         [0020]    Further, the test statistic may be proportional to 
         [0000]    
       
         
           
             
               
                 
                   ∑ 
                   
                     n 
                     = 
                     0 
                   
                   
                     N 
                     - 
                     1 
                   
                 
                  
                 
                   
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                     ^ 
                   
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                   2 
                 
               
               
                 
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                     n 
                     = 
                     0 
                   
                   
                     N 
                     - 
                     1 
                   
                 
                  
                 
                   
                     σ 
                     ^ 
                   
                   n 
                   2 
                 
               
             
             , 
           
         
       
       
         
           
             where {circumflex over (μ)} n  is the observed mean of the nth samples of each series, at a specific lag. 
           
         
       
     
         [0022]    Further, the test statistic may be equal to 
         [0000]    
       
         
           
             
               
                 ( 
                 
                   M 
                   - 
                   1 
                 
                 ) 
               
                
               
                 
                   
                     ∑ 
                     
                       n 
                       = 
                       0 
                     
                     
                       N 
                       - 
                       1 
                     
                   
                    
                   
                     
                       μ 
                       ^ 
                     
                     n 
                     2 
                   
                 
                 
                   
                     ∑ 
                     
                       n 
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                       0 
                     
                     
                       N 
                       - 
                       1 
                     
                   
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                       σ 
                       ^ 
                     
                     n 
                     2 
                   
                 
               
             
             , 
           
         
       
       
         
           
             where M is the number of series of data samples. 
           
         
       
     
         [0024]    Like other methods, this test statistic can be used to assess the similarity of two series. However, unlike other methods, the test statistic described above may also be used to assess the similarity of three or more series of data samples. 
         [0025]    Preferably, the test statistic has a known distribution, which for the test statistic equal to 
         [0000]    
       
         
           
             
               ( 
               
                 M 
                 - 
                 1 
               
               ) 
             
              
             
               
                 
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                     ^ 
                   
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         [0000]    will be Snedecor&#39;s F distribution. 
         [0026]    In a second aspect of the present invention, there is provided a test statistic for assessing the similarity of two or more series of signal data samples, wherein the test statistic is inversely proportional to a variation measure, the variation measure indicating the variation between the series. 
         [0027]    One application of the method and test statistic of the present invention is in angle of arrival estimation, wherein series of signal data samples are compared at a variety of different lags. At different lags, a data sample in one series will correspond to different data samples in other series, depending on the lag. For three of more series of signal data samples, lag combinations may be selected to correspond to possible angles of arrival. 
         [0028]    Accordingly, in a third aspect of the present invention, there is provided a method of aligning two or more series of signal data samples comprising:
       calculating a test statistic at a first lag combination, wherein the test statistic is inversely proportional to a variation measure, the variation measure indicating the variation between the series;   calculating the test statistic at one or more second lag combinations; and   selecting the lag combination for which the test statistic is greatest, thereby aligning the series.       
 
         [0032]    As will be understood, where there are only two series to be aligned, each lag combination will consist of only one lag. 
         [0033]    Of course, there are other applications for the present invention. Classification of a signal of interest is one other obvious example of where it may be desirable to compare two series of signal data samples (for instance, comparing a signal of interest to a template representing a particular class of signal). However, another problem to which the present invention may be applied is the detection of a signal of interest, which conventionally does not assess the similarity between different series of signal data samples. 
         [0034]    Accordingly, in a fourth aspect of the present invention, there is provided a method of detecting a signal of interest (SOI) by:
       obtaining a first series of signal data samples from a first sensor;   obtaining a second series of signal data samples from a second sensor;   calculating, from the first and second series, a test statistic which provides a measure of the similarity between the first and second series at a first lag of zero or more data samples;   calculating the test statistic at one or more second lags of zero or more data samples; and   declaring a detection if the largest calculated test statistic is more extreme than a threshold value.       
 
         [0040]    Where the test statistic has a known distribution, this means that the threshold value can be set to provide a desired CFAR. 
         [0041]    In further aspects, the present invention provides computer readable media encoded with data representing computer programs, that can be used to direct a programmable device to perform the above methods. 
         [0042]    In yet further aspects of the present invention, computer program elements are provided comprising computer program code means to make a programmable device execute the above methods. 
         [0043]    In yet further aspects of the present invention, there are provided computer processing devices configured to perform the above methods. 
         [0044]    A detailed description of one or more embodiments of the invention is provided below, along with accompanying figures that illustrate by way of example the principles of the invention. While the invention is described in connection with such embodiments, it should be understood that the invention is not limited to any embodiment. On the contrary, the scope of the invention encompasses numerous alternatives, modifications and equivalents. For the purpose of example, specific details are set forth in the following description in order to provide a thorough understanding of the present invention. 
         [0045]    For the purpose of clarity, technical material that is known in the technical fields related to the invention has not been described in detail so that the present invention is not unnecessarily obscured. 
     
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         [0046]    An illustrative embodiment of the present invention will be discussed with reference to the accompanying drawings wherein: 
           [0047]      FIG. 1  depicts an incoming signal to be received by a linear array of sensors; 
           [0048]      FIG. 2  is a flow diagram of one embodiment of the present invention; 
           [0049]      FIG. 3  is a plot of two series of signal data samples for a signal of interest, aligned at the most likely lag; 
           [0050]      FIG. 3A  is a plot of test statistics calculated for the data in  FIG. 2 , plotted as a function of lag for four candidate lag-estimation methods; 
           [0051]      FIGS. 4 ,  5 ,  6 ,  7 ,  8  and  9  are each a plot of two series of signal data samples for different signals of interest, aligned at the most likely lag; and 
           [0052]      FIGS. 4A ,  5 A,  6 A,  7 A,  8 A and  9 A are each a plot of test statistics calculated for the data in  FIGS. 4 ,  5 ,  6 ,  7 ,  8  and  9  respectively, plotted as a function of lag for four candidate lag-estimation methods. 
       
    
    
     DETAILED DESCRIPTION 
       [0053]    This embodiment of the present invention is best described in relation to the problem of angular localisation, although the present invention does have broader application to other circumstances where it is desirable to assess the similarity of two or more series of signal data samples. 
         [0054]    With an array of sensors, a signal of interest (SOI) arriving at an angle as shown in  FIG. 1  will be received by each sensor  12  at a different time. In  FIG. 1 , the right-most sensor receives the SOI first, the middle sensor receives the SOI after a time delay, and the left-most sensor only receives the SOI after another time delay. The angle of arrival (AOA) of the SOI can be determined by the duration of these time delays. For a three dimensional array (at least four sensors arranged in a non-coplanar configuration) the determined AOA can be uniquely determined in three dimensional space. 
         [0055]    As depicted in  FIG. 2 , series  22  of signal data samples can be obtained  20  from each sensor  12 , and may be measured in volts. To obtain  20  the series  22 , a continuous data segment can be extracted from the digitised data samples taken by each sensor  12 , for a given analysis window. The data within this analysis window will provide a series  22  of signal data samples, which can be used in the detection, classification or localisation of a signal of interest. 
         [0056]    According to the present invention, a statistical treatment of the series of data samples is performed with the null hypothesis being that there is no SOI and that the series contain only zero-mean Gaussian distributed noise. It is herein assumed that any DC offset in the data (e.g. sensor or DAQ bias) or frequencies that are of no interest (e.g. wind or self noise) have been removed by a pre-whitening stage. The test statistic for all possible lag combinations corresponding to all physically measurable angles is calculated  30 , and the largest calculated test statistic  32  is identified  40 . If the largest calculated test statistic is greater than a threshold value  42  (preferably set to give a specified CFAR) then the null hypothesis is rejected, and a detection is declared  50 . Furthermore, in this way the series  22  can be aligned, and the most likely direction of the source can be set equal to the angular coordinate for which the null hypothesis is least likely, i.e. the angle of arrival is estimated  60  by selecting the angle which corresponds to the lag combination for which the test statistic is maximised. 
         [0057]    The test statistic is a measure of similarity, applied to two or more series of signal data samples. In this embodiment, the signals are delayed acoustic sensor signals. 
         [0058]    The delay-and-sum (beam-formed) total of the two or more series is applied as: 
         [0000]    
       
         
           
             
               
                 
                   
                     y 
                      
                     
                       ( 
                       n 
                       ) 
                     
                   
                   = 
                   
                     
                       ∑ 
                       
                         m 
                         = 
                         0 
                       
                       
                         M 
                         - 
                         1 
                       
                     
                      
                     
                       
                         x 
                         m 
                       
                        
                       
                         ( 
                         n 
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   3 
                   ) 
                 
               
             
           
         
       
     
         [0059]    where x m (n) is the nth sample of the mth sensor at a specified lag of data samples, and y(n) is the delay-and-sum total. It is assumed that the data in the channel x m  has already been delayed to steer the beam in the desired direction. 
         [0060]    The noise statistics of each sensor are assumed to be the same, so the nth sample in each delay channel is assumed to be an independent observation of the random variable X n , giving a total of N different random variables, with M observations for each variable. Under the null hypothesis the variables have a Gaussian (or normal) distribution: 
         [0000]      X n ˜N(μ n ,σ n   2 )   (4) 
         [0061]    The Maximum Likelihood Estimates (MLE) {circumflex over (μ)} n  and {circumflex over (σ)} n , of the mean and variance are computed using: 
         [0000]    
       
         
           
             
               
                 
                   
                     
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                       ^ 
                     
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                          
                         
                           ( 
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                      
                     
                         
                     
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                     and 
                   
                 
               
               
                 
                   ( 
                   5 
                   ) 
                 
               
             
             
               
                 
                   
                     
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                       ^ 
                     
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                     2 
                   
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                       { 
                       
                         
                           
                             ∑ 
                             
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                               = 
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                       } 
                     
                   
                 
               
               
                 
                   ( 
                   6 
                   ) 
                 
               
             
           
         
       
     
         [0062]    Under the null hypothesis, the following relationships hold: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       a 
                       . 
                       
                           
                       
                        
                       If 
                     
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                           ( 
                           
                             
                               
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                               n 
                             
                           
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                         2 
                       
                       
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                         n 
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                         a 
                       
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                           ( 
                           1 
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                   ( 
                   7 
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                       b 
                       . 
                       
                           
                       
                        
                       If 
                     
                      
                     
                         
                     
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                       Z 
                       b 
                     
                   
                   = 
                   
                     M 
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                           σ 
                           ^ 
                         
                         n 
                         2 
                       
                       
                         σ 
                         n 
                         2 
                       
                     
                      
                     
                         
                     
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                     then 
                      
                     
                         
                     
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                         Z 
                         b 
                       
                       ~ 
                       
                         
                           χ 
                           2 
                         
                          
                         
                           ( 
                           
                             M 
                             - 
                             1 
                           
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   8 
                   ) 
                 
               
             
           
         
       
     
         [0063]    That is, the quantities Z a  and Z b  are distributed as χ 2  (chi-squared) variables with 1 and M−1 degrees of freedom respectively. 
         [0064]    Under the null hypothesis it is also assumed that the noise statistics of the sensor outputs are zero mean and time invariant so the parameters of each distribution are the same: 
         [0000]      μ i =μ 2 =, . . . ,=μ N =μ=0   (9) 
         [0000]      and 
         [0000]      σ 1   2 =σ 2   2 =, . . . ,=σ N   2 =σ 2    (10) 
         [0065]    Using the fact that the sum of χ 2  variables is a χ 2  variable, the following aggregate test statistics can be formed and analysed: 
         [0000]    
       
         
           
             
               
                 
                   
                       
                   
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                         c 
                         . 
                         
                             
                         
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                         If 
                       
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                         Z 
                         c 
                       
                     
                     = 
                     
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                               = 
                               0 
                             
                             
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                             n 
                             2 
                           
                         
                         
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                           2 
                         
                       
                        
                       
                           
                       
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                       then 
                        
                       
                           
                       
                        
                       
                         
                           Z 
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                         ~ 
                         
                           
                             χ 
                             2 
                           
                            
                           
                             ( 
                             N 
                             ) 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   11 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       
                         d 
                         . 
                         
                             
                         
                          
                         If 
                       
                        
                       
                           
                       
                        
                       
                         Z 
                         d 
                       
                     
                     = 
                     
                       M 
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                             ∑ 
                             
                               n 
                               = 
                               0 
                             
                             
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                               - 
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                               ^ 
                             
                             n 
                             2 
                           
                         
                         
                           σ 
                           2 
                         
                       
                        
                       
                           
                       
                        
                       then 
                        
                       
                           
                       
                        
                       
                         
                           Z 
                           d 
                         
                         ~ 
                         
                           
                             χ 
                             2 
                           
                            
                           
                             ( 
                             
                               N 
                                
                               
                                 ( 
                                 
                                   M 
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                                   1 
                                 
                                 ) 
                               
                             
                             ) 
                           
                         
                       
                     
                   
                    
                   
                       
                   
                 
               
               
                 
                   ( 
                   12 
                   ) 
                 
               
             
           
         
       
     
         [0066]    So far it has been assumed that the true variance (σ 2 ) of the white noise is known. This is an inconvenient and unnecessary assumption. It can be eliminated by dividing (11) by (12). Furthermore, if the numerator and the denominator are scaled by the inverse of their respective degrees of freedom then a variable distributed according to Snedecors&#39; F distribution results: 
         [0000]    
       
         
           
             
               
                 
                   
                     Z 
                     M 
                   
                   = 
                   
                     
                       
                         Z 
                         c 
                       
                       N 
                     
                     
                       
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                         d 
                       
                       
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                   ( 
                   13 
                   ) 
                 
               
             
             
               
                 
                   
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                     M 
                   
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                       ( 
                       
                         M 
                         - 
                         1 
                       
                       ) 
                     
                      
                     
                       
                         
                           ∑ 
                           
                             n 
                             = 
                             0 
                           
                           
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                             - 
                             1 
                           
                         
                          
                         
                           
                             μ 
                             ^ 
                           
                           n 
                           2 
                         
                       
                       
                         
                           ∑ 
                           
                             n 
                             = 
                             0 
                           
                           
                             N 
                             - 
                             1 
                           
                         
                          
                         
                           
                             σ 
                             ^ 
                           
                           n 
                           2 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   14 
                   ) 
                 
               
             
             
               
                 
                   
                     Z 
                     m 
                   
                   ~ 
                   
                     F 
                      
                     
                       ( 
                       
                         n 
                         , 
                         
                           n 
                            
                           
                             ( 
                             
                               m 
                               - 
                               1 
                             
                             ) 
                           
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   15 
                   ) 
                 
               
             
           
         
       
     
         [0067]    The Z M  value obtained can be used as a test statistic according to this embodiment of the present invention. The term 
         [0000]    
       
         
           
             
               ∑ 
               
                 n 
                 = 
                 0 
               
               N 
             
              
             
               
                 σ 
                 ^ 
               
               n 
               2 
             
           
         
       
     
         [0000]    accordingly provides a variation measure which indicates the variation between the series. 
         [0068]    Substituting the expressions for {circumflex over (μ)} n  and {circumflex over (σ)} n  into Eq. (14) yields: 
         [0000]    
       
         
           
             
               
                 
                   
                     Z 
                     
                       M 
                        
                       
                           
                       
                     
                   
                   = 
                   
                     
                       ( 
                       
                         M 
                         - 
                         1 
                       
                       ) 
                     
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                           1 
                           M 
                         
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                               n 
                               = 
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                               1 
                             
                           
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                               y 
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                                 ( 
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                               = 
                               0 
                             
                             
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                               1 
                             
                           
                            
                           
                             
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                                 n 
                                 = 
                                 0 
                               
                               
                                 N 
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                                 1 
                               
                             
                              
                             
                               
                                 
                                   x 
                                   m 
                                 
                                  
                                 
                                   ( 
                                   n 
                                   ) 
                                 
                               
                               2 
                             
                           
                         
                         - 
                         
                           
                             1 
                             M 
                           
                            
                           
                             
                               ∑ 
                               
                                 n 
                                 = 
                                 0 
                               
                               
                                 N 
                                 - 
                                 1 
                               
                             
                              
                             
                               
                                 y 
                                  
                                 
                                   ( 
                                   n 
                                   ) 
                                 
                               
                               2 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   16 
                   ) 
                 
               
             
           
         
       
     
         [0069]    Note that Eq. (16) appears to contain sum-of-squares terms that are equivalent to power measurements computed in the time domain. Alternatively, Eq. (16) may be written in terms of moments: 
         [0000]    
       
         
           
             
               
                 
                   
                     Z 
                     M 
                   
                   = 
                   
                     
                       ( 
                       
                         M 
                         - 
                         1 
                       
                       ) 
                     
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                                   n 
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                                     n 
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                               ] 
                             
                           
                         
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                                     n 
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                                 ] 
                               
                             
                             2 
                           
                         
                       
                     
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                     using 
                   
                 
               
               
                 
                   ( 
                   17 
                   ) 
                 
               
             
             
               
                 
                   
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                           ( 
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                   18 
                   ) 
                 
               
             
             
               
                 
                   
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                             n 
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                       ] 
                     
                   
                   = 
                   
                     
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                           = 
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                         and 
                       
                     
                   
                 
               
               
                 
                   ( 
                   19 
                   ) 
                 
               
             
             
               
                 
                   
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                      
                     
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                       ) 
                     
                   
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                         = 
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                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   20 
                   ) 
                 
               
             
           
         
       
     
         [0070]    When determining the AOA of a SOI, the Z M  test statistic can be computed efficiently using sliding windows, implemented in software as circular buffers, to estimate the moments using the delay line inputs x m (n). 
         [0071]    Furthermore, the final test statistic is the ratio of two squared quantities. If the square of the estimated mean is regarded as the signal power, and the variance the noise power, then Z M  may be converted into a signal to noise ratio (SNR) in dB using: 
         [0000]        SNR= 10 log 10 ( Z   M )   (21) 
         [0072]    Knowing that under the null hypothesis the Z M  test statistic is F distributed, allows a detection threshold  42  to be computed to give the desired probability of falsely rejecting the null hypothesis when it is indeed true (a false alarm). If the computed Z M  value exceeds the threshold then a localisable signal is instead assumed to be present. 
         [0073]    In practice the null hypothesis is rarely entirely true, and false alarms due to nuisance detections are common. Accordingly, a larger detection threshold  42  is usually appropriate, giving a negligible theoretical false alarm probability and an acceptable practical false alarm probability and probability of detection. 
         [0074]    The null hypothesis is rejected when the estimated means at each point ({circumflex over (μ)} n ) are a long way from zero and when the dispersion around the mean is small (as indicated by a small {circumflex over (σ)} n ). When the null hypothesis is indeed false, applying the appropriate lags results in large {circumflex over (μ)} n  values and small {circumflex over (σ)} n  s values because the waveforms in each delay channel are nearly identical (except for disturbances caused by additive white noise). 
         [0075]    In the presence of non-localisable signals (e.g. wind noise), or when the incorrect delays are applied, the {circumflex over (μ)} n  values are large but so are the {circumflex over (σ)} n  values, which ensures that Z M  remains small. It is this characteristic that makes this method an effective detector and localiser, with prominent peaks marking the directions to true sources set against a relatively flat and featureless background. 
         [0076]    The Z M  test statistic may be used, for example, for beam formation from multiple sensors. The test statistic may be formed for all angles, and the most likely lag used to declare a detection of a SOI and to calculate its AOA. It may be used on pairs of series of signal data samples obtained from respective sensor pairs, or alternatively it could be used on multiple sensors at once. 
         [0077]    With most planar or three dimensional sensor arrays, and with the limits of current processing technology, it is too computationally expensive, using software implementations, to continuously form beams in bearing and elevation, to cover the surveillance surface (hundreds of beams must be formed to give the desired angular resolution). Accordingly, a predetection stage may be performed such that the Z M  test statistics are only formed when processing data likely to contain a signal of interest. The predetector may, for example, look for a large and/or sudden increase in received signal power. The predetector could then select data segments of potential interest for further processing by a method according to the present invention. 
         [0078]    An alternative embodiment encompassed within the scope of the present invention will now be described, wherein a test statistic (in this case Z 12 ) is again inversely proportional to a measure of the variation between the series. Z 12  is a measure of similarity, between (exactly) two signals. This method will be described with reference to classification of signals, although it can also be applied to the problem of angular localisation by lag estimation. Similarly, methods utilising the Z M  test statistic may be applied to the problem of classification. 
         [0079]    For the Z 12  test statistic, it is assumed that the features being classified are impulsive or transient, i.e. they have a finite duration, typically with a rapid onset and a more gradual decay. A template is used to represent a given class, for example the type of weapon fired. Separate templates may, for example, be used for the ‘crack’ (the supersonic shockwave emitted by the passing bullet) and ‘thump’ (the muzzle blast) of a particular weapon. 
         [0080]    A sensor is used to obtain a series of signal data samples of a waveform for a signal of interest, which is then compared against the template. The comparison is done using a least-squares fit. Large fitting coefficients suggest that the received waveform may belong to the class represented by the template. Optionally, the significance of the fitting coefficient is assessed using the residual of the fit. The comparison may be done by identifying the start of the feature in the waveform being classified, or if this cannot be done reliably then the fit may be implemented for all possible displacements using a matched filter. 
         [0081]    This process may be formalised as follows. Let the template be the series x 1 (n), and the waveform containing the feature to be classified x 2 (n). It is assumed that the onset of the signature in the template and the waveform is at n=0 and that both x 1 (n) and x 2 (n) are of equal length. The null hypothesis is that there is no correlation between the two series. The signature in the waveform has been corrupted by additive white noise. The model is: 
         [0000]        x   2 ( n )=β 12   x   1 ( n )+ε  (22) 
         [0082]    We are assuming that the template is a perfect representation of the class, i.e. that it does not contain noise. This representation may be constructed from a theoretical model, from filtered or enhanced measurements, or simply set equal to raw measurements, which are assumed to be noise free. The most appropriate method depends on how well the signature can be measured and how well the feature can be modelled. It is also assumed here that the onset point of the signature in x 1 (n), can be identified, as misalignment will cause errors. When misalignment is unavoidable, the calculation described below may be repeated over a range of displacements, with the value of the best displacement used for decision-making purposes. 
         [0083]    The MLE estimate of β 12  is computed using: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       β 
                       ^ 
                     
                     12 
                   
                   = 
                   
                     
                       
                         
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                     or 
                   
                 
               
               
                 
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                   23 
                   ) 
                 
               
             
             
               
                 
                   
                     
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                       ^ 
                     
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                       · 
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                   ( 
                   24 
                   ) 
                 
               
             
           
         
       
     
         [0084]    where x=x 1 (n) and y=x 2 (n) 
         [0085]    If the template has been pre-normalised before it is used, so that x′·x=1, then: 
         [0000]    
       
         
           
             
               
                 
                   
                     
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                       ^ 
                     
                     12 
                   
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                       or 
                     
                   
                 
               
               
                 
                   ( 
                   25 
                   ) 
                 
               
             
             
               
                 
                   
                     
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                       ^ 
                     
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                       · 
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                     and 
                   
                 
               
               
                 
                   ( 
                   26 
                   ) 
                 
               
             
             
               
                 
                   
                     
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                       ~ 
                     
                     12 
                   
                   = 
                   
                     
                       
                         1 
                         
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                           - 
                           1 
                         
                       
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                   ( 
                   27 
                   ) 
                 
               
             
           
         
       
     
         [0086]    The following test statistic can be formed: 
         [0000]    
       
         
           
             
               
                 
                   
                     Z 
                     12 
                   
                   = 
                   
                     
                       
                         
                           
                             β 
                             ^ 
                           
                           12 
                         
                         - 
                         
                           β 
                           12 
                         
                       
                       
                         
                           
                             σ 
                             ~ 
                           
                           12 
                           2 
                         
                       
                     
                      
                     
                         
                     
                      
                     with 
                   
                 
               
               
                 
                   ( 
                   28 
                   ) 
                 
               
             
             
               
                 
                   
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                     12 
                   
                   ~ 
                   
                     t 
                      
                     
                       ( 
                       
                         N 
                         - 
                         1 
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   29 
                   ) 
                 
               
             
           
         
       
     
         [0087]    That is, Z 12  is distributed as a Student&#39;s t variable. It is also inversely proportional to the variation measure √{square root over ({circumflex over (σ)} 12   2 )} which provides a measure of the variation between the series. 
         [0088]    Classification can therefore be conducted by calculating the Z 12  test statistic for each template, and selecting the template with the best fit. 
         [0089]    Alternatively, Z 12  can be used for detection/localisation purposes wherein the null hypothesis is that the data contains only noise. A confidence level or false alarm rate can be set according to the magnitude of the Z 12  test statistic. 
         [0090]    When processing impulsive transients due to gunshots, that the duration (the time interval containing approximately 90% of the signals energy) of a ‘crack’ is approximately 0.001 seconds, or 20 samples at 20,000 samples per second. The duration of a ‘thump’ is approximately 0.01 seconds, or 200 samples. Ideally, the entire feature should be used for lag estimation and for classification. However, it has been found that coherence decreases, and variability increases, as the time after the initial wavefront onset time increases, primarily due to multi-path propagation and reverberation. It has also been found that it is useful to include sections of ‘quiet’ data in the series analysed, both before and after the feature of interest. The series analysed may therefore include an additional 25%-50% of the duration of the feature in quiet data before the feature, and an additional 50%-75% after the feature. 
       EXAMPLES 
       [0091]    By way of example, with respect to lag estimation and angular localisation,  FIG. 2  depicts a plot of two series of acoustic signal data samples for a signal of interest, obtained from two respective acoustic sensors. The data have been aligned as for the most likely lag—that is, the lag for which the test statistics according to the present invention are maximised.  FIGS. 3 ,  4 ,  5 ,  6 ,  7 , and  8  simply represent data series obtained for different signals of interest. 
         [0092]      FIG. 2A  is a plot of test statistics computed using four candidate lag estimation methods plotted as a function of the lag (or time-shift parameter), for the series shown in  FIG. 2 .  FIGS. 3A ,  4 A,  5 A,  6 A,  7 A and  8 A correspond to the series shown in the other respective figures. The four candidate methods are standard cross correlation (r 12 /R 12 ), normalised cross correlation (rho 12 ), Z M  test statistics according to the present invention, and Z 12  test statistics according to the present invention. As can be seen from the figures, the test statistics according to the present invention provide much sharper and more prominent peak at the most likely lag, with the Z M  test statistic providing the sharpest peaks of all. 
         [0093]    In  FIGS. 2A ,  3 A and  4 A, this makes little quantitative difference to the final selection of the best lag estimate. However, for  FIGS. 5A ,  6 A,  7 A and  8 A, the highest peak for the r 12  method is at the incorrect lag. This results in large angular errors being calculated for the incoming signal of interest. 
         [0094]    In addition to improved accuracy in the estimation of the lag between series, the Z M  and Z 12  variables have a known distribution, which allows a confidence level to be placed on the lag estimate. Poor lag estimates, for which the calculated test statistics are beneath a threshold value, can therefore be considered unreliable, and ignored. This improves accuracy by eliminating bad localisation results (or allowing for unknown classes). 
         [0095]    Whilst both methods according to the present invention give a greater ‘dynamic range’ than normalised cross correlation (giving greater difference between good and bad lag estimates), the Z M  method requires far less computational power than the Z 12  method. It can be calculated efficiently using sliding windows and, relative to normalised cross correlation, the Z M  method requires only a few more floating point operations per sample pair. On the other hand, the Z 12  method requires two loops through the data vector—the first to determine the MLE fitting coefficient, and the second to determine the variation measure. 
         [0096]    Yet another advantage of the Z M  method is that it can be extended to give the similarity of any number of series of signal data samples, whilst the other methods are only useful for two series at a time. 
         [0097]    Although an embodiment of the present invention has been described in the foregoing detailed description, it will be understood that the invention is not limited to the embodiment disclosed, but is capable of numerous rearrangements, modifications and substitutions without departing from the scope of the invention. Modifications and variations such as would be apparent to a skilled addressee are deemed within the scope of the present invention. For example, although particular reference has been made to acoustic signal analysis, the present invention has more broad application in the analysis of other types of signal, e.g. radar. Furthermore, although the Z M  and Z 12  methods were described in relation to localisation and classification respectively, both could be used for either application or for any situation where it is desirable to assess the similarity of series of signal data samples—i.e. the present invention may replace operations that require cross correlation, matched filtering or template matching giving greater expected performance (higher probability of detection and lower false alarm rate). Such operations are typically performed in radar and (active) sonar range processing, feature detection in images, classification of signals and passive direction finding of (intercepted) communication signals. 
         [0098]    It should be also appreciated that the present invention can be implemented in numerous ways, including as a process, an apparatus, a system, or a computer readable medium such as a computer readable storage medium or a computer network wherein program instructions are sent over optical or electronic communication links. Furthermore, the methods of the present invention could be implemented in hardware (as a computer processing device), on dedicated co-processing chips, possibly using Field Programmable Gate Arrays (FPGAs). This may enable real-time beam formation with 3-dimensional sensor arrays. It should be noted that the order of the steps of disclosed processes may be altered within the scope of the invention. 
         [0099]    Throughout this specification and the claims that follow unless the context requires otherwise, the words ‘comprise’ and ‘include’ and variations such as ‘comprising’ and ‘including’ will be understood to imply the inclusion of a stated integer or group of integers but not the exclusion of any other integer or group of integers. 
         [0100]    The reference to any prior art in this specification is not, and should not be taken as, an acknowledgment or any form of suggestion that such prior art forms part of the common general knowledge.