# Patent

## Title
Persymmetric parametric adaptive matched filters for detecting targets using space-time adaptive processing of radar signals

## Abstract

A method provides space-time adaptive processing (STAP) for target detection using adaptive matched filters (AMF). A generalized likelihood ratio test (GLRT) is determined where spatial and temporal correlation matrices Q and A are assumed. Then, the correlation matrices A and Q are replaced with maximum likelihood (ML) estimates obtained only from training signals subject to a persymmetric constraint.

## Background
FIELD OF THE INVENTION
This invention relates generally to signal processing, and in particular to space-time adaptive processing (STAP) for target detection using adaptive matched filters (AMF) applied to radar signals.
BACKGROUND OF THE INVENTION
Space-Time Adaptive Processing
Space-time adaptive processing (STAP) is frequently used in radar systems to detect a target. STAP has been known since the early 1970's. In airborne radar systems, STAP improves target detection when disturbances in an environment, e.g., noise, ground clutter and jamming, are a problem. The disturbances can be Gaussian or non-Gaussian. STAP can achieve order-of-magnitude sensitivity improvements in target detection.
Typically, STAP involves a two-dimensional filtering technique applied to signals received by a phased-array antenna with multiple spatial channels (antennas). Coupling the multiple spatial channels with time dependent pulse-Doppler waveforms leads to STAP. By applying statistics of interference of the environment, a space-time adaptive weight vector is formed. Then, the weight vector is applied to the coherent signals received by the radar to detect the target.
STAP Detector Filter
As shown in FIG. 1 for conventional STAP, observations include a test signal 101, a set of training signals 102, and a steering vector 103. In a homogeneous environment, the test signal shares the same covariance matrix with the training signals.
It is a known problem to detect a moving target from a moving platform in many applications including radar, wireless communications, and hyperspectral imaging acquired across the entire electromagnetic spectrum. STAP can to deal with strong spatially and temporally colored disturbances.
However, the conventional STAP is not suitable for many from practical applications due to excessive training requirements and a high computational complexity, for example, a covariance-matrix-based STAP detectors, which need K, JN training signals to ensure a full-rank estimate of the disturbance covariance matrix and have to invert a JN×JN matrix, where J denotes the number of antennas and N denotes the number of pulses.
As shown in FIG. 1, by modeling the disturbance as a multichannel auto-regressive (AR) process, a parametric STAP detector 100 decompose jointly spatio-temporal whitening of the covariance-matrix-based detectors into successive temporal 110 and spatial 120 whitening by using the estimate of the temporal correlation matrix A 130, and a spatial covariance matrix Q 140, given the test signal 101, training signals 102 and a steering vector 103, where (.)H 145 indicates a Hermitian transpose.
The results of the spatial whitenings 120 are multiplied 149 by each other to produce corresponding sums Σn=PN-1(.) 150, for which the magnitudes |.|2 170 are determined. The magnitudes are divided (/) 175 into each other to determine the test statistic TPAMF 180 of the parametric adaptive matched filter (PAMF). The test statistic 180 is compared to a threshold 190 to obtain a decision 195 whether a target is present or not in the test signal.
Matrix Estimation
As shown in FIG. 2, an estimated temporal correlation matrix A 130 and an estimated of spatial correlation matrix Q 140 are determined 200 as follows:

    
    
        a regression vector yk (n) 215 is constructed 210 from the training signals xk (n) 102;
        using the training signals 102, an estimate of an autocorrelation of the training signal {circumflex over (R)}xx 220 is determined;
        using the training signals and the regression vector, an estimate of a cross correlation of yk (n) 215 and xk (n) 102 is determined 240; using the regression vector 215, an estimate of an autocorrelation {circumflex over (R)}yy is determined 230;
        an estimate of the spatial covariance matrix Q 140 is determined 250 as
    
    






  
    
      
        
          
            Q
            ^
          
          =
          
            
              
                
                  R
                  ^
                
                xx
              
              -
              
                
                  
                    R
                    ^
                  
                  yx
                  H
                
                ⁢
                
                  
                    R
                    ^
                  
                  yy
                  
                    -
                    1
                  
                
                ⁢
                
                  
                    R
                    ^
                  
                  yx
                
              
            
            
              K
              ⁡
              
                (
                
                  N
                  -
                  P
                
                )
              
            
          
        
        ⁢
        
          
        
        ;
      
    
    
      130
    
  




and

    
    
        an estimate of the autoregressive coefficient matrix 130 is determined 260 Â=−{circumflex over (R)}yy−1{circumflex over (R)}yx.
    
    


SUMMARY OF THE INVENTION
The embodiments of our invention take advantages of both a multichannel-AR-based parametric adaptive matched filter (PAMF), and an underlying persymmetry constraint to provide a persymmetric PAMF (per-PAMF).
Our per-PAMF is a two-step method.

    
    
        1) First, we determine a generalized likelihood ratio test (GLRT) where spatial and temporal correlation matrices Q and A are assumed to be known.
        2) Then, we replace the correlation matrices A and Q with maximum likelihood (ML) estimates obtained only from training signals subject to the persymmetric constraint.
    
    





BRIEF DESCRIPTION OF THE DRAWINGS
FIG. 1 is a block diagram of a conventional parametric adaptive matched filter (PAMF);
FIG. 2 is a block diagram 200 of a procedure for estimating a temporal covariance matrix A and a spatial covariance matrix Q;
FIG. 3 is a block diagram of a persymmetric parametric adaptive matched filter (per-PAMF with adaptively estimated temporal and spatial covariance matrices according to embodiments of the invention;
FIG. 4 is a block diagram of a procedure for estimating the temporal covariance matrix and the spatial covariance matrix according to embodiments of the invention.
FIG. 5 is a graph comparing a probability of detection performances of the prior art PAMF and the persymmetric PAMF according to embodiments of the invention at various SINR 530, given five antenna elements (J=5), eleven pulses (N=11), eight training signals (K=8), regression order of two (P=2) and a probability of false alarm of 0.01 (Pf=0.01);
FIG. 6 is a graph of a probability of detection performances of the prior art PAMF and the persymmetric PAMF according to embodiments of the invention at various SINR 630, given five antenna elements (J=5), eleven pulses (N=11), four training signals (K=4), regression order of two (P=2) and a probability of false alarm of 0.01 (Pf=0.01); and
FIG. 7 is a graph of a probability of detection performances of the prior art PAMF and the persymmetric PAMF according to embodiments of the invention at various SINR 730, given five antenna elements (J=5), eleven pulses (N=11), 2 training signals (K=2), regression order of two (P=2) and a probability of false alarm of 0.01 (Pf=0.01).



DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS
Overview
As shown in FIG. 3, the embodiments of our invention provide a persymmetric parametric adaptive matched filter (per-PAMF) 300 with an adaptively estimated temporal correlation matrix A 330 and an adaptively estimated spatial covariance matrix Q 340 obtained from training signals 302.
The filter can also be implemented as a method performed in a processor connected to a memory and input/output interfaces as known in the art. As can be seen the general structure of our filter 300 is similar to the conventional filter 100 shown in FIG. 1.
The AMD 300 filter can be used to detect targets using space-time adaptive processing (STAP) of test signals 301, and a generalized likelihood ratio test (GLRT).
Differences from Prior Art
The major differences are in how the matrices Q and A are estimated, and the resulting test statistic Tper-PAMF from these two matrices, compare FIGS. 2 and 4.
Method Overview
Our per-PAMF method basically has two steps.
First, we determine a generalized likelihood ratio test (GLRT)




  
    T
    =
    
      
        
          max
          α
        
        ⁢
        
          
            p
            1
          
          ⁡
          
            (
            
              
                
                  x
                  0
                
                ;
                α
              
              ,
              A
              ,
              Q
            
            )
          
        
      
      
        
          p
          0
        
        ⁡
        
          (
          
            
              
                x
                0
              
              ;
              A
            
            ,
            Q
          
          )
        
      
    
  
  ,




where {Pi}i=01 are likelihood functions under a first hypothesis H0 that the target is present and a second hypothesis H1 that the target is not present, α is an unknown complex-values amplitude, x0 is the test signal, and max is a function that evaluates to a maximum, and the spatial and temporal correlation matrices Q and A are known.

Second, we replace the correlation matrices A and Q with maximum likelihood (ML) estimates {circumflex over (Q)} and Â obtained from training signals subject to a persymmetric constraint.
Test Hypothesis
Specifically, we use the following hypothesis testing problem:

H0:x0=d0,xk=dk,k=1, . . . , K, 

H1:x0=αs+d0,xk=dk,k=1, . . . , K,  (1)

where the hypothesis H0 is that the target is not present in the test signal x0 301, H1 is the hypothesis that the target is present, xk are target free training signals 302, x0 is the JN×1 test signal 301, which is a Kronecker () product between space and temporal steering vectors, s is a presumed known array response 303, α is an unknown complex-values amplitude, d0 and dk are noise terms with covariance matrices R0 and R, respectively.

Joint Decomposition
Like before, we jointly decompose spatio-temporal whitening of the covariance-matrix-based detector into successive temporal 310 and spatial 320 whitening by using the estimates of the temporal correlation matrix A 330 and a spatial covariance matrix Q 340, given the test signal 301, the training signals 302 and a steering vector 303, and where (.)H 145 indicates a Hermitian transpose.
The results of the spatial whitenings 320 are multiplied 349 by each other to produce corresponding sums




  
    
      ∑
      
        n
        =
        P
      
      
        N
        -
        1
      
    
    ⁢
    
      
        (
        .
        )
      
      ⁢
      
        
      
      ⁢
      350
    
  
  ,




for which the magnitudes |.|2 370 are determined, respectively. The magnitudes are divided (/) 375 into each other to determine our test statistic Tper-PAMF 380. Finally, the test statistic 380 is compared to a threshold 390 to obtain a decision 395 whether a target is present or not in the test signal.

We decompose jointly the spatio-temporal whitening of the covariance matrix-based detectors into successive the temporal 310 and the spatial 320 whitening by using the estimate of the temporal correlation matrix A 330, and a spatial covariance matrix Q 340, given the test signal 301, the training signals 302 and a steering vector 103, an auto-regressive (AR) process.
Let xk (n)εCJ×1 302 denote the nth temporal training signal given by




  
    
      
        
          
            
              x
              k
            
            ⁡
            
              (
              n
              )
            
          
          =
          
            
              -
              
                
                  ∑
                  
                    p
                    =
                    1
                  
                  P
                
                ⁢
                
                  
                
                ⁢
                
                  
                    
                      A
                      H
                    
                    ⁡
                    
                      (
                      p
                      )
                    
                  
                  ⁢
                  
                    
                      x
                      k
                    
                    ⁡
                    
                      (
                      
                        n
                        -
                        p
                      
                      )
                    
                  
                
              
            
            +
            
              
                ɛ
                k
              
              ⁡
              
                (
                n
                )
              
            
          
        
        ⁢
        
          

        
        ⁢
        where
        ⁢
        
          

        
        ⁢
        
          
            
              ɛ
              k
            
            ⁡
            
              (
              n
              )
            
          
          ∼
          
            C
            ⁢
            
              
            
            ⁢
            
              N
              ⁡
              
                (
                
                  0
                  ,
                  Q
                
                )
              
            
          
        
      
    
    
      
        (
        2
        )
      
    
  




is the Jth channel (antenna), which is temporally white but spatially colored Gaussian noise with Q denoting an unknown J×J temporal correlation matrix. The antenna elements are placed symmetrically with respect to a phase center, e.g., the middle of the antenna element. Thus, the spatial covariance matrix is Q imposed to be persymmetric by satisfying Q=EQ*E, where E denotes an exchange matrix, which is 1 at anti-diagonal elements and 0 otherwise.

Persymmetry
As defined herein, and as known in the art, a square matrix is persymmetric when the matrix is symmetric about its upper right to lower left diagonal. Generally, this type of matrix is also known as a Hankel matrix with constant (positive sloping) skew-diagonals:




  
    [
    
      
        
          a
        
        
          b
        
        
          c
        
        
          d
        
        
          e
        
      
      
        
          b
        
        
          c
        
        
          d
        
        
          e
        
        
          f
        
      
      
        
          c
        
        
          d
        
        
          e
        
        
          f
        
        
          g
        
      
      
        
          d
        
        
          e
        
        
          f
        
        
          g
        
        
          h
        
      
      
        
          e
        
        
          f
        
        
          g
        
        
          h
        
        
          i
        
      
    
    ]
  
  .




In the receiver, a tapped delay line with symmetrically spaced taps with respect to the mid delay value is used. In this case, temporal persymmetry occurs, that is

A(p)=EA*(p)E 

If the temporal and spatial covariance matrices Q and A are known, then maximum likelihood estimates of the matrices Â 330 and {circumflex over (Q)} 340 are obtained from the training signals according to the invention.
Our processing of the training signals differs from the conventional processing because we exploit persymmetric properties of the covariance matrices A and Q, i.e., the square matrices are symmetric along their main diagonal.
Joint Likelihood
A joint likelihood of the training signals 302 can be written as




  
    
      
        
          
            
              p
              ⁡
              
                (
                
                  
                    x
                    1
                  
                  ,
                  ...
                  ⁢
                  
                    
                  
                  ,
                  
                    
                      x
                      K
                    
                    ;
                    A
                  
                  ,
                  Q
                
                )
              
            
            =
            
              
                [
                
                  
                    1
                    
                      
                        π
                        J
                      
                      ⁢
                      
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                        Q
                        
                      
                    
                  
                  ⁢
                  
                    e
                    
                      -
                      
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                        ⁡
                        
                          (
                          
                            
                              Q
                              
                                -
                                1
                              
                            
                            ⁢
                            
                              Γ
                              0
                            
                          
                          )
                        
                      
                    
                  
                
                ]
              
              
                K
                ⁡
                
                  (
                  
                    N
                    -
                    P
                  
                  )
                
              
            
          
          ,
          
            

          
          ⁢
          where
        
        ⁢
        
          

        
        ⁢
        
          
            
              
                K
                ⁡
                
                  (
                  
                    N
                    -
                    P
                  
                  )
                
              
              ⁢
              
                Γ
                0
              
            
            =
            
              
                ∑
                
                  k
                  =
                  1
                
                K
              
              ⁢
              
                
              
              ⁢
              
                
                  ∑
                  
                    n
                    =
                    P
                  
                  
                    N
                    -
                    1
                  
                
                ⁢
                
                  
                
                ⁢
                
                  
                    
                      ɛ
                      k
                    
                    ⁡
                    
                      (
                      n
                      )
                    
                  
                  ⁢
                  
                    
                      ɛ
                      k
                      H
                    
                    ⁡
                    
                      (
                      n
                      )
                    
                  
                
              
            
          
          ,
          and
        
        ⁢
        
          

        
        ⁢
        
          
            
              ɛ
              k
            
            ⁡
            
              (
              n
              )
            
          
          =
          
            
              
                x
                k
              
              ⁡
              
                (
                n
                )
              
            
            +
            
              
                ∑
                
                  p
                  =
                  1
                
                P
              
              ⁢
              
                
              
              ⁢
              
                
                  
                    A
                    H
                  
                  ⁡
                  
                    (
                    p
                    )
                  
                
                ⁢
                
                  
                    
                      x
                      k
                    
                    ⁡
                    
                      (
                      
                        n
                        -
                        p
                      
                      )
                    
                  
                  .
                
              
            
          
        
      
    
    
      
        (
        3
        )
      
    
  




Alternatively,

K(N−P)Γ0 

can be written as





  
    
      
        
          
            
              K
              ⁡
              
                (
                
                  N
                  -
                  P
                
                )
              
            
            ⁢
            
              Γ
              0
            
          
          =
          
            
              
                R
                ^
              
              xx
            
            +
            
              
                A
                H
              
              ⁢
              
                
                  R
                  ^
                
                yx
              
            
            +
            
              
                
                  R
                  ^
                
                yx
                H
              
              ⁢
              A
            
            +
            
              
                A
                H
              
              ⁢
              
                
                  R
                  ^
                
                yy
              
              ⁢
              A
            
          
        
        ,
      
    
    
      
        
      
    
  
  
    
      where
    
    
      
        
      
    
  
  
    
      
        
          A
          ⁢
          
            =
            Δ
          
          ⁢
          
            
              [
              
                
                  
                    A
                    H
                  
                  ⁡
                  
                    (
                    1
                    )
                  
                
                ,
                
                  
                    A
                    H
                  
                  ⁡
                  
                    (
                    2
                    )
                  
                
                ,
                ...
                ⁢
                
                  
                
                ,
                
                  
                    A
                    H
                  
                  ⁡
                  
                    (
                    P
                    )
                  
                
              
              ]
            
            H
          
        
        ,
      
    
    
      
        
      
    
  
  
    
      
        
          
            
              
                R
                ^
              
              xx
            
            =
            
              
                ∑
                
                  k
                  =
                  1
                
                K
              
              ⁢
              
                
              
              ⁢
              
                
                  ∑
                  
                    n
                    =
                    P
                  
                  
                    N
                    -
                    1
                  
                
                ⁢
                
                  
                
                ⁢
                
                  
                    
                      x
                      k
                    
                    ⁡
                    
                      (
                      n
                      )
                    
                  
                  ⁢
                  
                    
                      x
                      k
                      H
                    
                    ⁡
                    
                      (
                      n
                      )
                    
                  
                
              
            
          
          ,
        
        ⁢
        
          
        
      
    
    
      421
    
  
  
    
      
        
          
            
              R
              ^
            
            yy
          
          =
          
            
              ∑
              
                k
                =
                1
              
              K
            
            ⁢
            
              
            
            ⁢
            
              
                ∑
                
                  n
                  =
                  P
                
                
                  N
                  -
                  1
                
              
              ⁢
              
                
              
              ⁢
              
                
                  
                    y
                    k
                  
                  ⁡
                  
                    (
                    n
                    )
                  
                
                ⁢
                
                  
                    y
                    k
                    H
                  
                  ⁡
                  
                    (
                    n
                    )
                  
                
              
            
          
        
        ,
      
    
    
      431
    
  
  
    
      
        
          
            
              R
              ^
            
            yx
          
          =
          
            
              ∑
              
                k
                =
                1
              
              K
            
            ⁢
            
              
            
            ⁢
            
              
                ∑
                
                  n
                  =
                  P
                
                
                  N
                  -
                  1
                
              
              ⁢
              
                
              
              ⁢
              
                
                  
                    y
                    k
                  
                  ⁡
                  
                    (
                    n
                    )
                  
                
                ⁢
                
                  
                    x
                    k
                    H
                  
                  ⁡
                  
                    (
                    n
                    )
                  
                
              
            
          
        
        ,
      
    
    
      441
    
  




and yk (n) is the regression vector 415 given by

yk(n)≡[xkT(n−1),xkT(n−2), . . . , xkT(n−P)]T 

see FIG. 4.

By exploiting the persymmetric property of the matrix Q according to the invention, it is possible to write




  
    tr
    ⁡
    
      (
      
        
          Q
          
            -
            1
          
        
        ⁢
        
          Γ
          0
        
      
      )
    
  
  =
  
    
      tr
      ⁡
      
        (
        
          
            Q
            
              -
              1
            
          
          ⁢
          
            
              
                Γ
                0
              
              +
              
                E
                ⁢
                
                  
                
                ⁢
                
                  Γ
                  0
                  *
                
                ⁢
                E
              
            
            2
          
        
        )
      
    
    .
  




This leads to the persymmetric estimate of the spatial matrix Q 340. As a result, the logarithm of the likelihood function in Equation (3) can be written as




  
    
      
        
          
            ln
            ⁢
            
              
            
            ⁢
            p
          
          ∝
          
            
              -
              ln
            
            ⁢
            
              
              
                
                  1
                  2
                
                ⁢
                
                  (
                  
                    
                      Γ
                      0
                    
                    +
                    
                      E
                      ⁢
                      
                        
                      
                      ⁢
                      
                        Γ
                        0
                        *
                      
                      ⁢
                      E
                    
                  
                  )
                
              
              
            
          
        
        ,
      
    
    
      
        (
        4
        )
      
    
  




Therefore, the persymmetric maximum likelihood estimate of the temporal correlation matrix A 330 can be determined by minimizing the determinant of

K(N−P)(Γ0+EΓ0*E)/2

To determine the persymmetric maximum likelihood estimate of the matrix A, we define

A=EBA*E, where

EB=IPE, where

where IP is a P by P identity matrix.

According to embodiments of the invention, the persymmetric estimate 420 of the autocorrelation of xk (n) 421 is given by

{circumflex over (R)}xx,P=({circumflex over (R)}xx+E{circumflex over (R)}xx*E)/2  (5)

According to the invention, the persymmetric estimate 430 of the autocorrelation of the regression vector yk (n) 431 is given by

{circumflex over (R)}yx,P=({circumflex over (R)}yx+EBE{circumflex over (R)}yx*E)/2  (6)

According to the invention, the persymmetric estimate 440 of the crosscorrelation of yk(n) and xk(n) 441 is given by

{circumflex over (R)}yy,P=({circumflex over (R)}yyEB{circumflex over (R)}yy*EB)/2  (7)

As a result,




  
    
      
        
          
        
        ⁢
        
          
            
              K
              ⁡
              
                (
                
                  N
                  -
                  P
                
                )
              
            
            ⁢
            
              
                (
                
                  
                    Γ
                    0
                  
                  +
                  
                    E
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Γ
                      0
                      *
                    
                    ⁢
                    E
                  
                
                )
              
              /
              2
            
          
          ⁢
          
            

          
          ⁢
          
            
          
          ⁢
          
            
              can
              ⁢
              
                
              
              ⁢
              be
              ⁢
              
                
              
              ⁢
              rewritten
              ⁢
              
                
              
              ⁢
              as
            
            ⁢
            
              

            
            ⁢
            
              
            
            =
            
              
                
                  
                    R
                    ^
                  
                  
                    xx
                    ,
                    P
                  
                
                +
                
                  
                    A
                    H
                  
                  ⁢
                  
                    
                      R
                      ^
                    
                    
                      yx
                      ,
                      P
                    
                  
                
                +
                
                  
                    
                      R
                      ^
                    
                    
                      yx
                      ,
                      P
                    
                    H
                  
                  ⁢
                  A
                
                +
                
                  
                    A
                    H
                  
                  ⁢
                  
                    
                      R
                      ^
                    
                    
                      yy
                      ,
                      P
                    
                  
                  ⁢
                  A
                
              
              ⁢
              
                

              
              =
              
                
                  
                    (
                    
                      
                        A
                        H
                      
                      +
                      
                        
                          
                            R
                            ^
                          
                          
                            yx
                            ,
                            P
                          
                          H
                        
                        ⁢
                        
                          
                            R
                            ^
                          
                          
                            yy
                            ,
                            P
                          
                          
                            -
                            1
                          
                        
                      
                    
                    )
                  
                  ⁢
                  
                    
                      
                        
                          R
                          ^
                        
                        
                          yy
                          ,
                          P
                        
                      
                      ⁡
                      
                        (
                        
                          
                            A
                            H
                          
                          +
                          
                            
                              
                                R
                                ^
                              
                              
                                yx
                                ,
                                P
                              
                              H
                            
                            ⁢
                            
                              
                                R
                                ^
                              
                              
                                yy
                                ,
                                P
                              
                              
                                -
                                1
                              
                            
                          
                        
                        )
                      
                    
                    H
                  
                
                +
                
                  
                    (
                    
                      
                        
                          R
                          ^
                        
                        
                          xx
                          ,
                          P
                        
                      
                      -
                      
                        
                          
                            R
                            ^
                          
                          
                            yx
                            ,
                            P
                          
                          H
                        
                        ⁢
                        
                          
                            R
                            ^
                          
                          
                            yy
                            ,
                            P
                          
                          
                            -
                            1
                          
                        
                        ⁢
                        
                          
                            R
                            ^
                          
                          
                            yx
                            ,
                            P
                          
                        
                      
                    
                    )
                  
                  .
                
              
            
          
        
      
    
    
      
        (
        8
        )
      
    
  




It follows that




  
    
      
        
          
            
              
                (
                
                  
                    Γ
                    0
                  
                  +
                  
                    E
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Γ
                      0
                      *
                    
                    ⁢
                    E
                  
                
                )
              
              /
              2
            
            ≥
            
              
                (
                
                  
                    Γ
                    0
                  
                  +
                  
                    E
                    ⁢
                    
                      
                    
                    ⁢
                    
                      Γ
                      0
                      *
                    
                    ⁢
                    E
                  
                
                )
              
              /
              2
            
          
          ⁢
          
            |
            
              A
              =
              
                
                  -
                  
                    
                      R
                      ^
                    
                    
                      yx
                      ,
                      P
                    
                    H
                  
                
                ⁢
                
                  
                    R
                    ^
                  
                  
                    yy
                    ,
                    P
                  
                  
                    -
                    1
                  
                
              
            
          
        
        =
        
          
            
              
                R
                ^
              
              
                xx
                ,
                P
              
            
            -
            
              
                
                  R
                  ^
                
                
                  yx
                  ,
                  P
                
                H
              
              ⁢
              
                
                  R
                  ^
                
                
                  yy
                  ,
                  P
                
                
                  -
                  1
                
              
              ⁢
              
                
                  R
                  ^
                
                
                  yx
                  ,
                  P
                
              
            
          
          
            K
            ⁡
            
              (
              
                N
                -
                P
              
              )
            
          
        
      
    
    
      
        (
        9
        )
      
    
  




The persymmetric estimate of the temporal correlation matrix A 330 is

ÂPML=−{circumflex over (R)}yy,P−1{circumflex over (R)}yx,P,  (10)

and the persymmetric estimate of the spatial correlation matrix Q 340 is





  
    
      
        
          
            Q
            ^
          
          PML
        
        =
        
          
            
              
                
                  R
                  ^
                
                
                  xx
                  ,
                  P
                
              
              -
              
                
                  
                    R
                    ^
                  
                  
                    yx
                    ,
                    P
                  
                  H
                
                ⁢
                
                  
                    R
                    ^
                  
                  
                    yy
                    ,
                    P
                  
                  
                    -
                    1
                  
                
                ⁢
                
                  
                    R
                    ^
                  
                  
                    yx
                    ,
                    P
                  
                
              
            
            
              K
              ⁡
              
                (
                
                  N
                  -
                  P
                
                )
              
            
          
          .
        
      
    
    
      
        (
        11
        )
      
    
  




The resulting test statistic 380 for the persymmetric PAMF is




  
    
      
        
          
            T
            
              Per
              -
              PAMF
            
          
          =
          
            
              
                
                  
                  
                    
                      ∑
                      
                        n
                        =
                        P
                      
                      
                        N
                        -
                        1
                      
                    
                    ⁢
                    
                      
                    
                    ⁢
                    
                      
                        
                          
                            
                              s
                              ~
                            
                            ^
                          
                          P
                          H
                        
                        ⁡
                        
                          (
                          n
                          )
                        
                      
                      ⁢
                      
                        
                          Q
                          ^
                        
                        PML
                        
                          -
                          1
                        
                      
                      ⁢
                      
                        
                          
                            
                              x
                              ~
                            
                            ^
                          
                          
                            0
                            ,
                            P
                          
                        
                        ⁡
                        
                          (
                          n
                          )
                        
                      
                    
                  
                  
                
                2
              
              
                
                  ∑
                  
                    n
                    =
                    P
                  
                  
                    N
                    -
                    1
                  
                
                ⁢
                
                  
                
                ⁢
                
                  
                    
                      
                        
                          s
                          ~
                        
                        ^
                      
                      P
                      H
                    
                    ⁡
                    
                      (
                      n
                      )
                    
                  
                  ⁢
                  
                    
                      Q
                      ^
                    
                    PML
                    
                      -
                      1
                    
                  
                  ⁢
                  
                    
                      
                        
                          s
                          ~
                        
                        ^
                      
                      P
                    
                    ⁡
                    
                      (
                      n
                      )
                    
                  
                
              
            
            ⁢
            
              
                ≷
                
                  H
                  0
                
              
              
                H
                1
              
            
            ⁢
            
              
                γ
                ⁢
                
                  
                
                ⁢
                Per
              
              -
              PAMF
            
          
        
        ,
      
    
    
      
        (
        12
        )
      
    
  




where γPer-PAMF is the threshold used by the threshold comparator 390. The threshold is a preset probability of false alarm.

The temporally whitened steering vector {tilde over (Ŝ)}P and the temporally whitened test signal {tilde over ({circumflex over (X)}0,P are obtained as




  
    
      
        
          
            
              
                s
                ~
              
              ^
            
            p
          
          =
          
            
              s
              ⁡
              
                (
                n
                )
              
            
            +
            
              
                ∑
                
                  p
                  =
                  1
                
                P
              
              ⁢
              
                
              
              ⁢
              
                
                  
                    
                      A
                      ^
                    
                    PML
                    H
                  
                  ⁡
                  
                    (
                    p
                    )
                  
                
                ⁢
                
                  s
                  ⁡
                  
                    (
                    
                      n
                      -
                      p
                    
                    )
                  
                
              
            
          
        
        ,
        
          

        
        ⁢
        and
      
    
    
      
        (
        13
        )
      
    
  
  
    
      
        
          
            
              x
              ~
            
            ^
          
          
            0
            ,
            p
          
        
        =
        
          
            
              x
              0
            
            ⁡
            
              (
              n
              )
            
          
          +
          
            
              ∑
              
                p
                =
                1
              
              P
            
            ⁢
            
              
            
            ⁢
            
              
                
                  
                    A
                    ^
                  
                  PML
                  H
                
                ⁡
                
                  (
                  p
                  )
                
              
              ⁢
              
                
                  
                    x
                    0
                  
                  ⁡
                  
                    (
                    
                      n
                      -
                      p
                    
                    )
                  
                
                .
              
            
          
        
      
    
    
      
        (
        14
        )
      
    
  




The test statistic 380 is compared to a threshold 390 to decide 395 whether the target is present or not.
Performances of the persymmetric PAMF according to embodiments of the invention are shown in FIGS. 5, 6 and 7.
In FIG. 5, the settings 510 are as follows: the number of antenna elements J=5, the number of pulses N=11, the number of training signals K=8, regression order of P=2, and the probability of false alarm is Pf=0.01. At signal to interference and noise ratio (SINR) 530 varying from −5 dB to 20 dB, the persymmetric PAMF has a higher probability of detection 520 performance than that of the prior art PAMF method.
in FIG. 6, the setting 610 as shown the number of antenna elements J=5, the number of pulses N=11, the number of training signals K=4, regression order of P=2 and the probability of false alarm is Pf=0.01. At signal to interference and noise ratio 630 varying from −5 dB to 20 dB, the persymmetric PAMF has a much higher probability of detection 620 performance than that of the prior art PAMF method. In this limited training case, the performance difference between per-PAMF and PAMF increases.
In FIG. 7, the setting 710 are as follows, the number of antenna elements J=5, the number of pulses N=11, the number of training signals K=2, regression order of P=2 and the probability of false alarm is Pf=0.01. At signal to interference and noise ratio 730 varying from −5 dB to 20 dB, the persymmetric PAMF has a dramatically higher probability of detection 720 performance than that of the prior art PAMF method.
EFFECT OF THE INVENTION
The invention improves a detection performance of multi-channel receivers, and, in particular, to improve the detection of signals masked by the presence of partially correlated Gaussian or non-Gaussian noise plus additive Gaussian thermal white noise. The present invention can be used in radar, sonar, geophysical, and biomedical applications.
The embodiments of the invention use a multichannel-AR-based parametric adaptive matched filter (PAMF), and an underlying persymmetry to provide a persymmetric PAMF (per-PAMF). The per-PAMF is determined from the PMF by replacing the correlation matrices A and Q with maximum likelihood (ML) estimates obtained from the training signals under the persymmetric constraints.
Although the invention has been described by way of examples of preferred embodiments, it is to be understood that various other adaptations and modifications can be made within the spirit and scope of the invention. Therefore, it is the object of the appended claims to cover all such variations and modifications as come within the true spirit and scope of the invention.

## Claims

1. An apparatus for detecting a moving target using radar signals, comprising:
a radar system emitting the radar signals as training signals and test signals; wherein the radar system further comprises:
a persymmetric parametric adaptive matched filter (per-PAMF) wherein the per-PAMF system is configured to determine an estimated spatial correlation matrix {circumflex over (Q)} and an estimated temporal correlation matrix Â using space-time adaptive processing, wherein a spatial correlation matrix A and a temporal correlation matrix Q are known from the training signals, and the per-PAMF is configured to determine a generalized likelihood ratio test (GLRT)





  
    T
    =
    
      
        
          max
          α
        
        ⁢
        
          
            p
            1
          
          ⁡
          
            (
            
              
                
                  x
                  0
                
                ;
                α
              
              ,
              A
              ,
              Q
            
            )
          
        
      
      
        
          p
          0
        
        ⁡
        
          (
          
            
              
                x
                0
              
              ;
              A
            
            ,
            Q
          
          )
        
      
    
  
  ,




where {Pi}1i=0 are likelihood functions under a first hypothesis H0 that the target is present and a second hypothesis H1 that the target is not present in the test radar signals, α is an unknown complex-values amplitude of the test radar signals, x0 represents the test signals, and max is a function that evaluates to a maximum, and the per-PAMF is configured to determine a maximum likelihood (ML) of the spatial correlation matrix Q as {circumflex over (Q)} and the temporal correlation matrix A as Â, wherein the correlation matrix Q is subject to a persymmetric constraint, and the per PAMF is configured to replace the spatial correlation matrices A and the temporal correlation matrix Q in the GLRT with {circumflex over (Q)} and Â.