# Patent

## Title
Doppler-compensated radar pulse compression processing system and method

## Abstract

A radar receiver system includes a receiver, a processor including a Doppler Compensated Adaptive Pulse Compressor (DCAPC) algorithm, possible other intermediate processing and a target detector. The DCAPC algorithm processes samples of a radar return signal, applies Minimum Mean Square Error (MMSE), or alternatively matched filtering, to the radar return signal to obtain initial radar impulse response estimates, computes power estimates, estimates a range cell Doppler shift for each range cell, computes range-dependent filters, applies the MMSE filters, and then repeats the cycle for subsequent reiterative stages until a desired length−L range window is reached, thereby resolving the scatterer from noise and other scatterers.

## Background
FIELD OF THE INVENTION
This invention relates to a radar processing method and system using adaptive pulse compression. More particularly, the invention relates to adaptive pulse compression radar processing that includes Doppler-mismatch compensation.
BACKGROUND OF THE INVENTION
Pulse compression allows a radar to obtain the range resolution of a short pulse without the need for very high peak transmit power. This is accomplished by transmitting a long pulse that is phase or frequency modulated. The modulated pulse, or waveform, is reflected back to the radar by scatterers that lie in the transmission path. This process can be viewed as the convolution of the transmitted waveform with an impulse response that is representative of the range profile illuminated by the radar. The purpose of pulse compression is then to estimate the range profile impulse response based upon the known transmitted waveform and the received radar return signal.
The traditional method of pulse compression is known as matched filtering which has been shown to maximize the received signal-to-noise ratio (SNR) for a solitary point scatterer. In the discreet domain, matched filtering can be expressed as

{circumflex over (x)}NIF(Λ)=sHy(Λ)  (1)

where {circumflex over (x)}NIF(Λ), for Λ=0, . . . , L−1, is the estimate of the Λth delayed sample (range cell index). s=[s1 s2 . . . sN]T is the length−N sampled version of the transmitted waveform. y(Λ)=[y(Λ) y(Λ+1) . . . y(Λ+N−1)]T is a vector of N contiguous samples of the received radar return signal, and (•)H is the complex-conjugate transpose (or Hermitian) operation. Matched filtering, however, is plagued by range sidelobes that can cause the presence of large targets to mask nearby small targets thus limiting radar sensitivity.

Numerous approaches have been developed to reduce the range sidelobes resulting from matched filtering. These approaches include optimal mismatched filtering and Least-Squares estimation, as well as numerous other variations of each. Adaptivity in pulse compression was first introduced by W. F. Gabriel. “Superresolution techniques in the range domain.” IEEE International Radar Conf., pp. 263-267. May 1990 and “Superresolution techniques and ISAR imaging,” Naval Research Laboratory Memorandum Report, No. 6714. Sep. 21, 1990. in which numerous (>200) pulses were employed to estimate a sample covariance matrix with which to cancel the range sidelobes.
In another approach described in U.S. Pat. No. 6,940,450. Blunt et al., issued Sep. 6, 2005, and incorporated herein by reference, adaptive pulse compression (APC) by way of Reiterative Minimum Mean-Square Error (RMMSE) estimation has been shown to be an effective ay to mitigate range sidelobes on a single pulse basis. However, while still superior to the matched filter, APC can suffer degradation due to Doppler mismatch for targets with high radial velocity with respect to the radar.
There is, therefore, a need for an adaptive pulse compression system with improved robustness, accuracy, and resolution.
SUMMARY OF THE INVENTION
According to the invention, a method for processing a received radar pulse from a moving target includes a) receiving a return signal; b) obtaining L+(2M−1)(N−1) samples y of the return signal, where y(Λ)={tilde over (x)}T(Λ)s+v(Λ); c) applying Minimum Mean Square Error (MMSE) estimation to each successive N samples to obtain initial impulse response estimates [{circumflex over (x)}1{−(M−1)(N−1)}, . . . , {circumflex over (x)}1{−1}, {circumflex over (x)}1{0}, . . . , {circumflex over (x)}1{L−1}, {circumflex over (x)}1{L}, . . . , {circumflex over (x)}1{L−1+(M−1)(N−1)}]; d) computing power estimates {circumflex over (ρ)}1(Λ)=|{circumflex over (x)}1(Λ)|2 for Λ=−(M−1)(N−1), . . . , L−1+(M−1)(N−1; e) estimating a range cell Doppler shift for each range cell above user-selected magnitude; f) computing MMSE filters according to w(Λ)=ρ(Λ)({tilde over (C)}(Λ)+R)−1[s∘e(Λ)], where ρ(Λ)=|x(Λ)|2 is the power of x(Λ). R=E[v(Λ)vH(Λ)] is the noise covariance matrix, and the range cell Doppler shifts are inserted; g) applying the MMSE filters to y to obtain [{circumflex over (x)}2{−(M−2)(N−1)}, . . . , {circumflex over (x)}2{−1}, . . . , {circumflex over (x)}2{−1}, {circumflex over (x)}2{L0}, . . . ,{circumflex over (x)}2{L−1}, {circumflex over (x)}2{L}, . . . , {circumflex over (x)}2{L−1+(M−2)(N−1)}]; and h) repeating (e)-(g) for subsequent reiterative stages until a desired length−L range window is reached, thereby resolving the scatterers from noise and other scatterers.
Alternatively matched filtering may be substituted for the initial MMSE in c), with the method otherwise being carried out as described for MMSE.
Also according to the invention, a radar receiver system includes a receiver, a processor including the Doppler Compensated Adaptive Pulse Compressor (DCAPC) algorithm, possible other intermediate pressing, and a target detector. The DCAPC algorithm processes samples of a radar return signal, initially applies Minimum Mean Square Error (MMSE), or alternatively matched filtering, to the radar return signal to obtain initial radar impulse response estimates, computes power estimates, estimates a range cell Doppler shift for each range cell, computers range-dependent filters, applies the MMSE filters, and then repeats the cycle for subsequent reiterative stages until a desired length-L range window is reached, thereby resolving the scatterer from noise and other scatterers.
The invention has obvious applications in military radar, and it is also useful for civilian radar, e.g. in airport radar system and in weather radar systems, e.g. Doppler radar, and other environmental radar applications. It may also find use in range profiling, image recognition for Synthetic Aperture Radar (SAR), and Inverse SAR (ISAR).
Additional features and advantages of the present invention will be set forth in, or be apparent from, the detailed description of preferred embodiments which follows.
BRIEF DESCRIPTION OF THE DRAWINGS
FIG. 1 is a schematic diagram of a radar processing system according to the invention.
FIG. 2 is a block diagram of a 3-stage RMMSE adaptive pulse compression algorithm according to the invention.
FIG. 3 is a graph showing a range profile estimate comparison of other processes to the invention for a stationary target.
FIG. 4 is a graph as in FIG. 3 except that the target is moving.
FIG. 5 is a graph as in FIG. 4 except there are three fast-moving targets in close proximity.


DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS
Referring now to FIG. 1, an adaptive pulse compression with Doppler compensation radar processing system 10 includes a transmitter 12 for transmitting a modulated pulse (or waveform) 14, that upon interacting with its transmission environment's unknown impulse response 15 (to be estimated, as is described further below) forms a signal 16 that is a convolution of waveform 14 and impulse response 15, an analog front-end receiver 18 for receiving signal 16, an optional analog-to-digital (A/D) converter 20, a processor 22 having a computer readable storage medium, such as a hard drive, RAM, or the like, for processing analog or digital signal 16, and a detector 24. The processor 22 includes an adaptive pulse compression with Doppler compensation algorithm, as is described below.
The new Doppler Compensated Adaptive Pulse Compressor (DCAPC) is based upon a minimum mean-square error (MMSE) formulation. We denote the discrete-time version of the transmitted waveform as the vector s having N samples. The Λth sample of the received radar return is defined as

{tilde over (y)}(Λ)={tilde over (x)}T(Λ)s+v(Λ)  (2)

where {tilde over (x)}(Λ) is modeled as

{tilde over (x)}(Λ)=x(Λ)∘e(Λ)  (3)

for Λ=0, . . . , L+N=2, where x(Λ)=[x(Λ) x(Λ−1) . . . x(Λ−N+1)]T is the vector of N contiguous samples of the range profile impulse response, v(Λ) is additive noise, (·)T is the transpose operation, and ∘ represents the Hadamard product (element-by-element multiplication). The vector e(Λ)=[1ejθt−1 . . . ej(N−1)θt−N−1] is comprised of the Doppler phase shifts for the same N contiguous samples as x(Λ) in which θΛ is the Doppler phase shift induced by a moving target at the Λth range cell between successive received samples. The collection of N samples of the received return signal {tilde over (y)}(Λ)=[{tilde over (y)}(Λ) {tilde over (y)}(Λ+1) . . . {tilde over (y)}(Λ+N−1)]T can be expressed as

{tilde over (y)}(Λ)=ÃT(Λ)s+v(Λ)  (4)

where v(Λ)=[v(Λ) v(Λ+1) . . . v(Λ+N−1)]T and





  
    
      
        
          
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Using the received signal model of (4) we minimize the MMSE cost function




  
    
      
        
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for w(Λ), Λ=0,1, . . . , L−1, where w(Λ) is the MMSE pulse compression filter used to estimate x(Λ) and E[•] is the expectation operator. Minimization of (6) yields

w(Λ)=(E[{tilde over (y)}(Λ){tilde over (y)}H(Λ)])1E[{tilde over (y)}(Λ)x•(Λ)]  (7)

where (·)• is the complex conjugate operation. Substituting for {tilde over (y)}(Λ) from (4), and assuming that the radar range returns are, in general, uncorrelated with one another and also uncorrelated with the noise, we obtain

w(Λ)=ρ(Λ) ({tilde over (C)}(Λ)+R)−1[s∘e(Λ)]  (8)

where ρ(Λ)=E[|x(Λ)|2] is the expected power of x(Λ), R=E[v(Λ) vH(Λ)] is the noise covariance matrix, and the matrix {tilde over (C)}(Λ)=E[ÃT(Λ)s sH Ã•(Λ)] can be simplified as





  
    
      
        
          
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in which {tilde over (s)}n(Λ+n) =[0 . . . 0 s0 s1ejθΛ+n . . . sN−n−1ej(N−n−1)θ1+n]T with n preceding zeros for n≧0 and {tilde over (s)}n(Λ+n)=[s−nej(−n)0t+n . . . sN−1ej(N−1)θt+n 0 . . . 0]T with n zeros at the end for n≦0.

The Doppler Compensated Adaptive Pulse Compressor (DCAPC) is implemented by inserting the current estimates of the respective range cell powers {circumflex over (p)}(Λ)=|{circumflex over (x)}(Λ)|2 and range cell Doppler shifts {circumflex over (θ)}Λ into (8) and (9) and then applying the resulting receive filter w(Λ) to the received signal {tilde over (y)}(Λ) to estimate the complex range cell amplitude x(Λ) for each Λ=0,1, . . . , L−1. This, then, provides a new estimate of the range cell complex amplitudes which can be used to repeat the process. The estimation of the individual Doppler shifts will be addressed shortly . The initialization of the range cell powers can be obtained from the output of the matched filter or by applying a simplified version of (8) and (9) as follows.
In its current stage, the MMSE pulse compression filter is a function of the powers of the surrounding range cells and their respective Doppler phage shifts, all of which are unavailable in practice. This lack of prior knowledge can be accounted for by setting all the initial range cell estimates to be equal to all the initial Doppler shift estimates to zero. In so doing, and treating the noise as negligible initially, the initial MMSE filter takes on the form

w1=C1−1s  (10)

where





  
    
      
        
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in which sn=[0 . . . 0 s0 s1 . . . sN−n−1]T with n preceding zeros for n≧0 and sn=[s−n . . . sN−1 0 . . . 0]T with n zeros at the end for n≦0. The initial MMSE pulse compression filter is invariant to the range cell delay Λ and can be pre-computed and then implemented in the same way as the traditional matched filter. Alternatively, the initial filtering can be accomplished by the matched filter.

Applying either the standard matched filter as in (1) or the MMSE filter w1 from (10) and (11) in place of s in (1) yields an initial estimate of the radar range profile. As such, either the MMSE filter or the matched filter can be used initially. The initial range cell estimates resulting from initial filtering (via either MMSE or matched filtering) then provide approximate information that can be employed within the MMSE formulation of (8) and (9) in order to re-estimate the appropriate receive filter to use for each individual range cell which is dependent upon the values of the surrounding range cells. In addition, the availability of the initial range cell estimates also enables the estimation of the Doppler phase shifts of range cells with large target returns which can then be incorporated into the MMSE formulation in (8) and (9) as well in order to further improve the accuracy of the receive filters.
Knowledge of target Doppler shifts can be used to more closely match the individual receive filters as well as to better null the range sidelobe interference from nearby large targets. Along with the range cell complex amplitudes, the range cell Doppler shifts can be estimated from the received data. Initially, the Doppler estimates are set to zero (or some assumed Doppler offset for a moving platform) because it can reasonably be assumed that the vast majority of range cell will not contain a moving target. Upon setting the initial Doppler phase shift estimates to zero, the received return signal for a single high SNR point target can be approximately expressed as the vector

y(Λ)=x(Λ)·[s0 s1ejθ1 . . . sN−1ej(N−1)θΛ].  (12)

Hence, the application of the matched filter s as in (1) yields

{circumflex over (x)}MF(Λ)=x(Λ)·(|s0|2+|s1|2eiθΛ+. . . +|sN−1|2ej(N−1)θΛ).  (13)

Based on (13) and assuming a constant modulus waveform (which is standard practice), it can be seen that the ratio of consecutive terms in the summation can be used as an estimate of the Doppler phase shift. Furthermore, the ratio of consecutive partial sums in (13) also yields an estimate of the Doppler phase shift while also exploiting some of the pulse compression integration gain to increase SNR. For example, one could estimate the Doppler by computing the phase angle of





  
    
      
        
          
            
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The denominator in (14) is the first N−1 elements in the summation of (13) and the numerator in (14) is the last N−1 elements from (13). The partial summations in (14) can be obtained by applying selected portions of the matched filter; i.e. the first N−1 elements of the matched filter and the last N−1 elements, respectively.

In a general sense, the partial matched filters can be applied as

g(Λ)=BMFHy(Λ).  (15)

in which





  
    
      
        
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with sm=[sm . . . sN−M−m−1]hu T where m=0,1, . . . , M for Mε[1, 2, . . . , N−1]. Hence, each partial matched filter in BMF has length N−M and g(Λ) will be comprised of M+1 elements from which M ratios as in (14) can be obtained. The Doppler phase shift θΛ can then be estimated either as





  
    
      
        
          
            
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where gm(Λ) is the mth element of g(Λ) and ∠[•] is the angle of the complex value of the argument. The estimate in (17) will be heavily influenced by any dominant values of the quotient resulting from overly small or large terms in g(Λ) (albeit it can be reasonably expected that all the terms will be of comparable magnitude). Conversely, the estimate in (18) is unaffected by magnitude variations but may be more sensitive to phase errors. Also, the formulation in (15) and (16) provides a choice (through the value of M) in the trade-off between almost full pulse compression integration gain (M=1) and the maximum number of samples over which to average the Doppler phase shift estimate (M=N−1). It has been found via simulation experimentation that values in the neighborhood of the middle (M≈N/2) tend to work best.

The estimation of the Doppler phase shift using the matched filter, while computationally efficient, only works well in cases where solitary point targets are present (i.e. targets sparsely populated in range). The matched filter based Doppler estimate degrades when one or more targets with comparable SNR are in relatively close proximity as the interference from neighboring targets acts to skew the Doppler estimates. To mitigate the effects of interference from other nearby targets on the estimation of θΛ, the MMSE formulation can again be employed in which BMF in (15) is replaced by




  
    
      
        
          
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The mth MMSE sub-filter wm(Λ) of length N−M is computed by partitioning Doppler-omitted versions of (8) and (9) as

 wm(Λ)=ρ(Λ)( Cm(Λ)+ R)−1 sm  (20)

where R is an (N−M)×(N−M) principal submatrix of R, the (N−M)×(N−M) matrix Cm(Λ) is the (m+1)st principal submatrix of





  
    
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            n
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                -
                N
              
              -
              1
            
          
          
            N
            -
            1
          
        
        ⁢
        
          
            ρ
            ⁡
            
              (
              
                l
                +
                n
              
              )
            
          
          ⁢
          
            s
            n
          
          ⁢
          
            s
            n
            H
          
          ⁢
          
            
          
          ⁢
          for
          ⁢
          
            
          
          ⁢
          m
        
      
      =
      0
    
  
  ,
  1
  ,
  ...
  ⁢
  
    
  
  ,
  
    M
    .
  




Note that Doppler phase shift information is not employed in the MMSE sub-filter formulation of (20). By nulling the interference from nearby targets, the MMSE formulation in (19) and (20) effectively approximates the Doppler estimate for a solitary point target as in (14) thereby significantly reducing the likelihood of mis-estimating θhd Λ. To further reduce the likelihood of mis-estimating θΛ, only the Doppler shifts of relatively large target returns are estimated. This greatly lessens the likelihood that nearby large targets may skew the estimate of θΛ for small targets, which in any case do not cause significant Doppler-induced range sidelobes. A variable threshold can be set by the operator or user according to the noise power and/or current local power estimates in order to determine which range cells are candidates for Doppler estimation. As the range sidelobes are driven down at each successive stage the threshold may be reduced somewhat in order to compensate for the Doppler mismatch of slightly lower target returns. In addition, as a robustness measure an upper limit may be placed on the value of the Doppler estimate such that it is maintained within practical bounds for the specific radar application and scenario.

The Doppler phase shift estimate obtained form either (17) or (18) can be incorporated into the MMSE filter formulation of (8) and (9) along with the current range cell power estimates in order to re-estimate the appropriate the receive filter for each individual range cell. It has been found based on simulation experimentation that this process of alternating the estimation of the range cell complex amplitudes and Dopplers with the estimation of the receive filters need only be repeated 2-4 times inclusive of the initialization stage.
The preferred embodiment of the invention includes the use of the MMSE sub-filter of (19) and (20) followed by Doppler estimation using either (17) or (18) along with the Adaptive Pulse Compression portion from (8) and (9). Also, the robustness parameter α as described in (U.S. patent app. NC 96.123) in which {circumflex over (ρ)}(Λ)=|{circumflex over (x)}(Λ)|2 is replaced with {circumflex over (ρ)}(Λ)=|{circumflex over (x)}(Λ)|α, for 1.1≦α≦1.8, would preferably be used in (8) and (9) so that the matrix {tilde over (C)}(Λ) does not become ill-conditioned as well as in (20) for the MMSE sub-filters so that Cm(Λ) is not ill-conditioned.
FIG. 2 illustrates the Reiterative MMSE (RMMSE) adaptive pulse compression algorithm for three iterations. In general, the RMMSE algorithm operates as follows:

    
    
        1) Collect the L+(2M−1)(N−1) samples of the received return signal [y|−(M−1)(N−1)|, . . . , y{−1}, y{0}, . . . , y{L−1}, y{L}, . . . , y{L−1+M(N−1)}], which comprise the length−L data window along with the (M−1)(N−1) samples prior to the data window and the M(N−1) samples after the data window.
    
    
    2) Apply either the initial MMSE filter from (13) or the matched filter as in (1) to obtain the initial impulse response estimates [{circumflex over (x)}1{−(M−1)(N−1)}, . . . , {circumflex over (x)}1{−1}, {circumflex over (x)}1{0}, . . . , {circumflex over (x)}1{L−1}, {circumflex over (x)}1{L}, . . . {circumflex over (x)}1{L−1+(M−1)(N−1)}].
    3) Compute the initial power estimates

{circumflex over (ρ)}1(Λ)=|{circumflex over (x)}1(Λ)|2 for

Λ=−(M−1)(N−1), . . . , L−1+(M−1)(N−1) which are used to compute the filters w1(Λ) as in (11),

and then apply to {tilde over (y)}(Λ) to

obtain [{circumflex over (x)}2{−(M−2)(N−1)}, . . . , {circumflex over (x)}2{−1},{circumflex over (x)}2{0}, . . . ,{circumflex over (x)}2{L−1},{circumflex over (x)}2{L}, . . . , {circumflex over (x)}2{L−1+(M−2)(N−1)}].

    4) Repeat 2) and 3), changing the indices where appropriate, until the desired length−L data window is reached.


Use of the Doppler Compensated Adaptive Pulse Compressor (DCAPC) can significantly reduce degradation resulting from severe Doppler mismatch as compared to matched filtering and the original APC algorithm, which has otherwise been shown to provide very good performance. In so doing, DCAPC enables the estimation of a high-speed target profile with considerably higher accuracy thereby facilitating greatly enhanced identification capability. As a simple example FIG. 3 depicts the range profile estimate of the matched filter the original APC algorithm (3 stages including initialization), and DCAPC algorithm (3 stages with M=20) for a range profile consisting of a large stationary (no Doppler) target and a small nearby target in which a length N=30 discrete Lewis-Kretschmer P3 code was employed as the transmitted waveform. As has been shown previously, the APC algorithm greatly outperforms the matched filter. Also, the new DCAPC algorithm performs just as well as APC: both uncover the small target that is masked when matched filtering is employed. In terms of Mean-Square Error (MSE), APC and DCAPC achieve MSE values of −72 dB and −68 dB, respectively, which are considerably better that the −29 dB attained by matched filtering.
By comparison with the stationary case, FIG. 4 depicts the results for matched filtering, APC (3 stages), and DCAPC (3 stages with M=20) for the same scenario except that the large target is moving at a speed that corresponds to a 25° Doppler phase shift over the length of the waveform. This Doppler shift is analogous to a Mach 3 target illuminated by a 1 μs pulse of an X-band radar. As FIG. 4 illustrates, while still superior to the matched filter the sensitivity of APC is significantly degraded due to the effects of Doppler-induced range sidelobes and the small target is now masked by the Doppler-mismatch induces range sidelobes. However, by estimating and subsequently compensating for the Doppler mismatch of the large target, DCAPC is able to estimate the range profile to the level of the noise floor thereby unmasking the small nearby target. In terms of MSE, the matched filter reaches −27 dB while APC degrades significantly to −32 dB. The DCAPC algorithm, on the other hand, achieves −57 dB, a 30 dB improvement over matched filtering and 25 dB better than APC.
For an even more stressing scenario consider FIG. 5 in which there are three large fast-moving targets in close proximity. The three large targets have Doppler phase shifts of +1.5°, −10°, and +5° over the extent of the waveform, respectively. Hence, the three targets are moving at significantly different velocities with the 1st and 3rd approaching the radar and the 2nd receding. Furthermore, there are 2 smaller stationary targets between the 3 large targets. As the results of FIG. 5 show, the matched filter again performs poorly. APC outperforms the matched filter yet is still degraded due to the Doppler-induced range sidelobes. DCAPC (with M=20) dramatically reduces the range sidelobes thereby unmasking the 2 smaller stationary targets. In terms of MSE, the matched filter reaches −23 dB. APC attains −30 dB, and DCAPC achieves −59 dB.
Obviously many modifications and variations of the present invention are possible in the light of the above teachings. It is therefore to be understood that the scope of the invention should be determined by referring to the following appended claims.

## Claims

1. A method for processing a received radar pulse from a moving target, comprising:
a) receiving a return signal;
b) obtaining L+(2M−1 (N−1) samples y of the return signal, where y(Λ)={tilde over (x)}T(Λ)s+v(Λ):
c) applying Minimum Mean Square Error (MMSE) estimation to each successive N samples to obtain initial impulse response estimates [{circumflex over (x)}1{−(M−1)(N−1)}, . . . , {circumflex over (x)}1{−1}, {circumflex over (x)}1{0}, . . . , {circumflex over (x)}1{L−1}, {circumflex over (x)}1{L}, . . . , {circumflex over (x)}1{L−1+(M−1)(N−1)}];
d) computing power estimates {circumflex over (ρ)}1(Λ)=|{circumflex over (x)}1(Λ)|2 for Λ=−(M−1)(N−1), . . . , L−1+(M−1)(N−1);
e) estimating a range cell Doppler shift for each range cell magnitude estimate above a user-selected threshold;
f) computing MMSE filters according to w(Λ)=ρ(Λ)({tilde over (C)}(Λ)+R)−1[s∘e(Λ], where ρ(Λ)=|x(Λ)|2 is the power of x(Λ), R=E[v(Λ) vH (Λ)] is the noise covariance matrix, and the range cell Doppler shifts are inserted;
g) applying the MMSE filters to y to obtain [{circumflex over (x)}2{−(M−2)(N−1)}, . . . , {circumflex over (x)}2{−1},{circumflex over (x)}20}, . . . , {circumflex over (x)}2(L−1}, {circumflex over (x)}2{L}, . . . , {circumflex over (x)}2{L−1+(M−2)(N−1)}]; and
h) repeating (e)-(g) for subsequent reiterative stages until a desired length−L range window is reached, thereby resolving the scatterers from noise and other scatterers.

2. A method as in claim 1, wherein the MMSE estimation is performed with a plurality of parallel processors.

3. A method as in claim 1, further comprising setting a nominal level for which the power estimates ρ(Λ)=|x(Λ)|2 are not allowed to fall below.

4. A method as in claim 1, wherein the y samples are obtained via analog/digital (A/D) conversion.

5. A method as in claim 1, wherein the method is applied in range profiling.

6. A method as in claim 1, wherein the method is applied for synthetic Aperture Radar (SAR).

7. A method as in claim 1, wherein the method is applied for Inverse SAR (ISAR).

8. A method as in claim 1, wherein the Doppler shift is determined from the equation


g(Λ)=BMFHy(Λ),


in which




  
    B
    MF
  
  =
  
    
      [
      
        
          
            
              
                s
                _
              
              0
            
          
          
            
              
            
          
          
            
              
            
          
          
            0
          
        
        
          
            
              
            
          
          
            
              
                s
                _
              
              1
            
          
          
            
              
            
          
          
            
              
            
          
        
        
          
            
              
            
          
          
            
              
            
          
          
            ⋰
          
          
            
              
            
          
        
        
          
            0
          
          
            
              
            
          
          
            
              
            
          
          
            
              
                s
                _
              
              M
            
          
        
      
      ]
    
    
      N
      -
      
        (
        
          M
          +
          1
        
        )
      
    
  




with sm=[sm . . . sN−M−m−1]T where m=0,1, . . . ,M for Mε[1, 2, . . . , N−1], and the Doppler phase shift θΛ is then estimated as




  
    
      
        θ
        ^
      
      i
    
    =
    
      ∠
      ⁡
      
        [
        
          
            ∑
            
              m
              =
              1
            
            M
          
          ⁢
          
            
              
                g
                m
              
              ⁡
              
                (
                l
                )
              
            
            
              
                g
                
                  m
                  -
                  1
                
              
              ⁡
              
                (
                l
                )
              
            
          
        
        ]
      
    
  
  ,




where gm(Λ) is the mth element of g(Λ) and ∠[•] is the angle of the complex value of the argument.

9. A method is in claim 1, wherein the Doppler shift is determined from the equation


g(Λ)=BMMSEHy(Λ)


in which




  
    
      
        B
        MMSE
      
      ⁡
      
        (
        l
        )
      
    
    =
    
      
        [
        
          
            
              
                
                  
                    w
                    _
                  
                  0
                
                ⁡
                
                  (
                  l
                  )
                
              
            
            
              
                
              
            
            
              
                
              
            
            
              0
            
          
          
            
              
                
              
            
            
              
                
                  
                    w
                    _
                  
                  1
                
                ⁡
                
                  (
                  l
                  )
                
              
            
            
              
                
              
            
            
              
                
              
            
          
          
            
              
                
              
            
            
              
                
              
            
            
              ⋰
            
            
              
                
              
            
          
          
            
              0
            
            
              
                
              
            
            
              
                
              
            
            
              
                
                  
                    w
                    _
                  
                  M
                
                ⁡
                
                  (
                  l
                  )
                
              
            
          
        
        ]
      
      
        N
        -
        
          (
          
            M
            +
            1
          
          )
        
      
    
  
  ,




the mth MMSE sub-filter wm(Λ) of length N−M is computed according to


 wm(Λ)=ρ(Λ)({tilde over (C)}m(Λ)+ R)−1 sm


where R is an (N−M×(N−M) principal submatrix of R, and the (N−M)×(N−M) matrix Cm(Λ) is the (m+1)m principal submatrix of




  
    
      C
      ⁡
      
        (
        l
        )
      
    
    =
    
      
        
          ∑
          
            n
            =
            
              
                -
                N
              
              -
              1
            
          
          
            N
            -
            1
          
        
        ⁢
        
          
            ρ
            ⁡
            
              (
              
                l
                +
                n
              
              )
            
          
          ⁢
          
            s
            n
          
          ⁢
          
            s
            n
            H
          
          ⁢
          
            
          
          ⁢
          for
          ⁢
          
            
          
          ⁢
          m
        
      
      =
      0
    
  
  ,
  1
  ,
  ...
  ⁢
  
    
  
  ,
  
    M
    .
  




and the Doppler base shift θΛ is then estimated as




  
    
      
        θ
        ^
      
      i
    
    =
    
      ∠
      ⁡
      
        [
        
          
            ∑
            
              m
              =
              1
            
            M
          
          ⁢
          
            
              
                g
                m
              
              ⁡
              
                (
                l
                )
              
            
            
              
                g
                
                  m
                  -
                  1
                
              
              ⁡
              
                (
                l
                )
              
            
          
        
        ]
      
    
  
  ,




where gm(Λ) is the mth element of g(Λ) and ∠[•] is the angle of the complex value of the argument.

10. A method as in claim 1, wherein the Doppler shift is determined from the equation


g(Λ)=BMFHy(Λ),


in which




  
    B
    MF
  
  =
  
    
      [
      
        
          
            
              
                s
                _
              
              0
            
          
          
            
              
            
          
          
            
              
            
          
          
            0
          
        
        
          
            
              
            
          
          
            
              
                s
                _
              
              1
            
          
          
            
              
            
          
          
            
              
            
          
        
        
          
            
              
            
          
          
            
              
            
          
          
            ⋰
          
          
            
              
            
          
        
        
          
            0
          
          
            
              
            
          
          
            
              
            
          
          
            
              
                s
                _
              
              M
            
          
        
      
      ]
    
    
      N
      -
      
        (
        
          M
          +
          1
        
        )
      
    
  




with sm=[sm . . . sN−M−m−1]T where m=0,1, . . . , M for Mε[1, 2, . . . , N−1], and the Doppler phase shift θΛ is then estimated as




  
    
      
        θ
        ^
      
      l
    
    =
    
      
        1
        M
      
      ⁢
      
        
          ∑
          
            m
            =
            1
          
          M
        
        ⁢
        
          ∠
          ⁡
          
            [
            
              
                
                  g
                  m
                
                ⁡
                
                  (
                  l
                  )
                
              
              
                
                  g
                  
                    m
                    -
                    1
                  
                
                ⁡
                
                  (
                  l
                  )
                
              
            
            ]
          
        
      
    
  
  ,



where gm(Λ) is the mth element of g(Λ) and ∠[•] is the angle of the complex value of the argument.

11. A method as in claim 1, wherein the Doppler shift is determined from the equation


g(Λ) =BMMSEHy(Λ),


in which




  
    
      
        B
        MMSE
      
      ⁡
      
        (
        l
        )
      
    
    =
    
      
        [
        
          
            
              
                
                  
                    w
                    _
                  
                  0
                
                ⁡
                
                  (
                  l
                  )
                
              
            
            
              
                
              
            
            
              
                
              
            
            
              0
            
          
          
            
              
                
              
            
            
              
                
                  
                    w
                    _
                  
                  1
                
                ⁡
                
                  (
                  l
                  )
                
              
            
            
              
                
              
            
            
              
                
              
            
          
          
            
              
                
              
            
            
              
                
              
            
            
              ⋰
            
            
              
                
              
            
          
          
            
              0
            
            
              
                
              
            
            
              
                
              
            
            
              
                
                  
                    w
                    _
                  
                  M
                
                ⁡
                
                  (
                  l
                  )
                
              
            
          
        
        ]
      
      
        N
        -
        
          (
          
            M
            +
            1
          
          )
        
      
    
  
  ,




the mth MMSE sub-filter wm(Λ) of length N−M is computed according to


 wm(Λ)=ρ(Λ) ({tilde over (C)}m(Λ)+ R)−1 sm


where R is an (N−M)×(N−M) principal submatrix of R, and the (N−M)×(N−M) matrix Cm(Λ) is the (m+1)st principal submatrix of




  
    
      C
      ⁡
      
        (
        l
        )
      
    
    =
    
      
        
          ∑
          
            n
            =
            
              
                -
                N
              
              -
              1
            
          
          
            N
            -
            1
          
        
        ⁢
        
          
            ρ
            ⁡
            
              (
              
                l
                +
                n
              
              )
            
          
          ⁢
          
            s
            n
          
          ⁢
          
            s
            n
            H
          
          ⁢
          
            
          
          ⁢
          for
          ⁢
          
            
          
          ⁢
          m
        
      
      =
      0
    
  
  ,
  1
  ,
  ...
  ⁢
  
    
  
  ,
  
    M
    .
  




and the Doppler phase shift θΛ is then estimated as




  
    
      
        θ
        ^
      
      l
    
    =
    
      
        1
        M
      
      ⁢
      
        
          ∑
          
            m
            =
            1
          
          M
        
        ⁢
        
          ∠
          ⁡
          
            [
            
              
                
                  g
                  m
                
                ⁡
                
                  (
                  l
                  )
                
              
              
                
                  g
                  
                    m
                    -
                    1
                  
                
                ⁡
                
                  (
                  l
                  )
                
              
            
            ]
          
        
      
    
  
  ,



where gm(Λ) is the mth element of g(Λ) and ∠[•] is the angle of the complex value of the argument.

12. A radar system, comprising:
a receiver;
a processor including a Doppler Compensated Adaptive Pulse Compressor (DCAPC) algorithm; and
a target detector.

13. A radar system as in claim 12, further comprising a transmitter for transmitting an initial radar signal toward a target.

14. A radar system as in claim 12, wherein the DCAPC algorithm comprises:
a) obtaining L+(2M−1)(N−1) samples y of a radar return signal, where y(Λ)={tilde over (x)}T(Λ)s+v(Λ);
b) applying Minimum Mean Square Error (MMSE) pulse compression to the radar return signal y to obtain initial radar impulse response estimates [{circumflex over (x)}1{−(M−1)(N−1)}, . . . , {circumflex over (x)}1{−1}, {circumflex over (x)}1{0}, . . . , {circumflex over (x)}1{L−1}, {circumflex over (x)}1{L}, . . . , {circumflex over (x)}1{L−1+(M−1)(N−1)}];
c) computing power estimates {circumflex over (ρ)}1(Λ)=|{circumflex over (x)}1(Λ)|2 for Λ=−(M−1)(N−1), . . . , L−1+(M−1)(N−1);
d) estimating a range cell Doppler shift for each range cell;
e) computing range-dependent filters according to w(Λ)=ρ(Λ) ({tilde over (C)}(Λ)+R)−1[s∘e(Λ)], where ρ(Λ)=|x(Λ)|2 is the power of x(Λ), and R=E[v(Λ) vH(Λ)| is the noise covariance matrix, and the range cell Doppler shifts are inserted;
f) applying the MMSE filters to y to obtain [{circumflex over (x)}2{−(M−2)(N−1)}, . . . , {circumflex over (x)}2{−1}, {circumflex over (x)}2{0}, . . . {circumflex over (x)}2(L−1), {circumflex over (x)}2{L}, . . . , {circumflex over (x)}2(L−1+(M−2)(N−1)}]; and
g) repeating (c)-(f) for subsequent reiterative stages until a desired length−L range window is reached, thereby resolving the scatterer from noise and other scatterers.

15. A radar system as in claim 14, further comprising a plurality of parallel processors for performing the MMSE pulse compression.

16. A radar system as in claim 14, wherein a nominal level is set for which the power estimates are not allowed to fall below.

17. A radar system as in claim 12, further comprising an analog-to-digital (A/D) converter.

18. A radar system as in claim 12, wherein the DCAPC algorithm comprises:
a) obtaining L+(2M−1)(N−1) samples y of a radar return signal, where y(Λ)={tilde over (x)}T(Λ)s+v(Λ);
b) applying matched filtering to the radar return signal y to obtain initial radar impulse response estimates [{circumflex over (x)}1{−(M−1)(N−1)}, . . . , {circumflex over (x)}1{−1}, {circumflex over (x)}1{0}, . . . {circumflex over (x)}1{L−1}, {circumflex over (x)}1{L}, . . . , {circumflex over (x)}1{L−1+(M−1)(N−1)}];
c) computing power estimates {circumflex over (ρ)}1(Λ)=|{circumflex over (x)}1(Λ)|2 for Λ=−(M−1)(N−1), . . . , L−1+(M−1(N−1);
d) estimating a range cell Doppler shift for each range cell;
e) computing range-dependent filters according to w(Λ)=ρ(Λ) ({tilde over (C)}(Λ)+R)−1[s∘e(Λ)], where ρ(Λ)=|x(Λ)|2 is the power of x(Λ), and R=E[v(Λ) vH(Λ)] is the noise covariance matrix, and the range cell Doppler shifts are inserted;
f) applying the MMSE filters to y to obtain [{circumflex over (x)}2{−(M−2)(N−1)}, . . . , {circumflex over (x)}2−1}, {circumflex over (x)}2{0], . . . , {circumflex over (x)}2(L−1), {circumflex over (x)}2{L}, . . . , {circumflex over (x)}2{L−1+(M−2)(N−1)}]; and
g) repeating (c)-(f) for subsequent reiterative stages until a desired length−L range window is reached, thereby resolving the scatterer from noise and other scatterers.

19. A radar system as in claim 18, further comprising a plurality of parallel processors for performing the MMSE pulse compression.

20. A radar system as in claim 18, wherein a nominal level is set for which the power estimates are not allowed to fall below.

21. A method for processing a received radar pulse from a moving target, comprising:
a) obtaining L+(2M−1)(N−1) samples y of a radar return signal, where y(Λ)={tilde over (x)}T(Λ)s+v(Λ);
b) applying matched filtering to the radar return signal y to obtain initial radar impulse response estimates [{circumflex over (x)}1{−(M−1)(N−1)}, . . . , {circumflex over (x)}1{−1}. . . {circumflex over (x)}1{0}, . . . , {circumflex over (x)}1{L−1}, {circumflex over (x)}1{L}, . . . {circumflex over (x)}1{L−1+(M−1)(N−1)}];
c) computing power estimates {circumflex over (ρ)}1(Λ)=|{circumflex over (x)}1(Λ)|2 for Λ=−(M−1)(N−1), . . . , L−1+(M−1)(N−1);
d) estimating a range cell Doppler shift for each range cell;
e) computing range-dependent filters according to w(Λ)=ρ(Λ)({tilde over (C)}(Λ)+R)−1[s∘e(Λ)], where ρ(Λ) =|x(Λ)|2 is the power of x(Λ), and R=E[v(Λ) vH(Λ)] is the noise covariance matrix, and the range cell Doppler shifts are inserted;
f) applying the MMSE filters to y to obtain [{circumflex over (x)}2{−(M−2)(N−1)}, . . . , {circumflex over (x)}2{−1}, {circumflex over (x)}2{0}, . . ., {circumflex over (x)}2{L−1}, {circumflex over (x)}2{L}, . . . , {circumflex over (x)}2{L−1+(M−2)(N−1)}]; and
g) repeating (c)-(f) for subsequent reiterative stages until a desired length−L range window is reached, thereby resolving the scatterer from noise and other scatterers.