Fast implementation for modified adaptive multipulse compression
 Xianxiang Yu^{1},
 Guolong Cui^{1}Email author,
 Meifang Luo^{2},
 Lingjiang Kong^{1} and
 Dan Ran^{3}
https://doi.org/10.1186/s1363401604232
© The Author(s) 2016
Received: 25 June 2016
Accepted: 9 November 2016
Published: 26 November 2016
Abstract
This paper deals with the estimation of rangeDoppler plane in pulse Doppler radar system, accounting both for clutterfree scenario and clutter scenario. A modified adaptive multipulse compression (MAMPC) algorithm including the estimation stages of range dimension and Doppler dimension is proposed for clutterfree scenario, where each stage is implemented based on the gain constraint adaptive pulse compression (GCAPC) algorithm. Additionally, the combination of whitening method removing the correlation of clutter component and MAMPC algorithm is presented for the considered clutter scenario. Numerical simulations are provided to validate the effectiveness of MAMPC in terms of estimation of rangeDoppler plane and computation burden.
Keywords
1 Introduction
Traditionally, the pulse Doppler radar systems repeat the same waveform to allow efficient pulse compression and Doppler processing technique to be used [1]. The traditional pulse compression method is matched filtering, in which the high range sidelobe of strong targets may interfere or even mask nearby weak targets. The Doppler processing technique, such as the moving target detection (MTD), also obtains Doppler sidelobe that results in the masking problem [2]. Consequently, suppressing the rangeDoppler sidelobe is meaningful for target detection.
Suppressing range or Doppler sidelobe has been received considerable attention. Summarizing, these works can be classified into three categories. The first category deals with the problem of adaptive range sidelobe suppression. In [3], iterative reweighted least squares (IRLS) algorithm was used to suppress range sidelobe. In [4], several binary pulse compression codes were designed to greatly reduce sidelobe meanwhile suffering only a small S/N loss. In [5], the adaptive pulse compression (APC) was proposed, which was shown to successfully suppress the range sidelobes over a variety of stressing scenarios. Li et al. [6] has demonstrated that gainconstraintAPC (GCAPC) [7] has better estimating performance especially for weak targets compared to original APC algorithm [5].
The second category focuses on addressing the problem of Doppler sidelobe suppression. As the mathematical model of the Doppler estimation for coherent multipulses is similar to direction of arrival (DOA) estimation, the studies related to DOA estimation can also be used in Doppler sidelobe suppression. For instance, the most wellknown methods for DOA estimation are MUSIC [8], rootMUSIC [9] and ESPRIT [10]. Reiterative super resolution (RISR) was studied in [11, 12], which was used to estimate DOA in array signal processing firstly.
The third category studies the sidelobe suppression problem by jointly suppressing rangeDoppler sidelobe [13, 14]. In [15], twodimensional reiterative minimum mean square error (MMSE) and 2D least square (LS) solutions that mitigate the sidelobe of both pulse compression processing and antenna radiation patterns are derived. In [16], a RISR algorithm was used in conjunction with Golay waveforms for rangeDoppler estimation. In [17], a recursive MMSEbased timerange adaptive processing was proposed for the purpose of jointly suppressing the rangeDoppler sidelobe. However, clutter scenario was not considered. In [18], the adaptive multipulse compression (AMPC) was presented to successfully suppress the rangeDoppler sidelobe over a variety of stressing scenarios. Unfortunately, the high computational cost of this method limits its usage in realtime systems. It is worth noting that these approaches based dimensionality reduction are well known in open literature as a means to facilitate practical solutions to computation problems. In [19], the fast adaptive pulse compression (FAPC) was proposed. In [20], the fast adaptive multipulse compression (FAMPC) was proposed based on fast adaptive pulse compression (FAPC) by segmenting the MMSE cost function into blocks. Of course, some inherent loss in performance can generally be expected by reducing dimensionality, though the attendant reduction in computation often easily justifies the tradeoff.
In this paper, we propose a modified adaptive multipulse compression (MAMPC) algorithm to obtain both good estimation performance and small amount of calculations. Unlike [21], we also consider clutter scenario assuming that some knowledge of clutter statistics is available. For clutterfree scenario, we implement MAMPC with two estimating stages by utilizing GCAPC algorithm. Specially, we obtain estimation in the range dimension using GCAPC. Then, based on the obtained results, we achieve the estimation of rangeDoppler plan in the Doppler dimension by exploiting GCAPC. In particular, for clutter scenario, the combination of whitening method removing the correlation of clutter component and MAMPC algorithm is proposed. Simulation results highlight that MAMPC is capable of achieving a close estimation performance with that of AMPC, while shares much less computational time than AMPC.
The rest of the paper is organized as follows. In Section 2, we give the signal model of rangeDoppler dimension. In Section 3 and Section 4, we present MAMPC algorithm for clutter free scenario and the combination of whitening method and MAMPC algorithm for clutter scenario, respectively. In Section 5, we evaluate the capabilities of MAMPC via numerical results. Finally, in Section 6, we provide some concluding remarks.
Notation: Vectors (matrices) are denoted by boldface lower (upper) case letters. Superscripts (·)^{ T },(·)^{∗}, and (·)^{ H } denote transpose, complex conjugate, and complex conjugate transpose, respectively. · denotes the modulus of a complex number. E(·) is the statistical expectation. \(\sum (\cdot)\) denotes the summation operation. I _{ N } is the identity matrix with N×N demension. diag(.) is an operation that creates a diagonal matrix by using the input vector as its diagonal. Finally, ⊗ denotes the Kronecker product.
2 Signal model

σ _{ T,m q } for q=1,⋯,Q, denote the complex parameters accounting for the target radar cross section (RCS), channel propagation effects, and other terms involved into the radar range equation. Assume that σ _{ T,m q }=σ _{ T,q } for all m=1,2,⋯,M, which are distributed as circular zeromean complex Gaussian random variables. In other words, the pdf of the amplitude A _{ q }=σ _{ T,q } is Rayleigh distributed, i.e.,$$p_{A_{q}}(x)=\frac{2x}{\bar{\sigma}_{T,q}^{2}}\exp\left\{\frac{x^{2}}{\bar{\sigma}_{T,q}^{2}}\right\},\,\,x\geq 0, $$

f _{ q }=2v _{ q } T _{ r }/λ denotes the normalized Doppler frequency of the qth target while v _{ q } is the radial velocity and λ is the carrier wavelength.

τ _{ T,q } and τ _{ c,m q } denote, respectively, the twoway time delays for the qth target and pth clutter scatterer for the mth pulse.

σ _{ c,m p } is the complex scattering parameter of the pth clutter scatterer at the mth pulse for m=1,⋯,M and p=1,⋯,P.

b _{ m }(t) denotes the zeromean circular complex Gaussian random process.

G is the N×(2N−1)dimensional linear transformation matrix, given by$$ \begin{aligned} \mathbf{G}= & \left[ \begin{array}{ccccccc} {s_{N}} & {s_{N1}} & \cdots & {s_{1}} & {} & {} & 0 \\ {} & {s_{N}} & \cdots & {s_{2}} & {s_{1}} & {} & {} \\ {} & {} & \ddots & \vdots & \vdots & \ddots & {} \\ 0 & {} & {} & {s_{N}} & {s_{N1}} & \cdots & {s_{1}} \\ \end{array} \right]. \\ \end{aligned} $$(6)

\(\bar {\mathbf {x}}_{l}[k]=\left [X[lN+1,k],\cdots,X\left [l+N1,k\right ]\right ]^{T}\) is the (2N−1)×1dimensional subvector accounting the scattering coefficients at the kth Doppler frequency in the rangeDoppler plane and \(\mathbf {X}[ l ]=\left [\bar {\mathbf {x}}_{l}[1],\cdots,\bar {\mathbf {x}}_{l}[K]\right ]\) is the (2N−1)×Kdimensional submatrix of X.

F is the discrete Fourier transform matrix, given by$$ \mathbf{F}= \left[ { \begin{array}{cccc} 1&1& \cdots &1\\ 1&{{e^{\,j\frac{{2\pi }}{K}1}}}&\cdots &{{e^{\,j\frac{{2\pi(K  1)}}{K}1}}}\\ \vdots & \vdots & & \vdots \\ 1&{{e^{\,j\frac{{2\pi }}{K}(M1)}}}& \cdots &{e^{\,j\frac{{2\pi(K1)}}{K}(M1)}} \end{array}} \right] $$(7)
and F ^{ T }[m] denotes the mth column of the matrix F ^{ T }.

c _{ m }[l]=[c _{ m }[l−N+1],⋯,c _{ m }[l+N−1]^{ T } is the (2N−1)×1 dimensional subvector accounting the clutter scattering coefficients.

b _{ m }[ l]=[b _{ m }[l],⋯,b _{ m }[l+N−1]]^{ T } is the N×1dimensional subvector of b _{ m }, which is distributed as the complex circular zeromean Gaussian random vector with identity covariance matrix \({\sigma _{n}^{2}}\mathbf {I}_{N}\).
where Y[ l]=[y _{1}[ l],⋯,y _{ M }[ l]], C[ l]=[c _{1}[ l],⋯,c _{ M }[ l]], and B[ l]=[b _{1}[ l],⋯,b _{ M }[l]].
3 Fast implementation of MAMPC for clutterfree scenario
3.1 The process in range dimension
3.2 The process in Doppler dimension
where \(\hat {\mathbf {\beta }}[\!l]\) and x[ l] are M×1 and K×1 vectors, respectively.
Finally, we conduct the same procedure for each range cell and can achieve the estimation of \(\hat {\mathbf {X}}\).
3.3 The joint procedure of MAMPC algorithm
In this subsection, the proposed procedure of MAMPC for the estimation of rangeDoppler plane X is summarized in Algorithm 1.
Finally, it is worth highlighting that in each iteration, Algorithm 1 shares the computational complexity of O(L M N ^{3}+L K M ^{3}). Additionally, we note that original AMPC [18] requires to conduct N M×N M matrix inversion for each individual rangeDoppler cell with corresponding to O((M N)^{3}) computation complexity. However, FAMPC [20] needs to implement K _{ l } K _{ p }×K _{ l } K _{ p } submatrix inversion for each rangeDoppler cell, which is order of O(R _{ p } R _{ l }(K _{ l } K _{ p })^{3}), where fulldimension model is divided into the R _{ l } segments in fast time domain and R _{ p } segments in slow time domain with K _{ l }=N/R _{ l } and K _{ p }=M/R _{ p }. Compared with AMPC, the computation load of FAMPC algorithm reduces a factor of (R _{ p } R _{ l })^{2}. In particular, MAMPC algorithm includes computational burden connected with M matrix inversions with size of N×N and one matrix inversion with size of M×M for each rangeDoppler cell, which is order of O(M N ^{3}+M ^{3}). We note that MAMPC algorithm has the same order of computation load with FAMPC when R _{ p }=M,R _{ l }=1. Consequently, we conclude that FAMPC and MAMPC can significantly reduce the computational complexity in comparison with fulldimension AMPC algorithm.
4 The application of MAMPC algorithm in the presence of clutter
where nonnegative number σ _{0} is the power of clutter, H is the positive semidefinite HermitianToeplitz matrix with size M×M, whose main diagonal elements 1.
with r _{ s }[ n],n=−N+1,⋯,N+1 being the autocorrelation value of s at delay n.
where we suppose that the energy of s equals to 1.
It is worth noting that decorrelation of observation matrix can be achieved by rightmultiplying the whitening matrix \(\boldsymbol {\Gamma }=(\sigma _{0}\mathbf {H}^{T} + {\sigma ^{2}_{n}}\mathbf {I}_{M})^{ 1/2}\), which decreases the computation load.
where \(\mathbf {Y}[\!l]\boldsymbol {\Gamma }=\mathbf {Y}_{1}[\!l], {\mathbf {F}^{T}}\boldsymbol {\Gamma }={\mathbf {F}_{1}^{T}}, \mathbf {G}\mathbf {C}[\!l]\boldsymbol {\Gamma }+{\mathbf {B}}[l]\boldsymbol {\Gamma }={\mathbf {B}}_{1}[l]\) with the same statistical characteristics as B[l].
Finally, we can estimate X[l] based on Y _{1}[l] using MAMPC Algorithm which is reported in 3.
5 Numerical results
Target parameters
Range cell index  Velocity (m/s)  Normalized Doppler  SNR (dB) 

frequency  
40  30  0.2  5 
30  35  0.233  –5 
45  –35  –0.233  10 
47  –40  –0.267  0 
60  35  0.233  –8 
55  30  0.2  5 
70  –30  –0.2  5 
25  –30  –0.2  –5 
20  22  0.147  –8 
30  62  0.413  –5 
5.1 Clutterfree scenario
In this subsection, we focus on the discussion of MAMPC in terms of achieved estimation of rangeDoppler plane X and computational burden accounting for clutterfree scenario. In particular, for comparison purpose, matched filter and MTD (MFMTD), AMPC, and FAMPC are also evaluated.
Iteration number and computation time (in seconds) of AMPC, FAMPC, and MAMPC for clutterfree scenario
Algorithm  AMPC  FAMPC  MAMPC 

n  6  4  7 
Time(s)  1014.7  91.6  7.7 
5.2 Clutter scenario
Hence, the (i,j)th element of the covariance matrix σ _{0} H is \(\sigma _{i,j}= {r_{c}}(i  jT_{PRT})/{\sigma _{c}^{2}}\) for i,j=1,…,M. Here, we assume the rms of clutter σ _{ v }=5 m/s and the power of clutter \(P={\sigma _{c}^{2}}=30\) dB, λ=c/f _{0}=0.3 m with c=3×10^{8} m/s being the velocity of light.
Iteration number and computation time (in seconds) of AMPC, FAMPC, and MAMPC for clutter scenario
Algorithm  AMPC  FAMPC  MAMPC 

n  8  4  6 
Time(s)  1400.5  105.8  7.1 
6 Conclusions
In this paper, we have addressed the estimation of rangeDoppler plane for pulse Doppler radar systems considering clutterfree scenario and clutter scenario. We have proposed MAMPC algorithm including the estimation stages of range dimension and Doppler dimension for clutterfree scenario, where each stage is implemented based on GCAPC. In addition, we also have presented the combination of whitening method removing the correlation of the clutter component and MAMPC algorithm for considered clutter scenario. We have designed numerical simulations to assess the ability of proposed algorithm. We have observed that the proposed MAMPC keeps the near same estimation performance of rangeDoppler plane X with that of AMPC, whereas FAMPC is likely to lose weak targets. Results have also impled that MAMPC shares much less computational time in comparision with AMPC. Possible future research tracks might concern the extension of the proposed framework to account for electronic jamming and nonhomogeneous characteristics clutter.
7 \thelikesection Endnote
^{1} We notice that the waveform with good autocorrelation has better estimation performance for rangeDoppler plane. Since the limitation on paper length, we here do not show the simulation results for the selection of waveform in simulation.
Declarations
Acknowledgements
This work was supported by the National Natural Science Foundation of China under Grants 61201276 and 61301266, the Fundamental Research Funds of Central Universities under Grants ZYGX2014J013 and ZYGX2014Z005.
Competing interests
The authors declare that they have no competing interests.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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