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Fast implementation for modified adaptive multipulse compression
EURASIP Journal on Advances in Signal Processing volume 2016, Article number: 127 (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.
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.
Signal model
Consider a stationary monostatic radar system which transmits M coherent pulses of train. Let s(t) be the baseband complex probing waveform. Assume that there are Q pointlike targets and P clutter scatterers in different range cells and with different radial velocities. The received signal y _{ m }(t) of the mth pulse can be represented as
where

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.
Let s=[s _{1},s _{2},...,s _{ N }]^{T} be the discrete version of the baseband waveform s(t), then, after sampling with the same rate, the discrete versions of the received signal y _{ m }(t) and the noise term b _{ m }(t) can be expressed respectively by
where L denotes the number of the range cells. Stacking the M pulses to be the columns of the L×Mdimensional data matrix Y, we have
In addition, let us divide uniformly the normalized frequency with K points, e.g., f _{ k }=(k−1)/K,k=1,⋯,K, and denote by f=[f _{1},f _{2},⋯,f _{ K }] the Kpoints Doppler frequency vector. Hence, the L×Kdimensional rangeDoppler plane X accounting for the scattering coefficients of the targets of interest can be expressed as
In this paper, we employ the adaptive pulse compression algorithm to estimate the components of the rangeDoppler plane X. To this end, we formulate the subvector y _{ m }[ l]=[y _{ m }[ l],⋯,y _{ m }[l+N−1]]^{T} for the mth pulse with length N as
where

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 \({\sigma _{n}^{2}}\mathbf {I}_{N}\).
Let us rewrite all the M subvectors y _{ m }[ l], for m=1,⋯,M, in terms of the N×Mdimensional matrix, we have
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]].
Fast implementation of MAMPC for clutterfree scenario
In this section, we focus on the estimation of the components of rangeDoppler plane X in the presence of white noise using the fast implementation of MAMPC. To this end, the signal model without considering clutter in Eq. (8) can be simplified as
The process in range dimension
In this subsection, we are devoted to the process in range dimension by exploiting the proposed fast algorithm. Specifically, Eq. (8) can be further recast as
while
where A[l]=[a _{1}[l],⋯,a _{ M }[l]] is a (2N−1)×Mdimensional matrix and a _{ m }[l]=[α _{ m }[l−(N−1)],...,α _{ m }[l+(N−1)]]^{T} denotes the mth column of A[l], for m=1,…,M. In particular, for the mth pulse, we have
Since observation y _{ m }[ l] is the linear combination of a _{ m }[ l],α _{ m }[ l] can be obtained by designing a filter for lth range cell employing GCAPC algorithm. In the following, we focus on the discussion of three cases of different scopes of range cell for the estimation of α _{ m }[ l]. Specifically, for l=N,⋯,L−(N−1), we minimize the output power of lth range cell by devising GCAPC filter coefficients w _{ m }[ l] accounting for {w _{ m }[l]}^{H} s=1, with corresponding to optimization problem formulated as
Using lagrangian multiplier method, the filter w _{ m }[l] for the estimation of \({{\hat {\alpha }}_{m}}[l]\) can be derived as
We further assume the independence between scattering coefficients. As a consequence, the covariance matrix of observation y _{ m }[l] can be written as
where \({\sigma _{n}^{2}}\) denotes the noise power and
with \(\tilde {\alpha }_{m}[\!l]\) the prior information of α _{ m }[ l], where we assume the target range profiles located different range cells are independent. Submitting Eq. (15) in Eq. (27), w _{ m }[l] can be rewritten as
Applying w _{ m }[l] to y _{ m }[l], the estimation of α _{ m }[l] is given by
As to the estimation of from 1th to (N−1)th range cells, i.e., \(\hat {\alpha }_{m}[l],l=1,\cdots,(N1)\), the observation of length N is expressed as
Similarly, the optimization problem can be given by,
where g _{ l } is the lth column of G. Consequently, the optimized filter w _{ m }[l] of lth range cell is derived as
We further obtain the estimation of α _{ m }[ l],l=1,⋯,(N−1),
As to the estimation of from (L−(N−2))th to Lth range cells, i.e., \(\hat {\alpha }_{m}[\!l],l=L(N2),\cdots,L\), we write the observation y _{ m }[L−(N−1)] as
The optimization problem is
Hence, the optimized filter coefficient w _{ m }[l] is given by
The estimation value of α _{ m }[ l],l=L−(N−2),⋯,L, can be computed as
Based on the above discussion, the estimation value \(\hat {\mathbf {a}}_{m}=\left [\hat {\alpha }_{m}[l],...,\hat {\alpha }_{m}[L)] \right ]^{T}, m=1,\cdots,M\) can be obtained. Hence, the estimation of \(\hat {\mathbf {A}}\in \mathbb {C}^{L\times M}\) can be expressed as
The process in Doppler dimension
In this subsection, we focus on the process of doppler dimension in order to estimate the elements of rangeDoppler plane X. Specifically, using the linear function in Eq. (8) and A[ l]=X[ l]F ^{T}, we have
where E is noise vector with covariation matrix I/s^{H}s. After transposition operation, Eq. (28) can be recast as,
Let \(\hat {\mathbf {\beta }}[l], \mathbf {x}[l]\) and e[l] denote the lth columns of \(\hat {\mathbf {A}}^{T}, \mathbf {X}^{\mathrm {T}}\) and E ^{T}, respectively. Hence, similar to the process of range dimension, we have
where \(\hat {\mathbf {\beta }}[\!l]\) and x[ l] are M×1 and K×1 vectors, respectively.
Let x _{ k }[l](k=1,...,K) and F _{ k } denote the kth element of x[l] and kth column of F, respectively. The GCAPC filter coefficients w _{ k } for the estimation of \(\hat {\mathbf {x}}_{k}[l]\) are expressed as
where
\({\tilde {\mathrm {x}}}_{k}[l]\) is the prior information of x_{ k }[l]. Hence, the estimation value \(\hat {\mathbf {x}}_{k}[l]\) can be obtained by
The estimation of \(\hat {\mathbf {x}}[l]\) can be expressed as
Finally, we conduct the same procedure for each range cell and can achieve the estimation of \(\hat {\mathbf {X}}\).
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.
The application of MAMPC algorithm in the presence of clutter
In this section, we focus on the estimation of rangeDoppler plane X in the presence of clutter employing MAMPC algorithm. According to the received signal model in Eq. (8), we here adopt whitening method to remove correlation of clutter. Specifically, let c[l]=[c _{1}[l],c _{2}[l],⋯,c _{ M }[l]]^{T} denote the range profile of the lth cell for M pulses; hence, we have
In particular, we further assume that c _{ m }[ j] and c _{ n }[ i]∀(m,n)∈{1,2,…,M}^{2},∀(i,j)∈{1,2,⋯,L}^{2},i≠j, are zeromean uncorrelated random variables, and c _{ m }[ j],c _{ n }[ i] with i=j are correlative random variables obeying zeromean Gaussian distribution with covariance matrix
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.
Our purpose is to remove correlation of clutter. In other words, the interference term including clutter and noise in Eq. (8) should show the same statistics feature as the noise term in Eq. (8) after whitening operation. In particular, we stack all the column of Y[l] with only considering the interference term in Eq. (8), denoted as \(\tilde {\mathbf {Y}}[\!l]\), i.e., \(\tilde {\mathbf { Y}}[\!l]=\left [\mathbf {y}_{1}^{T}[\!l],...,\mathbf {y}_{M}^{T}[\!l]\right ]^{T}\). Hence, we have
Exploiting the fact in Eq. (36) that G E[c _{ i }[l]c _{ j }[l]^{H}]G ^{H}=σ _{ i,j } G G ^{H},i,j∈{1,2,…,M}^{2}, the whitening matrix \(E{\left [ {\tilde {\mathbf {Y}}[l]{\tilde {\mathbf {Y}}^{H}}[l]} \right ]^{ \frac {1}{2}}}\) [22] can be given by,
where R=G G ^{H} is the correlation matrix for the range dimension, computed as
with r _{ s }[ n],n=−N+1,⋯,N+1 being the autocorrelation value of s at delay n.
Assuming that s possesses good autocorrelation property, i.e., r _{ s }[ n]≪r _{ s }[0],n=−(N−1),...,N−1,n≠0, we have
Submitting Eq. (40) into Eq. (38), we have
where we suppose that the energy of s equals to 1.
Based on the aforementioned discussion, the whitening result \(\tilde {\mathbf {Y}}_{W}[\!l]\) of \(\tilde {\mathbf {Y}}[\!l]\) can be expressed as follows
According to Eq. (42), we can obtain the whitening result of Y[ l] through some mathematical operations, given by
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.
Finally, exploiting Eq. (43), Eq. (8) can be expressed as
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.
Numerical results
In this section, we assess the performance of proposed algorithm for the estimation of rangeDoppler plane X in terms of clutterfree scenario and clutter scenario. To this end, we consider a multitargets case with corresponding to locations, velocities, and signaltonoise radio (SNR) of targets given in Table 1. Besides, we suppose the range processing window L=100 and the number of Doppler cell K=128 and consider the transmit signal^{1} is linear frequency modulation (LFM) phase coding with code length N=32, bandwidth B=4 MHz, pulse width T=4 μs, center frequency f _{0}= GHz, and PRT T _{ PRT }=1 ms. In partiuclar, we set the pulse number M=32. Finally, we consider the exit condition ε=10^{−6} for Algorithm 1. Besides, the running computation time is analyzed using Matlab 2010a version, running on a standard PC (with a 3.3 GHz Core i5 CPU and 8 GB RAM).
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.
Figure 1 exhibits the estimation of rangeDoppler plane X using traditional MFMTD. In particular, the locations of true targets are marked with circles. The results indicate that weak targets are possibly masked by the rangeDoppler sidelobes of strong targets. For example, the high sidelobe of target at the 45th range cell has a significant impact on parameter estimating (i.e., range profile) of nearby target located at the 47th range cell. It could be treated as a weak target leading to false alarm. Additionally, the mainlobes of the targets are expanded over the rangeDoppler plane.
Based on the APC and GCAPC, Figs. 2, 3, and 4 depict the estimations of rangeDoppler plane X utilizing AMPC, FAMPC, and MAMPC, respectively. In particular, we observe that, in Figs. 2 a, 3 a, and 4 a, the targets can be accurately estimated by exploiting AMPC, FAMPC, and MAMPC based on the accomplishment of GCAPC. However, the obtained results using FAMPC also exhibit the mainlobe energy of the targets can easily spread nearbyrange Doppler cells, as well as possess highrange Doppler sidelobes, low range, and Doppler resolutions in comparison with those optimized by AMPC and MAMPC. Additionally, in Figs. 2 b, 3 b, and 4 b, we assess the obtained estimation of rangeDoppler plane X exploiting AMPC, FAMPC, and MAMPC accomplished by APC. Interestingly, AMPC, FAMPC, and MAMPC achieve lower rangeDoppler sidelobe and narrower mainlobe in terms of estimation results of rangeDoppler plane X compared with MFMTD, whereas a portion of weak targets are missing for FAMPC.
In the following, we analyze the mean square error (MSE) performance of the estimation using MFMTD, AMPC, FAMPC, and MAMPC. In particular, the MSE is defined as
In Fig. 5, we plot the MSE curves of the X estimation versus iteration number exploiting MFMTD, AMPC, FAMPC, and MAMPC based on GCAPC for clutterfree scenario. Interestingly, AMPC and MAMPC both share the near performance and outperform MFMTD and FAMPC. This is a reasonable behavior since the optimized results by MFMTD and FAMPC show high rangeDoppler sidelobes (Figs. 1 and 3 a).
In Table 2, we report the iteration number and computation time of AMPC, FAMPC, and MAMPC for the implementation of estimation of rangeDoppler plane X in clutterfree scenario. As expected, MAMPC outperforms AMPC and FAMPC in terms of computation time. Specifically, MAMPC costs 7.7 s to the estimation, whereas AMPC and FAMPC require 1014.7 and 91.6 s, respectively. Finally, it is worth highlighting that MAMPC achieves the significant reduce of computational burden in comparison with AMPC and obtains more accurate estimation of rangeDoppler plane X in contrast to FAMPC.
Clutter scenario
In this subsection, we consider the estimation of rangeDoppler plane X in presence of clutter, where we suppose that colored Gaussian clutter is adopted with assuming internal motion of the clutter scatters due to, for example, wind affecting a forest or grassland. Thus, the temporal correlation of such clutter can be described by its power spectral density (PSD),
where σ _{ v } is the root of mean square(rms) of clutter velocity and λ is the length of waveform. Furthermore, the autocorrelation function of clutter is expressed as
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.
In Fig. 6, the obtained rangeDoppler plane X by traditional MFMTD for clutter scenario is plotted. In particular, we observe that the targets completely are masked by strong clutter, showing that the adopted method fails to the considered clutter scenario.
In Figs.7 a, 8 a, and 9 a, we plot the estimation results of rangeDoppler plane X obtained by AMPC, FAMPC, and MAMPC utilizing GCAPC, respectively. Interestingly, it can be seen that the weak target located at 20th range cell, cannot be found by AMPC, FAMPC, and MAMPC. Again, a wide mainlobe behavior can be observed for FAMPC. Based on the accomplishment of APC, we give the estimation results of rangeDoppler plane X obtained by AMPC, FAMPC, and MAMPC in Figs. 7 b, 8 b, and 9 b. Results again show that FAMPC is noneffective to a part of weak targets. In particular, by contrasting to MFMTD, AMPC, and MAMPC can attain to a much better estimation and significantly alleviate the impact of clutter.
In Fig. 10, the MSE curves of the X estimation versus iteration number exploiting MFMTD, AMPC, FAMPC, and MAMPC based on GCAPC for clutter scenario, are plotted. As expected, MAMPC exhibits a slightly better performance than AMPC, and outperform MFMTD and FAMPC due to low rangeDoppler sidelobes in Fig. 9(a).
Table 3 summarizes the behavior of computational time of AMPC, FAMPC, and MAMPC for clutter scenario. Again, MAMPC exhibits a lower computation burden than AMPC and FAMPC. Precisely, MAMPC spends 7.1s to implement the estimation, whereas AMPC and FAMPC need 1400.5 and 105.8 s, respectively. Finally, it is worth pointing out that the performance behaviors in Fig. 10 and in Table 2 reflect the capability of the proposed MAMPC that not only estimates rangeDoppler plane X accurately, but also can reduce significantly computation load.
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.
\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.
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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.
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The authors declare that they have no competing interests.
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Yu, X., Cui, G., Luo, M. et al. Fast implementation for modified adaptive multipulse compression. EURASIP J. Adv. Signal Process. 2016, 127 (2016) doi:10.1186/s1363401604232
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Keywords
 RangeDoppler plane
 Modified adaptive multipulse compression
 Range dimension
 Doppler dimension
 Gain constraint adaptive pulse compression