- Research
- Open Access
Two-dimensional SLIM with application to pulse Doppler MIMO radars
- Mohammad Jabbarian-Jahromi^{1}Email author and
- Mohammad Hossein Kahaei^{1}
https://doi.org/10.1186/s13634-015-0254-6
© Jabbarian-Jahromi and Kahaei. 2015
- Received: 23 December 2014
- Accepted: 20 July 2015
- Published: 5 August 2015
Abstract
A two-dimensional (2D) sparse signal model is developed for pulse Doppler MIMO radars. Using this model, we develop the 2D sparse learning via iterative minimization (2D SLIM) algorithm. Simulation results show that the 2D SLIM compared to the 1D SLIM drastically reduces the computational burden while both of them have the same performance. Also, for estimation of range-angle-Doppler parameters, the 2D SLIM outperforms the matched filter (MF), smoothed L0-norm (SL0), iterative adaptive approach (IAA), and spectral projected gradient for l _{1}-norm minimization (SPGL1) algorithms.
Keywords
- Pulse Doppler MIMO radar
- Sparse learning via iterative minimization
- Two-dimensional sparse signal model
1 Introduction
Multiple-input multiple-output (MIMO) radars by exploiting multiple transmitters and receivers have recently been introduced [1–3]. It is well known that in this structure due to making use of orthogonal (or highly uncorrelated) transmit signals, the received signals can easily be separated. MIMO radars are often divided into two categories based on antenna placement. In the first one, the transmit and receive antennas are widely separated, and thus, the targets are observed from different directions dealing with target fluctuation fading [4–7]. In the second category, however, antennas are collocated so that the different phases from received signals can be extracted by the receivers. In this case, due to the waveforms diversity, a higher spatial resolution is achieved compared to the traditional radars. Also, in MIMO radars, target detection and parameter estimation are improved by suitably designing transmit beam-pattern [8–13]. Here, we consider the second structure.
From a sparsity perspective, in most radar applications, the number of targets located in the radar surveillance area is much smaller than the whole number of range-angle-Doppler bins. Thus, a sparse model can be derived for the received signal, and accordingly sparse signal recovery algorithms can be used for estimating the target parameters including range, Doppler frequency, and angle [14–18]. The aim of using sparse solution in a radar system is to more accurately estimate the target parameters compared to the traditional methods such as matched filters (MF).
Compressed sensing (CS), which is rooted on the principles of sparsity theory, has recently received considerable attention in MIMO radars [19–21]. Although, the main goal of the CS problem is to reduce the sampling rate lower than the Nyquist criterion, here we mainly focused on achieving accurate estimates for target parameters with much lower computations. For this purpose, an efficient technique is the sparse learning via iterative minimization (SLIM) algorithm which is computationally simple compared to the iterative adaptive approach (IAA) and focal underdetermined system solver (FOCUSS) algorithms due to the use of the conjugate gradient least squares (CGLS) algorithm [22]. This algorithm, which is a 1D algorithm (or namely 1D SLIM), is a maximum a posteriori (MAP) estimator which maximizes a posteriori Bayesian model. An important characteristic of 1D SLIM is incorporation of l _{ q }-norm optimization (0 < q ≤ 1) in comparison with the l _{1}-norm in order to reach sparser solutions and more accurate estimates. In [22], the 1D SLIM has been developed for MIMO radars by using only one pulse. Based on [22], this algorithm estimates a sparser vector compared to the l _{1}-norm algorithm. On the other hand, the smoothed L0 (SL0) algorithm has been presented for two-dimensional (2D) sparse problems [23]. In this algorithm, a discontinuous l _{0}-norm function is approximated by a continuous one and then a sparse solution is reached using the steepest ascent algorithm followed by a projection onto a feasible set [24–27]. In [28], this algorithm has been applied to pulse Doppler radars with a lot of advantages such as target velocity extraction and pulse integration. However, this algorithm has been presented by using an approximated l _{0}-norm function for which we will later show that it achieves a lower performance in sparse signal recovery at low signal-to-noise ratios (SNRs) or for a small number of pulses compared to the SLIM algorithm. Also, the 2D IAA which is a nonparametric algorithm is presented in [29] for a general sparse solution. However, as it will be shown in the simulation section, its performance is poor at low SNRs and also for a small number of pulses.
In [30], a low-complexity CS approach is developed by decoupling the range, Doppler frequency, and angle parameters. It is assumed that the estimates of azimuth angles are obtained from one pulse by discretizing the angle space. Then, the Doppler estimates are extracted by combining the data of multiple pulses and using the initial estimated angles. Based on angle-Doppler estimates, the range is then estimated using frequency-varying received pulses. However, two problems are still of concern. First, the number of targets within the radar surveillance area is, in practice, so large that the angle space will not be sparse enough to apply the CS theory. In addition, a huge number of antennas are needed in a MIMO radar to discriminate among a large number of targets in the angle space with an acceptable resolution. Secondly, the SNR of one pulse is not sufficient for estimation of targets’ angles.
In this paper, we develop a 2D sparse model for pulse Doppler MIMO radar signals and find its relation with the Kronecker product factorization in the 1D model. To solve the 2D sparse signal equation, it can be converted to a 1D model and be recovered using 1D sparse recovery algorithms. However, this leads to a very large number of computations for which we will introduce here a new simpler technique. Therefore, a 2D SLIM algorithm is proposed for direct solution of a 2D sparse signal equation. In this approach, the Kronecker factorization is used to separate a large-dimension matrix into two smaller matrices. This procedure leads to reducing the number of products and therefore decreasing the computation cost and required memories.
Moreover, we develop the 2D version of the well-known 1D matched filter (1D MF) for comparison with the proposed 2D SLIM. Moreover, we compare the 2D SLIM algorithm with spectral projected gradient for l _{1}-norm minimization (SPGL1) algorithm which is appropriate for large-scale sparse recovery problems and complex-valued data [31].
List of notations
‖. ‖_{2} | l _{2} -norm |
‖. ‖_{ F } | Frobenius norm of matrix |
⊙ | Hadamard (element-wise) matrix product |
⊗ | Kronecker product |
(⋅)^{ T } | Transpose of a vector or matrix |
(⋅)^{ H } | Conjugate transpose of a vector or matrix |
(⋅)* | Conjugate of a vector or matrix |
⊘ | Element-wise matrix division |
\( {\left(\cdot \right)}_i^c \) | ith column of a matrix |
\( {\left(\cdot \right)}_i^r \) | ith row of a matrix |
vec(⋅) | Stacking the columns of a matrix on top of each other |
2 Signal model for pulse Doppler MIMO radars
is an (N _{ s } + N _{ R } − 1) × (N _{ s } + N _{ R } − 1) shift matrix for the rth range bin. Also, α _{ r,a,d } for r = 1, ⋯, N _{ R }, a = 1, ⋯, N _{ A }, and d = 1, ⋯, N _{ D } denote the return complex reflection coefficients of targets corresponding to the radar cross-section. In this scenario, since the number of range-angle-Doppler bins, in which actual targets are detected, is much smaller than the total number of radar bins, most of reflection coefficients are zero, and thus, the radar signal model can be assumed sparse. We assume that complex reflection coefficients during pulse repetitions are constant.
where \( \boldsymbol{X}=\left[\begin{array}{ccc}\hfill {\boldsymbol{x}}_1\hfill & \hfill \cdots \hfill & \hfill {\boldsymbol{x}}_{N_D}\hfill \end{array}\right] \) contains the complex reflection coefficients corresponding to the radar cross-section, and \( {\boldsymbol{\theta}}_p={\left[\begin{array}{ccc}\hfill {e}^{j{T}_R\left(p-1\right){\omega}_1}\hfill & \hfill \cdots \hfill & \hfill {e}^{j{T}_R\left(p-1\right){\omega}_{N_D}}\hfill \end{array}\right]}^T. \)
where \( \boldsymbol{E}=\left[\begin{array}{ccc}\hfill {\boldsymbol{e}}_1\hfill & \hfill \cdots \hfill & \hfill {\boldsymbol{e}}_{N_P}\hfill \end{array}\right] \) and \( \boldsymbol{\Theta} =\left[\begin{array}{ccc}\hfill {\boldsymbol{\theta}}_1\hfill & \hfill \cdots \hfill & \hfill {\boldsymbol{\theta}}_{N_P}\hfill \end{array}\right] \).
Equation (14) presents a 2D sparse signal model for pulse Doppler MIMO radars where \( \boldsymbol{Y}\in {\mathrm{\mathbb{C}}}^{M_r\left({N}_s+{N}_R-1\right)\times {N}_P}, \) \( \boldsymbol{A}\in {\mathrm{\mathbb{C}}}^{\left[{M}_r\left({N}_s+{N}_R-1\right)\right]\times \left[{N}_R{N}_A\right]}, \) \( \boldsymbol{\Theta} \in {\mathrm{\mathbb{C}}}^{N_D\times {N}_P}, \) and \( \boldsymbol{X}\in {\mathrm{\mathbb{C}}}^{N_R{N}_A\times {N}_D} \). Due to the underdetermined nature of (14) for a sparse model (i.e., M _{ r }(N _{ s } + N _{ R } − 1) < N _{ R } N _{ A } and N _{ P } < N _{ D }), it has no unique solution. Our goal is to find the sparsest matrix for X in which we have as many zero elements as possible.
where x = vec(X), y = vec(Y), e = vec(E), and Φ = Θ ^{ T } ⊗ A. Although x can be computed using 1D sparse recovery algorithms such as IAA [35] and SLIM [22], due to the large dimension of \( \boldsymbol{\Phi} \in {\mathrm{\mathbb{C}}}^{\left[{N}_P{M}_r\left({N}_s+{N}_R-1\right)\right]\times \left[{N}_R{N}_A{N}_D\right]}, \) 1D solutions are computationally extremely expensive. Accordingly, due to the smaller number of products appeared in (14) compared to (16), developing the 2D algorithm for direct solution of (14) leads to an extreme reduction of computational load compared to 1D one.
3 2D sparse signal recovery
In this section, at first, we give an overview of 1D SLIM algorithm. Then, a 2D SLIM algorithm is proposed for pulse Doppler MIMO radars by direct solution of (14), which leads to a lower computational cost compared to the 1D algorithms. In addition, for comparison purposes, the 2D SLIM is compared with the 2D SL0 [23] and 2D IAA [29] algorithms recently introduced for 2D sparse recovery problems. Moreover, we develop the 2D MF for comparison purposes.
3.1 Overview on 1D SLIM algorithm
- 1.
Iterative estimation of sparse vector x,
- 2.
Iterative estimation of noise power η,
3.2 2D SLIM
2D CGLS Algorithm
Initialization: |
U _{0} = 0, G _{0} = 0, T _{0} = − η ^{1/2} Y, R _{0} = − Y, P _{0} = Y |
1 W _{ l } = Σ ⊙ (A ^{ H } P _{ l } Θ ^{ H }) |
2 V _{ l } = η ^{1/2} P _{ l } |
3 \( {\alpha}_l={\left\Vert {\boldsymbol{R}}_l\right\Vert}_F^2/\left({\left\Vert {\boldsymbol{W}}_l\right\Vert}_F^2+{\left\Vert {\boldsymbol{V}}_l\right\Vert}_F^2\right) \) |
4 U _{ l + 1} = U _{ l } + α _{ l } P _{ l } |
5 G _{ l + 1} = G _{ l } + α _{ l } W _{ l } |
6 T _{ l + 1} = T _{ l } + α _{ l } V _{ l } |
7 R _{ l + 1} = A(Σ ⊙ G _{ l + 1})Θ + η ^{1/2} T _{ l + 1} |
8 \( {\beta}_l={\left\Vert {\boldsymbol{R}}_{l+1}\right\Vert}_F^2/{\left\Vert {\boldsymbol{R}}_l\right\Vert}_F^2 \) |
9 P _{ l + 1} = − R _{ l + 1} + β _{ l } P _{ l } |
Go to Step 1 until \( \frac{1}{K}{\left\Vert \boldsymbol{A}\left(\boldsymbol{\Gamma} \odot \left({\boldsymbol{A}}^H\boldsymbol{U}{\boldsymbol{\Theta}}^H\right)\right)\boldsymbol{\Theta} +\eta \boldsymbol{U}-\boldsymbol{Y}\right\Vert}_F<\varepsilon \) |
Final result U = U _{ l + 1} |
where Δ is a small positive constant.
where h(q) is the number of selected peaks in the output of the SLIM algorithm that is executed for a particular value of q. To explain how we choose the number of selected peaks (h(q)), at first, the absolute value of the SLIM algorithm output defined as |X| is sorted in a descending order. Then, the largest peak is selected and the other values of matrix X are set to zero to form the matrix \( \widehat{\boldsymbol{X}} \). Using (28), the BIC is computed for \( \widehat{\boldsymbol{X}} \) and h(q) = 1. In the round, the two largest peaks are selected and the other values of X are set to zero, and the related BIC is computed for \( \widehat{\boldsymbol{X}} \) and h(q) = 2 and so on. The value of h(q) is the number of the selected peaks that yields the lowest BIC. After running the 2D SLIM for a selected set of q and computing h(q), we choose that q which minimizes the BIC. The factor 5 in (28) shows the number of unknown parameters including range, angle, Doppler frequency, and the complex reflection coefficients of targets.
3.3 2D MF
where the elements of \( \boldsymbol{\Lambda} \in {\mathrm{\mathbb{C}}}^{N_R{N}_A\times {N}_D} \) are defined by Λ_{ ij } and ⊘ is the element-wise matrix division.
4 Computational complexity of 1D and 2D SLIM
In the 1D SLIM, the main computational cost in each iteration belongs to the product of Φ and a vector like x. For the 2D SLIM, this product is converted to a 2D form as AX Θ using the Kronecker factorization Φ = Θ ^{ T } ⊗ A.
The main difference between the 1D and 2D forms from a computational point of view is in the number of flops for calculating Φ x and AX Θ. The complexities of Φ x and AX Θ are O(N _{ P } M _{ r }(N _{ s } + N _{ R } − 1)N _{ R } N _{ A } N _{ D }) and O(M _{ r }(N _{ s } + N _{ R } − 1)N _{ R } N _{ A } N _{ D }) + O(N _{ P } M _{ r }(N _{ s } + N _{ R } − 1)N _{ D }), respectively. We have assumed that the product matrix AX is computed first, and then the result is multiplied by Θ. By comparing these two computational complexities, it is demonstrated that the ratio of the 2D processing load over that of its equivalent 1D is \( \frac{1}{N_P}+\frac{1}{N_R{N}_A} \). If it is assumed N _{ P } ≪ N _{ R } N _{ A }, then this ratio is equal to \( \frac{1}{N_P} \).
In addition, the Kronecker factorization can take advantage of multi-core processors [38] by which the 2D SLIM algorithm can be parallelized and solved. Then, the speed of this algorithm compared to the 1D one is approximately increased by a factor proportional to the number of processing cores.
5 Numerical examples
To show the computational efficiency of the 2D SLIM, which is the main goal of this work, we compare the running time of the 1D version of SLIM, IAA, SL0, and MF algorithms with their 2D versions. To make a fair comparison between 1D and 2D SLIM algorithms, we use the CGLS in the 1D SLIM to reduce the computational cost of matrix inversion. Moreover, we express the product of the diagonal matrix Π and the vector Φ ^{ H } u by the Hadamard product as x = ΠΦ ^{ H } u = ϑ ⊙ (Φ ^{ H } u) and do a similar procedure for the CGLS steps.
The number of targets is N _{ t } = 40 and the SNR of each target is 10 dB. We consider the transmit signal with a cyclic approach [39] with N _{ s } = 32. The transmit and receive antennas are uniform linear arrays with Δ_{ t } = 2.5λ _{0}, Δ_{ r } = 0.5λ _{0}, and M _{ t } = M _{ r } = 5. The carrier frequency, sub-pulses bandwidth, and the PRF are respectively f _{ c } = 1 GHz, B = 10 MHz, and f _{ r } = 2 kHz. The surveillance field is divided into N _{ R } = 20 range bins, N _{ A } = 31 angular bins between − 30° to 30° with 2° angular resolution, and N _{ D } = 40 Doppler bins with the resolution of 50 Hz. The lower and upper thresholds for limiting the amplitude of output signals are − 30 and 10 dB, respectively.
Runtime of different algorithms for N _{ P } = 20
SLIM | IAA | SL0 | MF | SPGL1 | |
---|---|---|---|---|---|
1D | 74.5 | 264.02 | 456.70 | 5.19 | 147.5 |
2D | 0.62 | 0.64 | 0.67 | 0.13 | N/A |
6 Conclusions
We derived the 1D and 2D sparse signal models for pulse Doppler MIMO radars. The 2D SLIM algorithm was derived by direction solution of the 2D sparse model. Due to using a lower number of products in the corresponding relationships, the computational cost of 2D SLIM compared to that of the 1D one was extremely reduced. Also, simulation results showed that the 2D SLIM outperforms the other algorithms in accurate estimation of range, angle, and Doppler parameters.
Declarations
Authors’ Affiliations
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