Joint Smoothed l₀-Norm DOA Estimation Algorithm for Multiple Measurement Vectors in MIMO Radar

Colocated multiple-input multiple-output (MIMO) radar has attracted a growing interest recently because it can achieve higher resolution and better parameter identification compared with conventional phased-array radar [1]. As one type of colocated MIMO radar, the monostatic MIMO radar is equipped with closely-located transmit and receive arrays, which result in the same angle for direction-of-departure (DOD) and direction-of-arrival (DOA) [2]. Aiming at DOA estimation, a large quantity of methods have been proposed, most of them are based on the signal and noise subspaces [3,4,5,6], such as the representative MUSIC (multiple signal classification) algorithm [3] and ESPRIT (estimation of signal parameters via rotational invariance techniques) algorithm [4].

In the area of sensor array signal processing, sparse representation is an important technique to estimate the parameters. When reconstructing the sparse signal, to avoid the NP-hard (non-deterministic polynomial-time hard) $l_0$-norm minimization, different methods such as those based on the relaxed constraint $l_1$-norm minimization [7], the focal underdetermined system solution (FOCUSS) [8] and the sparse Bayesian learning (SBL) [9] have been proposed. These algorithms were originally designed for the single measurement vector (SMV) problem. Then, to extend them to the multiple measurement vector (MMV) case, some algorithms, e.g., M-FOCUSS and M-SBL, were proposed [10,11]. Recently, the emerging application that exploits the sparse representation technique to achieve the DOA estimation has been increasingly attractive. Compared with the alternative methods, sparse representation-based DOA estimation methods can achieve higher angle resolution and better adapt to some challenging scenarios such as low signal-to-noise ratio (SNR), as discussed in [7]. Among the abovementioned sparse signal reconstruction methods, $l_1$-norm minimization is widely used for sparse DOA estimation because of its desirable solution accuracy and reasonable computing speed [7,10,12]. The DOA estimation algorithms using $l_1$-norm minimization include $l_1$-SRACV (sparse representation array covariance vectors) [13], real-valued (RV) $l_1$-SRACV [14], and the iterative algorithm with reweighting $l_1$-norm minimization [15]. All of these methods involve the SMV problem when recovering the signal. Aiming at the MMV case, the $l_1$-SVD (singular value decomposition) [7], RV $l_1$-SVD [16], reweighted $l_1$-SVD [17] and CMSR (covariance matrix sparse representation) [10] were proposed for DOA estimation. However, the computational complexity of the $l_1$-norm minimization is relatively large compared with that of the fast sparse representation method named smoothed $l_0$-norm algorithm, which was proposed in [18,19] under the SMV circumstance, and applied in [20,21] for the imaging and the power-line communications. To achieve fast sparse DOA estimation, by designing a reweighted continuous function in the SMV case, the reweighted smoothed $l_0$-norm (RSL0) algorithm was proposed in [22] based on the covariance vector. With enlarged array aperture and better angle estimation performance, the RSL0 algorithm [22] is about two orders of magnitude faster than the $l_1$-norm minimization based methods.

The previously mentioned DOA estimation methods are all proposed under the ideal circumstance with Gaussian white noise. However, the noise is often correlated in practical, thus the additive noise is spatially modeled as Gaussian colored noise rather than white noise. It has been verified that, with colored noise, the performance of the conventional algorithms including the methods mentioned above, is seriously degraded, except for the specially designed methods such as those based on fourth-order cumulants (FOC) [12,23,24]. These FOC-based methods are computationally expensive when they are applied to estimate the DOAs for MIMO radar. In [25], the reduced-dimension (RD) FOC-based sparse representation method (RD $l_1$-SRFOC) was proposed for DOA estimation in MIMO radar with array errors. RD $l_1$-SRFOC is an $l_1$-norm minimization based sparse representation method, which uses the high-order cumulants to deal with the colored noise and simultaneously solves the problem of mutual coupling.

It can be concluded that the existing sparse DOA estimation should improve the following problems: (I) for the $l_1$-norm minimization based methods, the computation complexities need to be lowered down and the computing time needs speeding up; (II) the fast sparse DOA estimation algorithm, i.e., the reweighted smoothed $l_0$-norm algorithm [22], is based on the covariance vector that is deemed the SMV problem. The fast sparse algorithm tailored for the MMV case needs to be studied and put forward; and (III) fast sparse algorithms for solving the colored Gaussian noise have not been available so far.

To solve the three problems stated above, a joint smoothed $l_0$-norm algorithm for DOA estimation in the MIMO radar is proposed in this paper. The new sparse algorithm first obtains a low-complexity FOC-based data matrix to eliminate the white or colored Gaussian noises. Secondly, the proposed algorithm designs a joint smoothed function tailored for the MMV case. Thirdly, with the MMV-based joint smoothed function, a joint smoothed $l_0$-norm sparse representation framework is constructed. Finally, the corresponding gradient-based sparse signal reconstruction is designed, and then DOA estimation is achieved. The proposed algorithm is a fast sparse DOA estimation algorithm, which can solve the multiple measurement vector problem, and perform well for both of the white and colored Gaussian noise environments. The proposed algorithm is about two orders of magnitude faster than the $l_1$-norm minimization based methods. This is because it approximates the $l_0$-norm by a joint smoothed function and performs the gradient-based steepest ascent scheme, which avoids the convex optimization problem involved in the $l_1$-norm minimization. The proposed algorithm can provide better DOA estimation performance than $l_1$-SVD, RV $l_1$-SVD and RV $l_1$-SRACV methods.

The rest of this paper is organized as follows. Section 2 presents the MIMO radar system and array signal models. In Section 3, the implementation process of the proposed algorithm is described in detail. Section 4 gives some related remarks and discussions regarding the advantages, the extended applications, the computational complexity and the noise elimination of the proposed algorithm. The experimental results of different methods are compared in Section 5. Finally, Section 6 concludes this paper.

Throughout the paper, the notations ${(·)}^{*}$, ${(·)}^{\mathrm{T}}$, ${(·)}^{\mathrm{H}}$, ${(·)}^{-1}$ and ${(·)}^{+}$ denote conjugation, transpose, conjugate transpose, inverse and pseudo-inverse, respectively. ${\mathbf{I}}_K$ represents a $K\times K$-dimensional unit matrix. $\left|\right|·{\left|\right|}_0$ and $\left|\right|·{\left|\right|}_2$ denote the $l_0$-norm and the $l_2$-norm, respectively. In addition, we use ⊗, $\mathrm{E}(·)$ and $\mathrm{vec}(·)$ to separately indicate the Kronecker product, the expectation and the vectorization operators.

A narrowband monostatic MIMO radar system is considered, and it is shown in Figure 1. The transmit and the receive arrays of this system are equipped with M transmit and N receive antennas, respectively. The arrays are both half-wavelength spaced uniform linear arrays (ULAs), whose effects of the array errors, including mutual coupling and gain-phase errors, can be ignored. The transmitting antennas transmit M orthogonal narrowband waveforms, such as BPSK (binary phase shift keying) modulated signal waveforms. In the far field, there are P uncorrelated targets regarded as point scatterers at the same range. In addition, it is assumed that $P\le M+N-2$ [22,25,26]. Thus, at the receive array, the N antennas are impinged by the echo signals reflected by the P targets. Because of the closely located arrays in the monostatic MIMO radar, for the pth target, DOA and DOD are the same and can be denoted by ${\theta }_p$. Then, by matched filtering operation, the $N\times 1$ dimensional complex envelop of the output of the mth carrier matched filter is expressed as [14,27] where ${\mathbf{a}}_r\left({\theta }_p\right)={\left[1,e^{j\pi sin\left({\theta }_p\right)},e^{j\pi 2sin\left({\theta }_p\right)},\dots ,e^{j\pi (N-1)sin\left({\theta }_p\right)}\right]}^{\mathtt{T}}$ is the receive steering vector, ${\mathbf{a}}_{tm}$ is the mth element of the transmit steering vector ${\mathbf{a}}_t\left({\theta }_p\right)={\left[1,e^{j\pi sin\left({\theta }_p\right)},e^{j\pi 2sin\left({\theta }_p\right)},\dots ,e^{j\pi (M-1)sin\left({\theta }_p\right)}\right]}^{\mathtt{T}}$, $s_p\left(t\right)$ contains the target reflection coefficient and the transmitted baseband signal such as non-circular signal, and ${\mathbf{n}}_m\left(t\right)$ is the noise vector after the mth matched filter. After all the matched filters, the received data vector, i.e., the vector composed of the outputs of the M matched filters, is given by [2,25,28] where $\mathbf{x}\left(t\right)\in {\mathbb{C}}^{MN\times 1}$ and $\mathbf{n}\left(t\right)={[{\mathbf{n}}_1^{\mathtt{T}}\left(t\right),\dots ,{\mathbf{n}}_M^{\mathtt{T}}\left(t\right)]}^{\mathtt{T}}\in {\mathbb{C}}^{MN\times 1}$ is the zero-mean Gaussian white or colored noise vector. $\mathbf{s}\left(t\right)={\left[s_1\left(t\right),s_2\left(t\right),\dots ,s_P\left(t\right)\right]}^{\mathtt{T}}\in {\mathbb{C}}^{P\times 1}$ is the reflected signal vector after the M carrier matched filters, and it is assumed to be statistically independent and non-Gaussian with zero-mean. The target model is considered as the classical Swerling case II, namely, the radar cross section (RCS) fluctuations are constant during a snapshot period and vary independently from snapshot to snapshot [29]. $\mathbf{n}\left(t\right)$ and $\mathbf{s}\left(t\right)$ are independent of each other. Moreover, $\mathbf{A}\in {\mathbb{C}}^{MN\times P}$ is the transmit–receive steering matrix, and its detailed expression is where $\mathbf{a}\left({\theta }_p\right)={\mathbf{a}}_t\left({\theta }_p\right)\otimes {\mathbf{a}}_r\left({\theta }_p\right)\in {\mathbb{C}}^{MN\times 1}$ is the transmit–receive steering vector for $p=1,2,\dots ,P$. Therefore, by collecting J snapshots, the $MN\times J$ dimensional received data matrix in monostatic MIMO radar is represented as where $\mathbf{S}=[\mathbf{s}\left(t_1\right),\dots ,\mathbf{s}\left(t_J\right)]\in {\mathbb{C}}^{P\times J}$ and $\mathbf{N}=[\mathbf{n}\left(t_1\right),\dots ,\mathbf{n}\left(t_J\right)]\in {\mathbb{C}}^{MN\times J}$ are the signal matrix and complex Gaussian noise matrix, respectively. Based on the received data $\mathbf{X}$, sparse DOA estimation can be viewed as the signal reconstruction that subjects to [14] where ${\mathbf{A}}_{\widehat{\theta }}=[\mathbf{a}({\widehat{\theta }}_1),\dots ,\mathbf{a}({\widehat{\theta }}_L)]$ is the complete dictionary with the discrete sample grid ${\{{\widehat{\theta }}_i\}}_{i=1}^L$, $L\gg P$. ${\mathbf{S}}_{\widehat{\theta }}$ has the same row support with $\mathbf{S}$. Let ${\mathbf{s}}_{\widehat{\theta }}=[\mathbf{s}({\widehat{\theta }}_1),\dots ,\mathbf{s}({\widehat{\theta }}_L)]$ be a sparse vector whose ith element can be equal to the $l_2$-norm of the ith row in ${\mathbf{S}}_{\widehat{\theta }}$. To obtain the sparsest solution of Equation (5), an ideal constraint is the $l_0$-norm method by minimizing the nonzero number of ${\mathbf{s}}_{\widehat{\theta }}$, which can be expressed as

Unfortunately, the $l_0$-norm minimization method in Equation (6) is NP-hard. To solve this problem, conventional sparse DOA estimation methods [7,10,13,14,15,17] can obtain the DOAs with the $l_1$-norm minimization. However, the computational complexity of the $l_1$-norm minimization is relatively large. In [22], by solving the array covariance vector based SMV sparse signal reconstruction problem, the reweighted smoothed $l_0$-norm algorithm greatly improves the computation speed. To develop a fast sparse DOA estimation algorithm tailored for the MMV problem existed in the MIMO radar systems, a joint smoothed $l_0$-norm algorithm is proposed in the following.

In this part, some simulations are implemented to evaluate the performance and the computation speed of the proposed method. The simulation results of the proposed method are compared with those of the $l_1$-SVD [7], RV $l_1$-SVD [16], RV $l_1$-SRACV [14] and reweighted smoothed $l_0$-norm (RSL0) [22] algorithms. In all of the simulations, a narrowband monostatic MIMO radar system with ULAs is considered. The antennas of the transmit array and the receive array are both half-wavelength spaced, and their numbers are $M=6$ and $N=6$, respectively. The transmitted signals are assumed to be BPSK modulated [27,30], moreover, they are mutually orthogonal and different from each other. The noise is white Gaussian with the covariance matrix $\mathbf{R}={\mathbf{I}}_{MN}$, or colored Gaussian with the elements in the covariance matrix being $\mathbf{R}(k_1,k_2)=0.{75}^{|k_1-k_2|}{e}^{j\pi (k_1-k_2)/2}$. By means of the minimum description length (MDL) principle or the Akaike information criterion (AIC) principle [31], the prior knowledge of the target number is assumed to be known as P.

When constructing the complete dictionary, the discrete sample grid is set from $-{90}^{\circ }$ to ${90}^{\circ }$ with the uniform interval distance $\Delta \widehat{\theta }=0.{05}^{\circ }$ for the proposed algorithm, as well as the $l_1$-SVD, RV $l_1$-SVD, RV $l_1$-SRACV and RSL0 algorithms. The parameters ${\sigma }_1$, $\mu$ and ${\sigma }_{off}$ are fixedly set at ${\sigma }_1=4\max \{|{\mathbf{U}}_0^{\widehat{\theta }}|\}$, $\mu =2.5$ and ${\sigma }_{off}=0.0007$ for the proposed algorithm and the RSL0 algorithm. The effects of the other parameters, i.e., the decreasing factor $\alpha$ and the iteration number Q, will be evaluated and given in detail in the following experiments. The definitions for the signal-to-noise ratio (SNR) and the root mean square error (RMSE) are shown, respectively, as follows: where ${\widehat{\theta }}_{p,i}$ is the angle estimation of the pth target DOA ${\theta }_p$ for the ith Monte Carlo trial.

Figure 2 depicts the spatial spectrum of the proposed method for white noise and colored noise, where SNR = 5 dB and SNR = −10 dB are considered. There are three targets located at ${\theta }_1=-8.2^{\circ }$, ${\theta }_2=0^{\circ }$ and ${\theta }_3={21}^{\circ }$. The spatial spectrum is plotted by computing $10{\log }_{10}\left[\right|{\overline{\ell }}_{\mathbf{p}}|/\max (|{\overline{\ell }}_{\mathbf{p}}\left|\right)]$ with ${\overline{\ell }}_{\mathbf{p}}$ being the final sparse solution. Figure 2 shows that the proposed joint smoothed $l_0$-norm minimization algorithm is effective for the MMV case of DOA estimation in monostatic MIMO radar. Moreover, it is suitable for the practical radar systems with colored Gaussian noise.

Figure 3a,b evaluates the effects of the parameters $\alpha$ and Q on the DOA estimation performance of the proposed method, with $Q=3$ and $\alpha =0.5$, respectively. Moreover, $J=1000$. Three uncorrelated targets with the DOAs ${\theta }_1=-{8.2}^{\circ }$, ${\theta }_2=0^{\circ }$ and ${\theta }_3={21}^{\circ }$ are considered. In Figure 3, it can be verified that the proposed method keeps the same angle estimation performance when the parameters $\alpha$ and Q vary from $\alpha =0.5$ to $\alpha =0.9$ and $Q=3$ to $Q=35$. In Figure 3b, the RMSE decreases with the increase of Q when $Q\le 3$, and it reaches an asymptotic value at $Q=3$. For the iterations of the steepest ascent algorithm, we just need a region near the maximum value of $F_{w\sigma }({\mathbf{T}}_{\widehat{\theta }}^{l_2})$; thus, a fixed small positive integer $Q=3$ is applicable.

With respect to Figure 3, the effects of the parameters $\alpha$ and Q on the average computation time of the proposed method are shown in Table 1 and Table 2. The simulation conditions keep the same with those of Figure 3. As the implementing processes of the analyzed methods are not affected by the type of the noise, the computation time is the same for the white and the colored Gaussian noises. It can be observed that the computation speed becomes slower when $\alpha$ and Q increase. However, larger $\alpha$ and Q are not able to improve the estimation performance of the proposed method, as shown in Figure 3a,b. As a result, small values of $\alpha$ and Q are enough to guarantee the estimation accuracy and the computation speed. In the following experiments, the parameters $\alpha$ and Q are set at $\alpha =0.5$ and $Q=3$.

Figure 4a,b verify the RMSE of DOA estimation versus SNR for white noise and colored noise, where $J=1000$. Consider three targets whose DOAs are ${\theta }_1=-8.2^{\circ }$, ${\theta }_2=0^{\circ }$ and ${\theta }_3={21}^{\circ }$. According to Figure 4a,b, we can conclude that the RSL0 algorithm achieves the best DOA estimation for white noise. For colored noise, the proposed algorithm and the RSL0 algorithm provide better angle estimation performance than the $l_1$-SVD, RV $l_1$-SVD and RV $l_1$-SRACV methods.

The processing time of signal reconstruction is compared in Table 3, where $J=1000$ and SNR = 0 dB. Different antenna numbers and target numbers are considered. For $p=2$, the DOAs are ${\theta }_2=0^{\circ }$ and ${\theta }_3={21}^{\circ }$. For $p=3$, they are ${\theta }_1=-8.2^{\circ }$, ${\theta }_2=0^{\circ }$ and ${\theta }_3={21}^{\circ }$. Table 3 verifies that, compared with the $l_1$-norm minimization based methods, the proposed algorithm has much lower computational complexity and is about two orders of magnitude faster.