Efficient Implementation for SBL-Based Coherent Distributed mmWave Radar Imaging

Abstract

In a distributed frequency-modulated continuous waveform (FMCW) radar system, the echo data collected are not continuous in the azimuth direction, so the imaging effect of the traditional range-Doppler (RD) algorithm is poor. Sparse Bayesian learning (SBL) is an optimization algorithm based on Bayesian theory that has been successfully applied to high-resolution radar imaging because of its strong robustness and high accuracy. However, SBL is highly computationally complex. Fortunately, with FMCW radar echo data, most of the time-consuming SBL operations involve a Toeplitz-block Toeplitz (TBT) matrix. In this article, based on this advantage, we propose a fast SBL algorithm that can be used to obtain high-angular-resolution images, in which the inverse of the TBT matrix can be transposed as the sum of the products of the block lower triangular Toeplitz matrix and the block circulant matrix by using a new decomposition method, and some of the matrix multiplications can be quickly computed using the fast Fourier transform (FFT), decreasing the computation time by several orders of magnitude. Finally, simulations and experiments were used to ensure the effectiveness of the proposed algorithm.

1. Introduction

Since millimeter-wave imaging radars have the advantages of small size, high resolution, a long detection range, and insensitivity to light and weather [1], they have received increasing attention in various civil fields, such as automatic driving [2,3,4,5,6], traffic surveillance [7,8], UAV obstacle avoidance [9,10], security surveillance [11,12], health monitoring [13,14] and so on.

As is well-known, a radar’s range resolution is proportional to the bandwidth of the radar’s transmit signal and can be improved by increasing the bandwidth of the transmit signal. The angular resolution of the radar is proportional to the size of the radar antenna aperture [15]. The synthetic aperture technique is an effective method that can be used to increase the effective aperture of a radar and, thus, improve the angular resolution, but it requires the radar to have a cross-range motion relative to the imaging scene. This requirement makes this technology unsuitable for forward-looking imaging applications where the radar and the imaging scene are relatively static or do not have relative cross-range motion, such as automatic driving and traffic surveillance [2,15,16]. In these situations, real aperture imaging technology based on a multichannel array radar is required to improve the angular resolution. For example, the millimeter-wave real aperture imaging radar MMWAVCAS-RF-EVM produced by Texas Instruments Incorporated adopts 12 transmitter and 16 receiver multiple-input multiple-output (MIMO) arrays to synthetize 192 channel array apertures with an angular resolution of 1.2 degrees [17]. However, such an angular resolution is still unsatisfactory for many practical applications. If we continue to increase the antenna size and the number of channels in a single radar, it will cause the cost, power consumption, signal transmission loss and the difficulty of manufacturing and installing the radar to increase significantly and become unacceptable [2].

The coherent distributed aperture technique is an effective method to solve the above problems. Coherent distributed aperture radar involves the coherent synthesis of the return signals of multiple small aperture radars placed separately through signal processing algorithms, increasing the signal-to-noise ratio and the measurement accuracy of the target return and improving the angular resolution of the imaging system [18,19,20,21]. In [19], the signal-to-noise ratio of the target return was improved by 9 dB through coherent distributed transmission and reception with two radars. In [20], the target positioning accuracy was improved by using the weighted summation of the imaging results of multiple vehicle-mounted radars. In [2,21], coherent distributed millimeter-wave imaging radar systems with two monistric radars were proposed to improve the angular resolution for automotive vehicles. However, neither provided the specific distributed coherent processing methods. Similar to [2,21], we addressed the problem of improving the angular resolution of real aperture imaging through coherent processing of multiple distributed millimeter-wave radars in this study.

From the perspective of signal processing, the problem of improving angular resolution through the coherent processing of distributed radars is a spectrum estimation problem with gapped measured data. There are many signal processing methods that can realize spectral estimation based on gapped measured data. Coherent processing of gapped frequency band data based on the multiple signal classification (MUSIC) algorithm was described in [22], and approaches to spectral estimation using gapped data based on the Capon and the amplitude and phase estimation (APES) algorithms were given in [23]. However, all these algorithms were designed for multiple snapshots. For the single-snapshot spectral estimation problem for imaging, spatial smoothing is required to obtain the covariance matrix for the measured data. Unfortunately, the spatial smoothing inevitably leads to resolution loss. To overcome this problem, a method for gapped data spectral estimation based on the iterative adaptive approach (IAA) algorithm was proposed in [24], and the data covariance matrix was constructed from the single-snapshot gapped data iteratively. Thus, the resolution loss caused by spatial smoothing was avoided. Recently, the sparse reconstruction algorithm has become a promising way to deal with gapped data spectrum estimation in the case of single-snapshot data [25].

Sparse reconstruction algorithms can usually be divided into three categories: orthogonal matching pursuit [26], basis pursuit [27] and sparse Bayesian learning (SBL) algorithms [28,29,30]. Compared with the other two kinds of algorithms, SBL has the advantages of less strict requirements for the signal sparsity, good noise resistance, high resolution, high reconstruction accuracy, and insensitivity to manual parameter settings, making it widely used [28,29,30,31,32,33,34]. Consequently, we implemented coherent distributed radar imaging via the SBL algorithm in this study.

However, the conventional SBL approach involves the calculation of the matrix inversion and the matrix multiplication in each iteration, which requires a large amount of computation [29,30,31,32,33,34,35]. Several strategies have been developed to reduce the computation complexity of the SBL algorithm. Fast SBL algorithm technologies, adding or removing the basis vectors sequentially based on the variation [29] or marginal likelihood maximization [30], have been proposed. However, the reconstruction accuracies of these kinds of fast SBL algorithms degenerate and the reductions in their computation complexities are not significant in real applications. Other kinds of fast SBL algorithms have been developed based on approximated message passing (AMP) [31] and Gaussian belief propagation (GBP) [32]. However, these methods only work well in cases in which the sensing matrix follows a zero-mean sub-Gaussian distribution. Recently, inverse-free fast SBL algorithms based on majorization–minimization technology have been proposed [33,34], but they have slow convergence rates and become invalid with high-correlation dictionaries.

In contrast to the abovementioned fast SBL algorithms that have made certain approximations in some of their formulations, in this paper, we propose a novel fast SBL algorithm without any approximations, which is referred to as LC-SBL. Our fast SBL algorithm was developed based on the signal model of the coherent distributed aperture millimeter-wave radar and uses the structural characteristics of the Toeplitz-block Toeplitz (TBT) matrix for its data covariance matrix. Since there are no extra approximations, the convergence rate and the reconstruction accuracy of LC-SBL are the same as those of the original SBL algorithm, but the computation complexity is reduced significantly. The main contributions of this paper are as follows:

• A signal model for the coherent distributed aperture imaging radar is established, and a forward-looking imaging method based on SBL is proposed;
• A novel form of low-displacement-rank decomposition with a block triangle and block cyclic matrix for the TBT matrix is established, and a fast SBL algorithm for gapped measured data based on the novel low-displacement-rank decomposition of the TBT matrix is proposed;
• A coherent distributed millimeter-wave forward-looking imaging radar system composed of three 77GHz millimeter-wave imaging radars was designed. The improvements in both the angular resolution of the coherent distributed radar imaging system and the performance of the LC-SBL algorithm were verified through experiments.

The rest of this paper is organized as following. First, the notations used in this article are explained. The distributed FMCW MIMO radar system and SBL are briefly described in Section 2. In Section 3, we describe the new decomposition method for the inverse of a TBT matrix and the development of a new fast SBL algorithm based on this decomposition. Section 4 describes the several simulations and experiments used to verify the effectiveness of applying the proposed algorithm to a distributed radar system. Finally, the conclusions are drawn in Section 5.

Notations

${(•)}^{-1}$ represents the matrix inverse operation. $diag(•)$ represents the main diagonal elements of a matrix. $conj(•)$ represents the complex conjugate. Matrices/vectors composed of submatrices with the same dimension are called block matrices/vectors. We use boldface lowercase letters to denote vectors and block vectors, boldface uppercase letters to denote matrices and block matrices and uppercase letters to denote the submatrices of block matrices. ${(•)}^H$ represents the conjugate transpose of a vector/matrix. ${(•)}^T$ represents the transpose of a vector/matrix. ${(•)}^{\prime }$ represents the transpose of a block vector/matrix. ${(•)}^{\ast }$ represents the conjugate transpose of a block vector/matrix. For example, if a block vector a = $\left(a_0,a_1,\cdots ,a_{n-1}\right)$, in which $a_i(i<n)$ is the submatrix and has a smaller dimension than a, then we have:

$${\mathbf{a}}^T=\left[\begin{array}{c}{a}_0^T\\ a_1^T\\ ⋮\\ a_{n-1}^T\end{array}\right]\hspace{1em}\hspace{1em}{\mathbf{a}}^H=conj\left[\begin{array}{c}{a}_0^T\\ a_1^T\\ ⋮\\ a_{n-1}^T\end{array}\right]\hspace{1em}\hspace{1em}{\mathbf{a}}^{\prime }=\left[\begin{array}{c}{a}_0\\ a_1\\ ⋮\\ a_{n-1}\end{array}\right]\hspace{1em}\hspace{1em}{\mathbf{a}}^{\ast }=conj\left[\begin{array}{c}{a}_0\\ a_1\\ ⋮\\ a_{n-1}\end{array}\right]$$

Additionally, I represents the identity matrix. $(\widetilde{•})$ represents the flipping of the matrix upside down and left and right. We use $\otimes$ and $\mathsf{\Theta }$ to denote the circular convolution and the linear convolution, respectively. $\mathcal{C}\mathcal{N}$ and $\mathcal{G}$ denote the Gaussian distribution and the Gamma distribution, respectively. $ℱ$ and $ℐℱ$ denote the employment of the fast Fourier transform (FFT) and the inverse FFT (IFFT) with a vector, respectively. ${ℱ}_2$ denotes the employment of the two-dimensional (2D) FFT with a matrix. For a vector $\mathbf{x}=(x_0,x_1,\cdots x_{n-1})$, $x_i$ indicates its (i + 1)th element.

2. Signal Model for a Coherent Distributed Radar System and the SBL Algorithm

2.1. Distributed Radar System

An FMCW MIMO radar transmits a continuous, linear, frequency-modulated microwave signal. The signal emitted from a transmit antenna (Tx) can be represented as

$$s_t(t)=exp\left\{j\left(2\pi f_{0}t+\pi k{t}^2\right)\right\}$$

(1)

where t is the time variable; 0 < t < T, with T being the chirp duration; $f_0$ is the start frequency; and k is the chirp rate, defined as B/T, with B being the bandwidth of the radar. The signal reflected from K objects and received by a receive antenna (Rx) can be represented as

$$s_r(t)={\displaystyle \sum _{\mathrm{m}=1}^K\exp \left\{j\left(2\pi f_0\left(t-{\tau }_m\right)+\pi k{\left(t-{\tau }_m\right)}^2\right)\right\}}$$

(2)

where ${\tau }_{\mathrm{m}}=2R_m/c$ is the propagation delay, $R_{\mathrm{m}}$ is the distance between the radar and the mth target and c is the speed of light. The received signal is multiplied by $s_t$ and then passed through a lowpass filter to obtain the beat signal, and the beat signal is represented as

$$s_b(t)={\displaystyle \sum _{m=1}^K\exp \left\{2\pi j\left(\frac{2k{R}_m}{c}t+\frac{2f_0{R}_m}{c}\right)\right\}}$$

(3)

From Equation (3), we can only obtain distance information from a single channel, so more channels are needed for the directions of targets. Traditional single-input multiple-output (SIMO) radars are radar devices with a single transmitter and multiple receivers. The angular resolution of an SIMO radar depends on the value of Rx; i.e., $N_{rx}$. However, it is usually not feasible to increase $N_{rx}$ because of the limitations of the radar size. When more Tx antennas are added to a radar, it is essential that every Rx antenna is able to separate the signals corresponding to the different Tx antennas. Time division multiplexing (TDM) [35,36] and binary phase modulation (BPM) [37] technologies have been widely used in radar systems to overcome this problem.

In contrast to SIMO radars, multiple-input multiple-output (MIMO) radars can provide more elements while reducing the radar size. For example, a MIMO radar with $N_{tx}=3$ and $N_{rx}=$8 can synthesize 23 virtual array elements. The relationship between the real Tx/Rx antenna positions and the virtual array positions is shown in Figure 1.

In Figure 1, the distances between the receivers and the transmitters are $\lambda$ and $3\lambda$, respectively. Using this special physical antenna configuration, we can generate a virtual linear array consisting of 23 elements with the interelement space $\lambda /2$. When the signal from the Tx antenna is reflected by an object at the angle $\theta$ with regard to the radar, the phase difference between elements is $\pi \sin \theta$. Assuming that the aperture size of a single radar is w, then the signal of the ith element is

$$s_b(t,i)={\displaystyle \sum _{m=1}^K\exp \left\{2\pi j\left(f_{m}t+i\sin {\theta }_m/2\right)+j{\phi }_m\right\}}\hspace{1em}i\in (1,w)$$

(4)

where

$$f_m=2k{R}_m/c$$

(5)

$${\phi }_{\mathrm{m}}=4\pi f_0{R}_m/c$$

(6)

Taking the 2D-FFT along the t and i dimensions from Equation (4), we obtain

$$\begin{array}{l}\mathrm{u}\left(f,K\right)={ℱ}_2\left(s_b(t,i)\right)\\ \hspace{1em}\hspace{1em}=Tw{\displaystyle \sum _{m=1}^k\mathrm{sinc}\left\{T\pi \left(f-f_m\right)\right\}\cdot \hspace{0.17em}}\mathrm{sinc}\left\{w\pi \left(K-\sin {\theta }_m/2\right)\right\}\cdot \exp \left\{j{\phi }_m\right\}\end{array}$$

(7)

where u is the imaging result of $s_b(t)$. It is clear that Equation (7) includes both the distance and the angle information in a single radar, so we can image according to the formula. Furthermore, we expected that the angular resolution would be improved by using multiple radars. To build a distributed radar system, we put three radars in a line. The virtual array elements of the distributed radar system are shown in Figure 2, where $d_t$ is the virtual array length of a single radar and $d_s$ is the gap length between the radars. In particular, we set $d_s$ as equal to $d_t$ here. The virtual array is discontinuous because $d_s\gg$$\lambda /2$. Therefore, the traditional range-Doppler imaging method would result in problems, such as ghosting, grating lobes and side lobes, thereby reducing the image resolution.

The signal in Equation (4) is still continuous in the t dimension, which can be represented as below after range compression

$$\begin{array}{l}r\left(f,i\right)=ℱ\left(s_b(t,i)\right)=T{\displaystyle \sum _{m=1}^k\mathrm{sinc}\{T\pi \left(f-f_m\right)\}\cdot }\exp \{j\pi i\sin {\theta }_m\}\cdot \exp \left\{j{\phi }_m\right\}\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}i\in (1,w)\cup (2w+1,3w)\cup (4w+1,5w)\end{array}$$

(8)

For a distributed radar system, $s_b(t)$ in Equation (8) can only complete the range compression if the FFT is used because y is discontinuous in the i dimension. Next, we tried to complete the azimuth compression. In matrix form, the relationship between the imaging result u and range compression result r can be rewritten as

$$\mathbf{r}=\mathbf{H}\mathbf{u}+\mathsf{\eta }$$

(9)

where $\mathbf{r}\in {ℂ}^{2nw\times 1}$ is the vector of the range compression and $\mathbf{u}\in {ℂ}^{2nw\times 1}$ is the imaging result. $\mathsf{\eta }\in {ℂ}^{2nw\times 1}$ denotes Gaussian noise with variance $\beta$ and $\mathbf{H}\in {ℂ}^{nw\times 2nw}$ is the discontinuous Fourier dictionary matrix related to i, which has the form below

$$\begin{array}{l}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{0.17em} \mathbf{H}={\left({\mathbf{H}}_1,{\mathbf{H}}_2,\dots ,{\mathbf{H}}_n\right)}^T\hspace{1em}\\ {\mathbf{H}}_i=\left[\begin{array}{ccccc}1& 1& 1& \cdots & 1\\ 1& e^{j\frac{(\mu +1)\pi }{nw}}& e^{j2\frac{(\mu +1)\pi }{nw}}& \cdots & e^{j(2nw-1)\frac{(\mu +1)\pi }{nw}}\\ 1& e^{j\frac{(\mu +2)\pi }{nw}}& e^{j2\frac{(\mu +2)\pi }{nw}}& \cdots & e^{j(2nw-1)\frac{(\mu +2)\pi }{nw}}\\ ⋮& ⋮& ⋮& \ddots & ⋮\\ 1& e^{j\frac{(\mu +w-1)\pi }{nw}}& e^{j2\frac{(\mu +w-1)\pi }{nw}}& \cdots & e^{j(2nw-1)\frac{(\mu +w-1)\pi }{nw}}\end{array}\right]\hspace{1em}\end{array}$$

(10)

where $\mu =2(\mathrm{i}-1)w$, and n is the number of radars.

In this context, the imaging problem can be converted into the problem of estimating u given r and H. For radar applications, the scene of interest is usually predominantly sparse. Due to the sparsity of u, the crux of the problem is to find a value for $\widehat{\mathbf{u}}$ that satisfies Equation (9) and has as few non-zero elements as possible. By introducing the $l_0$ norm, $\widehat{\mathbf{u}}$ can be estimated via the following optimization problem

$$\widehat{\mathbf{u}}=\arg \underset{\mathbf{u}}{\min }\left|\right|\mathbf{r}-\mathbf{D}\mathbf{u}|{|}_2^2+\lambda |\left|\mathbf{u}\right|{|}_0$$

(11)

where $\lambda$ is the parameter to control the sparsity, and $\left|\right|\mathbf{u}|{|}_0$ defines the number of non-zero elements in u.

2.2. Sparse Bayesian Learning Algorithm

Equation (9) is known as compressed sensing (CS) [27]. The problem to be solved with CS is that there exists a vector x and observation matrix H in advance, and we need to obtain u as accurately as possible. In this study, we needed to obtain the imaging vector u from the range compression vector r. Let M and N equal the number of rows and columns of H. Given the fact that M < N, Equation (9) is an ill-posed problem that has infinite solutions.

Sparse Bayesian learning has been widely used to solve CS reconstruction problems [38,39,40]. It assumes that r obeys a Gaussian likelihood $\mathcal{C}\mathcal{N}(r|\mathbf{u},\beta )$, and $\beta$ is the variance in $\mathsf{\eta }$, so the problem is estimating the maximum likelihood of u and β. To avoid overfitting, a hierarchical Gaussian prior $\mathcal{C}\mathcal{N}(\mathbf{u}|\mathbf{0},{\mathsf{\alpha }}^{-1})$ is applied to u, where $\mathsf{\alpha }$ is a vector of N hyperparameters corresponding to u. Next, let $\mathsf{\alpha }$ and $\beta$ have Gamma densities $\mathcal{G}({\alpha }_i|a,b)$ and $\mathcal{G}(\beta |c,d)$. Through a series of transformations based on Bayesian theory, the solution $\mathrm{p}(\mathbf{u}|\mathbf{r},\mathsf{\alpha },\beta )$ can be obtained. The steps of the SBL algorithm are shown in Algorithm 1.

Algorithm 1 Sparse Bayesian Learning (SBL algorithm)

1:   Input r, H, a, b, c and d. We set a = b =c = d = ${10}^{-6}$.       Let M (N) equal the number of rows (columns) of H. 2:   Initialize parameter $\mathsf{\Lambda },\mathbf{Q},\mathsf{\Sigma }$. We set $\mathsf{\Lambda }=\mathsf{{\rm I}}$ (${\alpha }_i=1$), and Q and $\mathsf{\Sigma }$ are determined according to Equations (12) and (13). 3:  Set a threshold $\epsilon$ to determine if $\widehat{\mathbf{u}}$ satisfies Equation (9). 4:  while $\left|\right|\mathbf{r}-\mathbf{H}\mathbf{u}|{|}_2^2>\epsilon$ do 5:        for i = 1: N do 6:            ${\gamma }_i=1-{\alpha }_i{\mathsf{\Sigma }}_{i,i}$ 7:            ${\alpha }_i=\frac{{\gamma }_i+a}{u_i^2+b}$ 8:        end 9:        $\beta =\frac{\left|\right|\mathbf{y}-\mathbf{D}\mathbf{u}|{|}^2+d}{M-{\displaystyle \sum _{i=1}^M{\gamma }_i+c}}$ 10:        $\mathsf{\Lambda }=diag(1/\mathsf{\alpha })$$\mathrm{where}\mathsf{\alpha }={({\alpha }_0,{\alpha }_1,\dots ,{\alpha }_{N-1})}^T$ 11:        $\begin{array}{c}\mathbf{Q}=\mathbf{I}+\beta \mathbf{H}\mathsf{\Lambda }{\mathbf{H}}^H\hfill \end{array}$    (12) 12:        $\begin{array}{c}\mathsf{\Sigma }=\mathsf{\Lambda }-\beta \mathsf{\Lambda }{\mathbf{H}}^H{\mathbf{Q}}^{-1}\mathbf{H}\mathsf{\Lambda }\hfill \end{array}$ (13) 13:        $\begin{array}{c}\widehat{\mathbf{u}}=\beta \mathsf{\Lambda }{\mathbf{H}}^H{\mathbf{Q}}^{-1}\mathbf{r}\hfill \end{array}$                (14) 14:  end 15:  Return $\widehat{\mathbf{u}}$

From Algorithm 1, it is obvious that SBL needs a significant amount of computation owing to the iterative process. In particular, steps 12–14 account for almost all the computation and involve the inverse of the matrix Q.

3. LC-SBL Fast SBL Algorithm

Thanks to the Fourier matrix H, many of the multiplications of the matrices can be quickly figured out using the FFT process. Focusing on steps 12–14, we established a new decomposition of ${\mathbf{Q}}^{-1}$ and developed LC-SBL algorithm.

In this section, we first give the equation for fast calculation of Q using the FFT. Next, we introduce the 2D Levinson–Durbin (L-D) algorithm to solve the linear system $\mathbf{Q}x={\mathbf{E}}_0$, where ${\mathbf{E}}_i$ is a block vector for which the (i + 1)th element is the identity matrix I, while the rest of the elements are 0. Then, we reconstruct the inverse of Q by using x and s, where $\mathbf{s}=\mathbf{x}\cdot x_0^{-1}$. Finally, the Vara algorithm and Wxper algorithm are employed to solve Equations (13) and (14) quickly.

3.1. Calculation Method for Matrix Q and 2D L-D Algorithm

Let $\mathbf{Q}=I+\beta \mathbf{F}$, where $\mathbf{F}=\mathbf{H}\mathsf{\Lambda }{\mathbf{H}}^H$. F is a Hermitian TBT matrix owing to the Fourier dictionary H, which has the following form:

$$\mathbf{F}=\left[\begin{array}{cccc}{\mathrm{F}}_0& {\mathrm{F}}_1^H& \cdots & {\mathrm{F}}_{n-1}^H\\ {\mathrm{F}}_1& {\mathrm{F}}_0& \ddots & ⋮\\ ⋮& \ddots & \ddots & {\mathrm{F}}_1^H\\ {\mathrm{F}}_{n-1}& \cdots & {\mathrm{F}}_1& {\mathrm{F}}_0\end{array}\right]$$

(15)

Each submatrix is a w × w Toeplitz matrix

$${\mathrm{F}}_i=\left[\begin{array}{cccc}{r}_0& r_{-1}& \cdots & r_{1-w}\\ r_1& r_0& \ddots & ⋮\\ ⋮& \ddots & \ddots & r_{-1}\\ r_{w-1}& \cdots & r_1& r_0\end{array}\right]$$

(16)

$r_c$ in ${\mathbf{F}}_i$ can be computed as

$$r_c={\displaystyle \sum _{k=0}^{N-1}{\alpha }_k^{-1}\exp \left\{-j\frac{2\pi k}{N}(c+i\cdot w)\right\}}$$

(17)

where ${\alpha }_k$ is the diagonal element of $\mathsf{\Lambda }$, $\mathsf{\alpha }={({\alpha }_0,{\alpha }_1,\dots ,{\alpha }_{N-1})}^T$. From Equation (20), we can see that $r_c$ can be quickly computed by using the N-points FFT with respect to the vector ${\mathsf{\alpha }}^{-1}$, so the matrix Q can be obtained directly through the FFT process. In addition, the main diagonal elements of Q are real numbers.

Let

$$\mathbf{x}={\left(x_0,x_1,\cdots ,x_{n-1}\right)}^{\prime }, \hspace{1em}\overline{\mathbf{x}}={\left({\overline{x}}_0,{\overline{x}}_1,\cdots ,{\overline{x}}_{n-1}\right)}^{\prime }$$

(18)

$$\mathbf{y}={\left(y_0,y_1,\dots ,y_{n-1}\right)}^{\prime },\hspace{1em}\overline{\mathbf{y}}={\left({\overline{y}}_0,{\overline{y}}_1,\dots ,{\overline{y}}_{n-1}\right)}^{\prime }$$

(19)

satisfy these equations

$$\hspace{0.17em} \mathbf{Q}\mathbf{x}={\mathbf{E}}_0\hspace{0.17em}\hspace{1em}\hspace{0.17em} {\overline{\mathbf{x}}}^{\prime }\mathbf{Q}={\mathbf{E}}_{n-1}^{\prime }$$

(20)

$$\mathbf{Q}\mathbf{y}={\mathbf{E}}_{n-1}\hspace{1em}{\overline{\mathbf{y}}}^{\prime }\mathbf{Q}={\mathbf{E}}_0^{\prime }$$

(21)

Then, x, $\overline{\mathbf{x}}$, y and $\overline{\mathbf{y}}$ have the following relationships

$${\overline{\mathbf{x}}}^{\prime }={\tilde{\mathbf{x}}}^{ T}$$

(22)

$$\mathbf{y}=conj\left(\tilde{\mathbf{x}}\right)$$

(23)

$${\overline{\mathbf{y}}}^{\prime }={\mathbf{x}}^H$$

(24)

It is easy to conform Equations (25)–(27) by using the equation ${\mathbf{Q}}^{-1}\mathbf{Q}=I$. We can obtain s and x efficiently using the 2D L-D algorithm. $\overline{\mathbf{x}}$, y and $\overline{\mathbf{y}}$ can easily be obtained from x, and we can use them to decompose ${\mathbf{Q}}^{-1}$. The 2D L-D algorithm is summarized in Algorithm 2.

Algorithm 2 Two-dimensional (2D) L-D algorithm

1:   Input ${\mathrm{Q}}_0$ and ${\mathrm{Q}}_1$ (similar to ${\mathrm{F}}_0$$\mathrm{and}{\mathrm{F}}_1$ in F). 2:   Initialize parameters W and Y as follows        $\mathbf{W}=-{\mathrm{Q}}_0^{-1}{\mathrm{Q}}_1$ $\mathbf{Y}={\mathrm{Q}}_0-{\mathrm{Q}}_1^H{\mathrm{Q}}_0^{-1}{\mathrm{Q}}_1$ 3:   for i = 2: w−1 do 4:   $\mathbf{L}={\tilde{\mathbf{W}}}^T{\left({\mathrm{Q}}_2,\dots {\mathrm{Q}}_i\right)}^{\prime }+{\mathrm{Q}}_{i+1}$ 5:   ${\mathbf{W}}^{new}=\left[\begin{array}{c}\mathbf{W}\\ \mathbf{O}\end{array}\right]-\left[\begin{array}{c}conj(\tilde{\mathbf{W}})\\ \mathbf{I}\end{array}\right]{\tilde{\mathbf{Y}}}^{-T}\mathbf{L}$ O is a zero matrix. 6:   ${\mathbf{Y}}^{new}=\mathbf{Y}-{\mathbf{L}}^H{\tilde{\mathbf{Y}}}^{-T}\mathbf{L}$ 7:   end 8:   $\mathbf{s}=\left[\begin{array}{c}\mathbf{I}\\ \mathbf{W}\end{array}\right]$ $x_0={\mathbf{Y}}^{-1}$ $\mathbf{x}=\mathbf{s}\cdot x_0$ 9:   Return s, x

3.2. Decomposition of Matrix $Q^{-1}$

The 2D L-D algorithm can be used to efficiently calculate x and s, which means that $\overline{\mathbf{x}}$, y and $\overline{\mathbf{y}}$ can also be obtained from x through Equations (25)–(27) without extra computation. Next, we use these block vectors to reconstruct the inverse of Q.

Let Z be the block cyclic lower shift matrix defined by

$$\mathbf{Z}=\left[\begin{array}{ccccc}\mathbf{0}& \cdots & \cdots & \mathbf{0}& \mathbf{I}\\ \mathbf{I}& \mathbf{0}& \ddots & \ddots & \mathbf{0}\\ \mathbf{0}& \mathbf{I}& \ddots & \ddots & ⋮\\ ⋮& \ddots & \ddots & \ddots & ⋮\\ \mathbf{0}& \cdots & \mathbf{0}& \mathbf{I}& \mathbf{0}\end{array}\right]$$

(25)

The cyclic displacement of $\mathbf{Q}$ [41] is given as

$$\nabla (\mathbf{Q})=\mathbf{Q}-\mathbf{Z}\mathbf{Q}{\mathbf{Z}}^T$$

(26)

Since Q is a TBT matrix with a special structure, we can easily determine that

$$\begin{array}{l}\nabla (\mathbf{Q})=\left[\begin{array}{cccc}\mathbf{0}& {\mathrm{Q}}_1^H-{\mathrm{Q}}_{n-1}& \cdots & {\mathrm{Q}}_n^H-{\mathrm{Q}}_1\\ {\mathrm{Q}}_1-{\mathrm{Q}}_n^H& \mathbf{0}& \cdots & \mathbf{0}\\ ⋮& ⋮& \ddots & ⋮\\ {\mathrm{Q}}_n-{\mathrm{Q}}_1^H& \mathbf{0}& \cdots & \mathbf{0}\end{array}\right]\\ \hspace{1em}\hspace{1em}={\mathbf{E}}_{\mathbf{0}}{\left[\begin{array}{c}-{\mathrm{Q}}_n\\ {\mathrm{Q}}_1^H-{\mathrm{Q}}_{n-1}\\ ⋮\\ {\mathrm{Q}}_{n-1}^H-{\mathrm{Q}}_1\end{array}\right]}^{\prime }+\left[\begin{array}{c}{\mathrm{Q}}_n\\ {\mathrm{Q}}_1-{\mathrm{Q}}_{n-1}^H\\ ⋮\\ {\mathrm{Q}}_{n-1}-{\mathrm{Q}}_1^H\end{array}\right]{\mathbf{E}}_{\mathbf{0}}^{\prime }\end{array}$$

(27)

${\mathrm{Q}}_n$ could be an arbitrary w × w matrix. It is easy to observe that $\nabla (\mathbf{Q})=0$ if Q is a block circulant matrix. There exists a common relationship between the displacement of Q and the displacement of ${\mathbf{Q}}^{-1}$, and the connection is as follows

$$\nabla ({\mathbf{Q}}^{-1})=-{\mathbf{Q}}^{-1}\nabla (\mathbf{Q})\mathbf{Z}{\mathbf{Q}}^{-1}{\mathbf{Z}}^T$$

(28)

Moreover, Equation (31) and the cyclic displacement allow us to calculate the inverse of Q. We obtain the following theorem [42].

THEOREM: If the cyclic displacement of Q can be rewritten as the sum

$$\nabla (\mathbf{Q})={\displaystyle \sum _{i=1}^{\alpha }{\mathbf{g}}_i\cdot {\mathbf{h}}_i^{\prime }}$$

(29)

where ${\mathbf{g}}_i$,${\mathbf{h}}_i$$(i\hspace{0.17em}=1, \hspace{0.17em}2\hspace{0.17em},\hspace{0.17em}\dots \hspace{0.17em},\hspace{0.17em}\alpha )$ are given block vectors, the following formula exists

$$\mathbf{Q}={\mathbf{C}}_{lr}+{\displaystyle \sum _{i=1}^{\alpha }L({\mathbf{g}}_i)\cdot \mathrm{C}{({\mathbf{h}}_i)}^{\prime }}$$

(30)

Let $\mathbf{z}={(z_0,z_1,\dots ,z_{n-1})}^{\prime }$. The matrix L(z) is a block lower triangular matrix, the form of which is defined by

$$\mathrm{L}(\mathbf{z})=\left[\begin{array}{cccc}{z}_0& 0& 0& 0\\ z_1& z_0& 0& 0\\ ⋮& \ddots & \ddots & 0\\ z_{n-1}& \cdots & z_1& z_0\end{array}\right]$$

(31)

C(z) is a circulant matrix with the form

$$\mathrm{C}(\mathbf{z})=\left[\begin{array}{cccc}{z}_0& z_{n-1}& \cdots & z_1\\ z_1& z_0& \ddots & ⋮\\ ⋮& \ddots & z_0& z_{n-1}\\ z_{n-1}& \cdots & z_1& z_0\end{array}\right]$$

(32)

${\mathbf{C}}_{lr}$ is a circulant matrix that has the same last row as Q.

Let

$$\mathsf{\mu }={({\mu }_0,{\mu }_1,\dots ,{\mu }_{n-1})}^{\prime }\hspace{1em}\overline{\mathsf{\mu }}={({\overline{\mu }}_0,{\overline{\mu }}_1,\dots ,{\overline{\mu }}_{n-1})}^{\prime }$$

(33)

satisfy the following equations

$$\mathbf{Q}\mathsf{\mu }={\left({\mathrm{Q}}_n,{\mathrm{Q}}_1-{\mathrm{Q}}_{n-1}^H,\dots ,{\mathrm{Q}}_{n-1}-{\mathrm{Q}}_1^H\right)}^{\prime }$$

(34)

$${\overline{\mathsf{\mu }}}^{\prime }\mathbf{Q}=\left({\mathrm{Q}}_{n-1}-{\mathrm{Q}}_1^H,\dots ,{\mathrm{Q}}_1-{\mathrm{Q}}_{n-1}^H,{\mathrm{Q}}_n\right)$$

(35)

With the help of Equation (31),$\nabla ({\mathbf{Q}}^{-1})$ can be rewritten as

$$\begin{array}{l}\nabla ({\mathbf{Q}}^{-1})=-{\mathbf{Q}}^{-1}{\mathbf{E}}_0{\left[\begin{array}{c}-{\mathrm{Q}}_n\\ {\mathrm{Q}}_1^H-{\mathrm{Q}}_{n-1}\\ ⋮\\ {\mathrm{Q}}_{n-1}^H-{\mathrm{Q}}_1\end{array}\right]}^{\prime }\mathbf{Z}{\mathbf{Q}}^{-1}{\mathbf{Z}}^T-{\mathbf{Q}}^{-1}\left[\begin{array}{c}{\mathrm{Q}}_n\\ {\mathrm{Q}}_1-{\mathrm{Q}}_{n-1}^H\\ ⋮\\ {\mathrm{Q}}_{n-1}-{\mathrm{Q}}_1^H\end{array}\right]\cdot {\mathbf{E}}_0^T\mathbf{Z}{\mathbf{Q}}^{-1}{\mathbf{Z}}^T\\ =-\mathbf{x}\cdot (-{\mathrm{Q}}_n,{\mathrm{Q}}_1^H-{\mathrm{Q}}_{n-1},\dots ,{\mathrm{Q}}_{n-1}^H-{\mathrm{Q}}_1)\cdot \mathbf{Z}{\mathbf{Q}}^{-1}{\mathbf{Z}}^T-\mathsf{\mu }\cdot (\mathbf{I},0,\dots ,0)\cdot \mathbf{Z}{\mathbf{Q}}^{-1}{\mathbf{Z}}^T\\ =\mathbf{x}\cdot ({\mathrm{Q}}_{n-1}-{\mathrm{Q}}_1^H,\dots ,{\mathrm{Q}}_1-{\mathrm{Q}}_{n-1}^H,{\mathrm{Q}}_n)\cdot {\mathbf{Q}}^{-1}{\mathbf{Z}}^T-\mathsf{\mu }\cdot (0,\dots ,0,\mathbf{I})\cdot {\mathbf{Q}}^{-1}{\mathbf{Z}}^T\\ =\left[\begin{array}{c}{x}_0\\ x_1\\ ⋮\\ x_{n-1}\end{array}\right]({\overline{\mu }}_{n-1},{\overline{\mu }}_0,\dots ,{\overline{\mu }}_{n-2})-\left[\begin{array}{c}{\mu }_0\\ {\mu }_1\\ ⋮\\ {\mu }_{n-1}\end{array}\right]({\overline{x}}_{n-1},{\overline{x}}_0,\dots ,{\overline{x}}_{n-2})\end{array}$$

(36)

According to the previous theorem, the matrix ${\mathbf{Q}}^{-1}$ can be represented as

$${\mathbf{Q}}^{-1}={\mathbf{C}}_{lr}+\mathrm{L}(\mathbf{x})\mathrm{C}(\mathbf{J}\overline{\mathsf{\mu }})-\mathrm{L}(\mathsf{\mu })\mathrm{C}(\mathbf{J} \overline{\mathbf{x}})$$

(37)

where

$$\mathbf{J}=\left[\begin{array}{cccc}& & & \mathbf{I}\\ & & \mathbf{I}& \\ & ⋰& & \\ \mathbf{I}& & & \end{array}\right]$$

(38)

Given the fact that

$${\mathbf{C}}_{lr}=\mathrm{C}{(\mathbf{Z} \overline{\mathbf{x}})}^{\prime }=\mathrm{C}(\mathbf{J} \overline{\mathbf{x}})$$

(39)

Equation (44) can be simplified as

$$\begin{array}{l}{\mathbf{Q}}^{-1}=\mathrm{L}(\mathbf{x})C(\mathbf{J}\overline{\mathsf{\mu }})-(\mathrm{L}(\mathsf{\mu })-\mathbf{I})\mathrm{C}(\mathbf{J} \overline{\mathbf{x}})\\ \hspace{1em}\hspace{0.17em}\hspace{0.17em} =\mathrm{L}(\mathbf{x})\mathrm{C}(\mathbf{J}\overline{\mathsf{\mu }})-\mathrm{L}(\mathsf{\mu }-{\mathbf{E}}_0)\mathrm{C}(\mathbf{J} \overline{\mathbf{x}})\end{array}$$

(40)

Solving Equations (41) and (42) directly requires significant amounts of computation. A fast method to obtain $\mathsf{\mu }$ and $\overline{\mathsf{\mu }}$ exists.

We obtain

$$\begin{array}{l}\mathbf{Q}(\mathbf{Z}\mathbf{y})y_{n-1}^{-1}=\mathbf{Q} {(y_{n-1},0,\dots ,0)}^{\prime }\cdot y_{n-1}^{-1}+\mathbf{Q} {(0,y_0,\dots ,y_{n-2})}^{\prime }\cdot y_{n-1}^{-1}\\ \hspace{1em}={\left({\displaystyle \sum _{i=1}^{n-1}{\mathrm{Q}}_i^H{y}_{i-1}{y}_{n-1}^{-1}},-{\mathrm{Q}}_{n-1}^H,\dots ,-{\mathrm{Q}}_1^H\right)}^{\prime }+{({\mathrm{Q}}_0,{\mathrm{Q}}_1,\dots ,{\mathrm{Q}}_{n-1})}^{\prime }\\ \hspace{1em}={({\mathrm{Q}}_n,{\mathrm{Q}}_1-{\mathrm{Q}}_{n-1}^H,\dots {\mathrm{Q}}_{n-1}-{\mathrm{Q}}_1^H)}^{\prime }\end{array}$$

(41)

$$\begin{array}{l}{\overline{\mathbf{y}}}_0^{-1}({\overline{\mathbf{y}}}^{\prime }\mathbf{Z})\mathbf{Q}={\overline{y}}_0^{-1}(0,0,\dots ,{\overline{y}}_0)\mathbf{Q}+{\overline{y}}_0^{-1}({\overline{y}}_1,\dots ,{\overline{y}}_{n-1},0)\mathbf{Q}\\ \hspace{1em}\hspace{1em}\hspace{1em}=\left(-{\mathrm{Q}}_1^H,\cdots ,-{\mathrm{Q}}_{n-1}^H,{\overline{y}}_0^{-1}{\displaystyle \sum _{i=1}^{n-1}{\overline{y}}_i{\mathrm{Q}}_{n-i}^H}\right)+({\mathrm{Q}}_{n-1},\dots ,{\mathrm{Q}}_1,{\mathrm{Q}}_0)\\ \hspace{1em}\hspace{1em}\hspace{1em}=\left({\mathrm{Q}}_{n-1}-{\mathrm{Q}}_1^H,\dots ,{\mathrm{Q}}_1-{\mathrm{Q}}_{n-1}^H,{\mathrm{Q}}_n\right)\end{array}$$

(42)

Reference [42] provides the following property for any invertible TB matrix

$${\displaystyle \sum _{i=1}^{n-1}{\mathrm{Q}}_i^H{y}_{i-1}{y}_{n-1}^{-1}}={\overline{y}}_0^{-1}{\displaystyle \sum _{i=1}^{n-1}{\overline{y}}_i{\mathrm{Q}}_{n-i}^H}$$

(43)

As a consequence, $\mathsf{\mu }=\mathbf{Z}\mathbf{y}{y}_{n-1}^{-1}$ and ${\mathsf{\mu }}^{\prime }=y_0^{-1}{\overline{\mathbf{y}}}^{\prime }\mathbf{Z}$ are the solutions for Equations (41) and (42). Furthermore, we obtain

$$\begin{array}{ll}{\overline{\mathsf{\mu }}}^{\prime }& ={\overline{y}}_0^{-1}{\overline{\mathbf{y}}}^{\prime }\mathbf{Z}={\overline{y}}_0^{-1}({\overline{y}}_1,\dots ,{\overline{y}}_{n-1},{\overline{y}}_0)\\ & ={(x_0^H)}^{-1}(x_1^H,\dots ,x_{n-1}^H,x_0^H)\\ & ={\mathbf{s}}^H\cdot \mathbf{Z}\end{array}$$

(44)

and

$$\begin{array}{ll}\mathsf{\mu }& =\mathbf{Z}\mathbf{y}{y}_{n-1}^{-1}={(y_{n-1},y_0,\dots ,y_{n-2})}^{\prime }{y}_{n-1}^{-1}\\ & =conj{({\tilde{x}}_0,{\tilde{x}}_{n-1},\dots ,{\tilde{x}}_1)}^{\prime }\cdot conj{({\tilde{x}}_0)}^{-1}\\ & =conj\left[\begin{array}{c}\mathbf{I}\\ {\tilde{s}}_{n-1}\\ ⋮\\ {\tilde{s}}_1\end{array}\right]\end{array}$$

(45)

Therefore, we can reconstruct the inverse of Q by using x, $\overline{\mathbf{x}}$, y and $\overline{\mathbf{y}}$. Due to the relationships between them, Equation (47) can be written as

$$\begin{array}{l}{\mathbf{Q}}^{-1}=\left[\begin{array}{cccc}{x}_0& & & \\ x_1& x_0& & \\ ⋮& \ddots & \ddots & \\ x_{n-1}& \dots & x_1& x_0\end{array}\right]\left[\begin{array}{cccc}\mathrm{I}& s_1^H& \cdots & s_{n-1}^H\\ s_{n-1}^H& \mathrm{I}& \ddots & ⋮\\ ⋮& \ddots & \ddots & s_1^H\\ s_1^H& \cdots & s_{n-1}^H& \mathrm{I}\end{array}\right]-\\ conj\left[\begin{array}{cccc}0& & & \\ {\tilde{s}}_{n-1}& 0& & \\ ⋮& \ddots & \ddots & \\ {\tilde{s}}_2& \cdots & {\tilde{s}}_{n-1}& 0\end{array}\right]\left[\begin{array}{cccc}{\tilde{x}}_0^T& {\tilde{x}}_{n-1}^T& \cdots & {\tilde{x}}_1^T\\ {\tilde{x}}_1^T& {\tilde{x}}_0^T& \ddots & ⋮\\ ⋮& \ddots & \ddots & {\tilde{x}}_{n-1}^T\\ {\tilde{x}}_{n-1}^T& \cdots & {\tilde{x}}_1^T& {\tilde{x}}_0^T\end{array}\right]\end{array}$$

(46)

3.3. Wxpet Algorithm

From the previous section, we know that the inverse of Q can be decomposed into the sum of products of block lower triangular Toeplitz matrices and block circulant matrices. With this method, we can calculate Equations (13) and (14) by using linear convolution and circulant convolution. Q is a n × n block matrix, the elements of which are a w × w square matrix. As $\mathrm{w}\gg n$, directly bringing Q into Equations (13) and (14) will lead to numerous FFT steps, and every vector operated with the FFT will be short. To reduce FFT operations and accelerate the algorithm, we can define the permutation matrix S by

$$\mathbf{S}=({\mathbf{e}}_0,{\mathbf{e}}_n,\dots ,{\mathbf{e}}_{(w-1)n},{\mathbf{e}}_1,\dots ,{\mathbf{e}}_{(w-1)n+1},\dots ,{\mathbf{e}}_{wn-1})$$

(47)

where ${\mathbf{e}}_{\mathrm{i}}$ is the (i +1)th column of I. Let

$$\mathbf{R}=\mathbf{S}{\mathbf{Q}}^{-1}{\mathbf{S}}^T$$

(48)

Here, R is a w × w Hermitian TBT matrix, the elements of which are n × n square matrices. We can use the previous method to decompose it. According to Equation (53), R can be rewritten briefly as

$$\mathbf{R}={\mathbf{L}}_1{\mathbf{C}}_1-{\mathbf{L}}_2{\mathbf{C}}_2$$

(49)

Then,${\mathbf{Q}}^{-1}$ can be expressed as

$${\mathbf{Q}}^{-1}={\mathbf{S}}^T{\mathbf{L}}_1{\mathbf{C}}_1\mathbf{S}-{\mathbf{S}}^T{\mathbf{L}}_2{\mathbf{C}}_2\mathbf{S}$$

(50)

We use ${\mathbf{a}}_i$ to denote the ith column of ${\mathbf{S}}^T\mathbf{L}$ and ${\mathbf{b}}_i$ to denote the ith row of $\mathbf{C}\mathbf{S}$

$${\mathbf{S}}^T\mathbf{L}=({\mathbf{a}}_0,{\mathbf{a}}_1,\dots {\mathbf{a}}_{mk-1})$$

(51)

$$\mathbf{C}\mathbf{S}={({\mathbf{b}}_0,{\mathbf{b}}_1,\dots ,{\mathbf{b}}_{mk-1})}^T$$

(52)

The matrix ${\mathbf{S}}^T\mathbf{L}\mathbf{C}\mathbf{S}$ can be rewritten as

$$\begin{array}{ll}{\mathbf{S}}^T\mathbf{L}\mathbf{C}\mathbf{S}& ={\displaystyle \sum _{i=0}^{n-1}({\mathbf{a}}_i,{\mathbf{a}}_{\mathrm{n}+i},\dots ,{\mathbf{a}}_{(k-1)n+i}){({\mathbf{b}}_i,{\mathbf{b}}_{n+i},\dots ,{\mathbf{b}}_{(k-1)n+i})}^T}\\ & ={\displaystyle \sum _{i=0}^{n-1}\left[\begin{array}{c}{\mathbf{T}}_{i1}\\ {\mathbf{T}}_{i2}\\ ⋮\\ {\mathbf{T}}_{in}\end{array}\right]}({\mathbf{M}}_{i1},{\mathbf{M}}_{i1},\dots ,{\mathbf{M}}_{in})\end{array}$$

(53)

where T represents the w × w lower triangular Toeplitz matrix

$$\mathbf{T}=\left[\begin{array}{cccc}{t}_1& & & \\ t_2& t_1& & \\ ⋮& \ddots & \ddots & \\ t_{\mathrm{w}}& \cdots & t_2& t_1\end{array}\right]$$

(54)

and M represents the w × w circulant matrix

$$\mathbf{M}=\left[\begin{array}{cccc}{\mathrm{m}}_1& m_{\mathrm{w}}& \cdots & m_2\\ m_2& m_1& \ddots & ⋮\\ ⋮& \ddots & \ddots & m_w\\ m_w& \cdots & m_2& m_1\end{array}\right]$$

(55)

T and M can be obtained from x and s using Equations (57)–(60). Consequently, the multiplication of the inverse of Q by r in Equation (14) can be computed as several T matrices multiplied by r and several M matrices multiplied by r

$$\begin{array}{l}\mathbf{v}={\mathbf{Q}}^{-1}\mathbf{r}={\displaystyle \sum _{a=0}^{n-1}\left[\begin{array}{c}{\mathbf{T}}_{a1}\\ {\mathbf{T}}_{a2}\\ ⋮\\ {\mathbf{T}}_{an}\end{array}\right]}({\mathbf{M}}_{a1},{\mathbf{M}}_{a1},\dots ,{\mathbf{M}}_{an})\mathbf{r}-{\displaystyle \sum _{b=0}^{n-1}\left[\begin{array}{c}{\mathbf{T}}_{b1}\\ {\mathbf{T}}_{b2}\\ ⋮\\ {\mathbf{T}}_{bn}\end{array}\right]}({\mathbf{M}}_{b1},{\mathbf{M}}_{b1},\dots ,{\mathbf{M}}_{bn})\mathbf{r}\\ \hspace{0.17em}\hspace{0.17em} ={\displaystyle \sum _{a=0}^{n-1}\left[\begin{array}{c}{\mathbf{T}}_{a1}\\ {\mathbf{T}}_{a2}\\ ⋮\\ {\mathbf{T}}_{an}\end{array}\right]}\left({\displaystyle \sum _{j=1}^n{\mathbf{M}}_{aj}{r}_j}\right)-{\displaystyle \sum _{b=0}^{n-1}\left[\begin{array}{c}{\mathbf{T}}_{b1}\\ {\mathbf{T}}_{b2}\\ ⋮\\ {\mathbf{T}}_{bn}\end{array}\right]}\left({\displaystyle \sum _{j=1}^n{\mathbf{M}}_{bj}{r}_j}\right)\end{array}$$

(56)

Linear convolution and circular convolution can be used to calculate v, which only requires the first rows of T and M. Additionally, due to the Fourier dictionary H, we can divide v into n equal segments and fill each segment with m zeros to perform the IFFT and obtain $\widehat{\mathbf{u}}$

$$\begin{array}{l}\widehat{\mathbf{u}}=\beta \mathsf{\Lambda }\left({\mathbf{H}}_1^H,{\mathbf{H}}_2^H,\cdots ,{\mathbf{H}}_n^H\right)\mathbf{v}\\ \hspace{1em}=\beta \mathsf{\Lambda }N\cdot ℐℱ{({\mathbf{v}}_1,\mathbf{0},{\mathbf{v}}_2,\mathbf{0},\dots ,{\mathbf{v}}_n,\mathbf{0})}^T\end{array}$$

(57)

The sizes of the zero vector and $v_i(i\in (1,n))$ depend on $d_t$ and $d_s$. The fast algorithm for the calculation of $\widehat{\mathbf{u}}$ is summarized below (Algorithm 3).

Algorithm 3 Compute $\widehat{\mathbf{u}}$ (Wxpet algorithm)

1:   Input x, s and r. 2:   for t = 1:2;          $\mathbf{b}\leftarrow {\mathbf{C}}_t{}^{(1)}$(the first block column of ${\mathbf{C}}_t$),$\mathbf{a}\leftarrow {\mathbf{L}}_t{}^{(1)}$ (the first block column of ${\mathbf{L}}_t$) 3:        for i = 1:n 4:              for j = 1:n                      ${\mathbf{r}}_j\leftarrow \mathbf{r}(k\ast (j-1):jk)$ ${\mathbf{b}}_j\leftarrow \mathbf{b} (i:n:end,j)$                      ${\mathbf{w}}_j={\mathbf{r}}_j\mathsf{\Theta }\hspace{0.17em}{\mathbf{b}}_j$ $\mathbf{w}=\mathbf{w}+{\mathbf{w}}_j$ 5:              end 6:              for j = 1:n                      ${\mathbf{a}}_j\leftarrow \mathbf{a}(j:n:end,i)$                      ${\mathsf{\alpha }}_j={\mathbf{a}}_j\otimes \hspace{0.17em}{\mathbf{w}}_j$ discard redundant convolution results                      ${\mathsf{\alpha }}_t={({\mathsf{\alpha }}_t,{\mathsf{\alpha }}_j(1:k))}^T$ 7:              end 8:        ${\mathsf{\alpha }}_t+={\mathsf{\alpha }}_t$ 9:        end 10:   end 11:   $\mathbf{v}={\mathsf{\alpha }}_1+{\mathsf{\alpha }}_2$ 12:   $\widehat{\mathbf{u}}=\beta \mathsf{\Lambda }N\cdot ℐℱ{({\mathbf{v}}_1,\mathbf{0},{\mathbf{v}}_2,\mathbf{0},\dots ,{\mathbf{v}}_n,\mathbf{0})}^T$ 13:   Return $\widehat{\mathbf{u}}$

3.4. Vara Algorithm

For a block matrix, we define the sum of matrices on the diagonal of ${\mathbf{Q}}^{-1}$ as $\mathrm{K}{({\mathbf{Q}}^{-1})}_c$, and the index c is 0 when the diagonal is the main diagonal and it is plus 1 when the diagonal moves up one step. The rule also applies to simple matrices. In this way, the number $\mathrm{K}{({\mathbf{Q}}^{-1})}_{c,d}$ denotes the sum of the dth diagonal of the matrix $\mathrm{K}{({\mathbf{Q}}^{-1})}_c$.

As ${\mathbf{Q}}^{-1}$ is a Hermitian matrix, we have

$${\mathrm{K}}_{c,d}=conj({\mathrm{K}}_{-c,-d})$$

(58)

The sum of all diagonals in the upper half of matrix ${\mathbf{Q}}^{-1}$ can be computed by

$$\begin{array}{l}{\mathrm{K}}_{\ast ,d}={{\displaystyle \sum _{a=1}^{n-1}\left(\mathrm{K}{({\mathbf{T}}_{a1}{\mathbf{M}}_{an})}_d,\dots ,\mathrm{K}{({\displaystyle \sum _{p=1}^{n-1}{\mathbf{T}}_{ap}{\mathbf{M}}_{a(p+1)}})}_d,\mathrm{K}{({\displaystyle \sum _{p=1}^{n-1}{\mathbf{T}}_{ap}{\mathbf{M}}_{a(p+1)}})}_d\right)}}^T+\\ {{\displaystyle \sum _{b=1}^{n-1}\left(\mathrm{K}{({\mathbf{T}}_{b1}{\mathbf{M}}_{bn})}_d,\dots ,\mathrm{K}{({\displaystyle \sum _{p=1}^{n-1}{\mathbf{T}}_{bp}{\mathbf{M}}_{b(p+1)}})}_d,\mathrm{K}{({\displaystyle \sum _{p=1}^{n-1}{\mathbf{T}}_{ap}{\mathbf{M}}_{a(p+1)}})}_d\right)}}^{\mathbf{T}}\hspace{1em}d\in (1-w,k-w)\end{array}$$

(59)

For matrix $\mathbf{T}\cdot \mathbf{M}$, the sums of all diagonals can be computed as

$$\mathrm{K}(\mathbf{T}\mathbf{M})={(\mathrm{T}({\mathbf{z}}_1){\mathbf{v}}_1,\mathbf{J}\mathrm{T}({\mathbf{z}}_2){\mathbf{v}}_2+\mathrm{T}({\mathbf{z}}_3){\mathbf{v}}_1\cdot \mathbf{q})}^T$$

(60)

where $\mathrm{T}({\mathbf{z}}_i)$ has the form shown in Equation (61), the first column of which is ${\mathbf{z}}_{ i }$, and q is a coefficient vector determined by k

$$\hspace{0.17em}\hspace{0.17em}{\mathbf{z}}_1={\left(t_1,t_2,\cdots ,t_w\right)}^T\hspace{1em}\hspace{1em}\hspace{1em}\hspace{0.17em}\hspace{1em}\hspace{1em}{\mathbf{v}}_1={\left(x_2,\cdots ,(w-1)x_w,w{x}_1\right)}^T$$

(61)

$${\mathbf{z}}_2={\left(t_w,2t_{w-1},\cdots ,(w-1)t_2\right)}^T \hspace{1em}\hspace{0.17em}{\mathbf{v}}_2={\left(x_1,x_w,\cdots ,x_3\right)}^T$$

(62)

$${\mathbf{z}}_3={\left(t_1,t_2,\cdots ,t_{w-1}\right)}^T\hspace{1em}\hspace{0.17em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{0.17em}\hspace{0.17em}{\mathbf{v}}_3={\left(x_2,x_3,\cdots ,x_w\right)}^T$$

(63)

$$\mathbf{q}={\left(w-1,w-2,\dots ,1\right)}^T$$

(64)

Let

$$\mathbf{h}=diag({\mathbf{H}}^H{\mathbf{Q}}^{-1}\mathbf{H})$$

(65)

We need to pad zero in K, and then we can get

$${\mathbf{c}}_1={({\mathrm{K}}_{0,0},{\mathrm{K}}_{0,1},\cdots ,{\mathrm{K}}_{0,\mathrm{n}-1},0,{\mathrm{K}}_{1,0},\cdots ,0,\cdots ,{\mathrm{K}}_{n-1,n-1})}^T$$

(66)

$${\mathbf{c}}_2=\mathit{conj}{(0,{\mathrm{K}}_{0,1},\cdots ,{\mathrm{K}}_{0,n-1},0,\cdots ,{\mathrm{K}}_{1,n-1},\cdots {\mathrm{K}}_{n-1,n-1})}^T$$

(67)

We just need to add one zero number in terms of $d_s$ = $d_t$, and then h can be quickly computed using the FFT and IFFT

$$\mathbf{h}=ℱ({\mathbf{c}}_1)+N\times ℐℱ({\mathbf{c}}_2)$$

(68)

The Vara algorithm is shown below (Algorithm 4). Diag(b, a) means the diagonal elements of the multiplication of circulant matrix T and Toeplitz matrix M should be calculated according to Equations (67)–(71), in which the first row of T is b and the first row of M is a.

Algorithm 4 Compute h (Vara Algorithm)

1:   Input x and s 2:      for t = 1:2          $\mathbf{b}\leftarrow {\mathbf{C}}_t{}^{(1)}$ the first block column of ${\mathbf{C}}_t$, $\mathbf{a}\leftarrow {\mathbf{L}}_t{}^{(1)}$ the first block column of ${\mathbf{L}}_t$ 3:            for i = 1:n 4:                  for j = 1:n                         $a_{ji}\leftarrow \mathbf{a}(j:m:end,i)$ $b_{ji}\leftarrow \mathbf{b}(i:m:end,j)$ 5:                  end 6:                  for s = 1:n 7:                        ${\mathbf{W}}_i$ += [zeros(2m−1, t−1) Diag $(b_{ji},a_{si})$] 8:                  end 9:                        ${\mathbf{K}}_t+={\mathbf{W}}_i$ 10:                 end 11:                        $\mathbf{K}={\mathbf{K}}_1-{\mathbf{K}}_2$ 12:   We obtain ${\mathbf{c}}_1$ and ${\mathbf{c}}_2$ by adding zeros in vector K 13:   $\mathbf{h}=ℱ({\mathbf{c}}_1)+N\ast ℐℱ({\mathbf{c}}_2)$ 14:   Return h

We focus on multiplication only because addition and subtraction operations are much faster than multiplication operations. We can calculate h and v directly by multiplying the matrices, requiring $2O(n^3{t}^3)+2O(n^2{t}^2)$ computations. In comparison, the 2D L-D algorithm, Wxpet algorithm and Vara algorithm need about $O(n^3{t}^2)/3+O(n^2{t}^2)/2$$+O(n{t}^3)$ computations in total. In the next section, we describe the simulation run with these methods, and we compare the number of calculations required and the computational time.

4. Simulation and Measured-Data Processing Results

We used simulation and measured data to demonstrate the advantages of LC-SBL in improving imaging resolution and saving computation time. In addition, the orthogonal matching pursuit (OMP) [26] algorithm, the fast IAA (FIAA) [24] algorithm and Student’s T prior-based efficient SBL (S-ESBL) [34] algorithm were emulated for comparison.

We defined normalized root mean square errors (nRMSE) to evaluate the reconstruction performance of the algorithms in simulation. The nRMSE is computed as

$$\mathrm{nRMSE}=\frac{\left|\right|\widehat{\mathbf{u}}-\mathbf{u}\left|\right|}{\left|\right|\mathbf{u}\left|\right|}$$

(69)

We simulated a radar system with 516 elements composed of six radars; each radar had 86 array elements and the spacing between adjacent elements was half the wavelength. We made the center of the 516 elements the origin. To obtain the gapped data, we manually abandoned the second, fourth and sixth radars’ data after collecting it. The radar parameters were as follows: the central frequency was 77 GHz, the bandwidth was 3 GHz, the sampling rate was 16,000 kHz, the number of ADC samples was 800 and the signal-to-noise ratio (SNR) was set to 15 dB.

4.1. Simulation of Angular Resolution of Several Algorithms

In the first simulation, all observed data were obtained from 4 sets of points 30 m away. The intervals between groups were 0.5 m, and the gapped distances between points in each group were 0.05 m, 0.1 m, 0.15 m and 0.2 m, respectively. For the SBL, S-ESBL and LCSBL algorithms, the hyperparameters were set as a = b = c = d = ${10}^{-6}$, and γ and β were initialized as $\gamma =1$ and $\beta =1$. The maximum number of iterations was set to 1000. For the SBL and LC-SBL algorithms, the convergence δ was set to 10⁻³. In particular, δ was set to 10⁻⁴ in the S-ESBL algorithms due to the slow convergence speed. Generally speaking, the FIAA algorithm converges to the best case when the number of iterations reaches 15 [24], so we set 15 iterations here.

Figure 3 shows three sets of simulation results obtained with conventional SBL to assess angular resolution. Figure 3b–g shows the images of the 86 elements, 516 elements and distributed radar system. In addition, the one-dimensional azimuth profiles of the 16 targets are displayed. Due to the small amount data for single radars, the resolution of Figure 3b is very poor, and the points of the fourth group cannot even be perfectly distinguished. In contrast, the data imaging effect of the 516 elements was excellent. It can also be seen that the points of each group were successfully identified, and even the points with a distance of 0.05 m can be distinguished accurately. In addition, the distributed radar system also identified every point successfully. With the conventional SBL algorithm, the angular resolution of the distributed radar system was comparable to that of a radar with 516 array elements.

In this experiment, we wanted to evaluate the imaging performance of the distributed radar system with the FIAA, OMP and S-ESBL algorithms. Figure 4 demonstrates the imaging results and the azimuth profiles. From Figure 4, we can see that the FIAA and S-ESBL algorithms achieved high accuracy. They clearly identified targets in both the second and third groups, but only three targets were displayed in the first group. This shows that their resolutions at 30 m were not as good as that of a 516 array element radar. Moreover, although the imaging result for the OMP algorithm had no side lobes, it was very unstable, and the target image with low sparsity was blurred. The LC-SBL algorithm is a fast version of the SBL algorithm, and the result shown in Figure 4g is as clear as that in Figure 3f.

4.2. nRMSE Values and Running Times of the Algorithms

To determine the nRMSE and running time of each algorithm, we performed 1000 Monte Carlo experiments with different signal-to-noise ratios.

Table 1. The average computational times of the OMP, FIAA, SBL, S-ESBL and LC-SBL algorithms with a 10 dB SNR.

	LC-SBL	SBL	FIAA	OMP	S-ESBL	FFD-SBL
Time (s)	0.431	11.319	1.327	0.019	0.910	0.442

and Figure 5 illustrate the results. In this experiment, we randomly selected 30 random frequencies from 516 azimuth profiles in the same range profile and transformed the frequency domain data to the time domain to obtain the data collected by the distributed radar system.

From Table 1 and Figure 5, we can see that the OMP algorithm was extremely fast, but its nRMSE value remained around 0.8 and hardly changed with different signal-to-noise ratios. Among the other algorithms, the LC-SBL algorithm not only had the lowest nRMSE but also required the shortest time. The FFD-SBL algorithm [43] is also a kind of fast SBL algorithm based on G-S decomposition of a TBT matrix without any approximations. It had the same nRMSE as the LC-SBL algorithm but took a little longer.

4.3. Measured Data Imaging

Finally, we used measured data from two scenes to demonstrate the performance of the LC-SBL algorithm in improving the angular resolution, with scene one being composed of five corner reflectors and scene two of town normal bikes. An inevitable millimeter-level error was expected in the placement of the radar, which would have caused a certain phase error for millimeter waves. Therefore, we applied the self-focus algorithm [44,45,46] after imaging. The distributed radar system consisted of three MMWAVCAS-RF-EVM development kits produced by Texas Instruments. The width of each radar was $d_t$ and the distance between radars was $d_s$; thus, we had $d_t=d_s$= 17.2 cm. Each radar board had 9 transmitting antennas and 12 receiving antennas, forming 86 apertures in the azimuth direction. The size of each aperture equaled 0.5$\lambda$. We calculated the angular resolution as 1.35° using the formula

$$\vartheta =\frac{\lambda }{D}$$

(70)

where $\lambda$ is the wavelength and D is the length of the whole aperture array.

In the experiment, we set the start frequency as 77 GHz, the bandwidth as 1.5 GHz, the sampling frequency as 16,000 KHz and the number of sampling points for one chirp as 512. Figure 6a shows the distributed radar system, and Figure 6b shows the measurement targets for scene one. We measured five corner reflections, imaging at a distance of 35.5 m from the radar system, with an interval between pairs of corner reflectors of 14 cm. The resolution of the single radars at 35.5 m was about 80 cm. The virtual aperture array length of the distributed radar system was six times that of a single radar, so it could theoretically achieve a resolution of 13.4 cm. Figure 6c,e show the images of the whole scene and the targets imaged by the single-radar system. Figure 6e,f show the distributed radar system imaging results for the five corner reflections obtained with the FIAA, OMP, S-ESBL and LC-SBL algorithms. From Figure 6, we can see that the single-radar SBL could not distinguish corner reflectors at all. In the distributed radar system, the OMP algorithms could only image four targets, and the imaging results for the FIAA and S-ESBL algorithms were somewhat blurred; they could not clearly show the five targets. In contrast, the LC-SBL algorithm successfully identified every corner reflector in the same scene.

The next scene depicted two bikes next to each other, as shown in Figure 7b. The parameters of the radar were set the same as scene 1. From Figure 7, we can see that the imaging result of the single-radar SBL was very poor, and the number and shapes of the bicycles cannot be distinguished because of the low resolution. For the distributed radar system, the results displayed by the OMP algorithm were obviously not as good as the other algorithms. The FIAA algorithm displayed the image as clearly as the S-ESBL algorithm, and the performance of the LC-SBL algorithm was the best, the silhouettes of the two bikes being perfectly distinguished.

5. Conclusions

In this study, a novel fast SBL algorithm, referred to as the LC-SBL algorithm, was developed for distributed FMCW MIMO radar system imaging to obtain high angular resolution. To address the problem of high computational complexity in SBL, we proposed a new decomposition method for the inverse of a TBT matrix. With this method, several important parameters can be quickly calculated using the FFT. The performance of the LC-SBL algorithm was much better than that of the OMP algorithm. Compared to the FIAA and S-ESBL algorithms, LC-SBL not only performed better but also required less time. Moreover, the LC-SBL algorithm was more efficient than the FFD-SBL algorithm.