A Sliding Window Data Compression Method for Spatial-Time DOA Estimation

Abstract

This paper presents a sliding window data compression method for spatial-time direction-of-arrival (DOA) estimation using coprime array. The signal model is firstly formulated by jointly using the temporal and spatial information of the impinging sources. Then, a sliding window data compression processing is performed on the array output matrix to realize fast calculation of time average function, and the computational burden has been reduced accordingly. Based on the concept of sum and difference co-array (SDCA), the vectorized conjugate augmented MUSIC is adopted, with which more sources than twice of the physical sensors can be resolved. Additionally, the sparse array robustness to sensor failure has been evaluated by introducing the concept of essential sensors. The theoretical analysis and numerical simulations are provided to confirm the effectiveness performance of the proposed method.

This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

Direction-of-arrival (DOA) estimation has been a crucial topic in various practical applications, such as radar, navigation, and wireless communication [1–4], where the antenna arrays are utilized for collecting the spatial sampling of impinging electromagnetic waves. In comparison to the typical uniform linear array (ULA) [5, 6], the emerging sparse arrays [7–9] have remarkable advantages in terms of sensor layout, degrees of freedom (DOF), and virtual array aperture, with which more sources than the number of the physical sensors can be resolved.

Nested array (NA) [10] and coprime array (CPA) [11] are the two most typical sparse array configurations, which have closed form expressions for array geometry and achievable DOF as compared with the minimum redundancy array (MRA) and the minimum hole array (MHA). NA is composed of two concatenated subarrays with increasing element spacing, which is capable of providing $\mathcal{O}{L}^2$ DOF with $\mathcal{O}L$ physical sensors, but the existing small interelement spacing in the subarray of NA would cause severe mutual coupling. As for the CPA, the mutual coupling can be alleviated for the configuration is constructed by a pair of coprime ULAs, which can offer $\mathcal{O}MN$ DOF with $\mathcal{O}M+N$ sensors, but has holes in the virtual co-array. Recently, lots of modified versions have been further developed, such as super NA [12], enhanced NA [13], augmented NA [14], generalized CPA [15], and thinned CPA [16], to increase the consecutive DOF and reduce the mutual coupling. The sparse arrays mentioned above construct virtual co-array from the view of difference co-array (DCA) and realize the multitarget DOA estimation by exploiting vector MUSIC method or compressed sensing (CS) approach; nevertheless, the number of resolvable sources cannot exceed twice of the physical aperture.

Motivated by the sum co-array (SCA) originating from active sensing [17–19], the concept of sum and difference co-array (SDCA) has provided a new perspective for DOA estimation [20], where the vectorized conjugate augmented MUSIC (VCAM) is presented by jointly using the temporal and spatial information of the impinging sources. The CPA configuration based on SDCA is firstly proposed for spatial-time DOA estimation [21]. Following this, a modified NA configuration named as sum-diff NA (SdNA) is proposed in [22] for the purpose of resolving more sources. In [23], combining with the VCAM method, the unfold CPA configuration on the basis of SDCA is developed to provide more DOF and larger array aperture, and the DOA estimation accuracy is improved accordingly. However, the above spatial-time DOA estimation methods based on SDCA involve high-dimensional data processing of multiple pseudo snapshots, which has high computational load. Additionally, another limitation of the above methods is that the case of sensor failure [24–26] always occurring in the actual direction-finding system has been ignored. The failure of some sensors or receiving channels may destroy the virtual array structure of sparse array and reduce the number of consecutive DOF and the effective array aperture, which results in the performance degradation.

To tackle these problems, a sliding window data compression method for spatial-time DOA estimation is proposed in this paper. The coprime array is adopted from the perspective of SDCA, which can provide more DOF and larger effective array aperture. Then, a sliding window data compression processing is performed on array output matrix to realize fast calculation of time average function. Afterwards, the vectorized conjugate augmented MUSIC is adopted by jointly using the temporal and spatial information of the impinging sources. Moreover, the sparse array robustness to sensor failure has been evaluated by introducing the concept of essential sensors.

Notations: vectors and matrices are denoted by lowercase and uppercase bold-face letters, respectively. ${\cdot }^T$, ${\cdot }^H$, and ${\cdot }^{\ast }$ and $\cdot$ denote transpose, conjugate transpose, conjugate, and the norm of the embraced matrix, respectively. ${\mathbf{0}}_{m\times n}$ represents an $m\times n$ null matrix and ${\mathbf{I}}_m$ represents an $m\times m$ identity matrix. The symbol $\otimes$ denotes the Kronecker product and $\odot$ denotes the Khatri–Rao product.

2. Problem Formulation

2.1. Sparse Array Configuration

Referring to [11], a prototype CPA is composed of two uniform linear subarrays: one is a N-element ULA with interspacing being Md (d denotes the unit interspacing) and the other is an M-element ULA with interspacing being Nd, where N and M are coprime integers satisfying $M<N$. Intuitively, an example of 8-element CPA with $M=4$ and $N=5$ is illustrated in Figure 1, where white circles and black circles, respectively, denote the sensor locations of subarray 1 and subarray 2 and shaded circles denote the overlapping sensor between subarray 1 and subarray 2. For the reason that the two uniform linear subarrays share the same antenna sensor placing at zero position, the CPA have $M+N-1$ physical sensors, whose locations are given by$$\begin{array}{}\text{(1)}& {\mathbf{\mathbb{P}}}_{CPA}=Mnd|n\in \mathbb{Z}N-1\cup Nmd|m\in \mathbb{Z}M-1,\end{array}$$where $\mathbb{Z}m$ denotes a set of positive integers ranging from 1 to m and the unit interspacing $d$ is generally set to be half of the wavelength.

[figure omitted; refer to PDF]

Since j varies from 1 to $L$, the conjugate augmented vector can be obtained as$$\begin{array}{}\text{(6)}& \mathbf{g}\tau ={{\mathbf{g}}_x^H-\tau \text{,\hspace{0.17em}}{\mathbf{g}}_x^T\tau }^T={{\mathbf{A}}^H,{\mathbf{A}}^T}^T{\mathbf{g}}_s\tau ,\end{array}$$where the time average vector $\mathbf{g}\tau$ and its mirrored version ${\mathbf{g}}_x-\tau$ are given as$$\begin{array}{}\text{(7)}& \begin{array}{c}{\mathbf{g}}_x\tau =\mathbf{A}{\mathbf{g}}_s\tau ={R_{11}^{\ast }\tau ,R_{12}^{\ast }\tau ,\dots ,R_{1L}^{\ast }\tau }^T,\\ {\mathbf{g}}_x-\tau =\mathbf{A}{\mathbf{g}}_s-\tau ={R_{11}^{\ast }-\tau ,R_{12}^{\ast }-\tau ,\dots ,R_{1L}^{\ast }-\tau }^T,\end{array}\end{array}$$with ${\mathbf{g}}_s\tau ={R_{s_1^{\ast }{s}_1}\tau ,R_{s_2^{\ast }{s}_2}\tau ,\dots ,R_{s_K^{\ast }{s}_K}\tau }^T$ and ${\mathbf{g}}_s-\tau ={R_{s_1^{\ast }{s}_1}-\tau ,R_{s_2^{\ast }{s}_2}-\tau ,\dots ,R_{s_K^{\ast }{s}_K}-\tau }^T$.

Then, for the $T_s$ pseudo snapshots, the pseudo data matrix can be constructed as$$\begin{array}{}\text{(8)}& \mathbf{G}={\mathbf{g}{n}_s{P}_s}_{n_s=1}^{T_s}={{\mathbf{A}}^H,{\mathbf{A}}^T}^T\mathbf{E}\mathbf{\Omega },\end{array}$$where $P_s$ is the pseudo sampling period satisfying Nyguist sampling principle, $\mathbf{E}$ is a diagonal matrix with main diagonal elements being ${E_k}^2$ and zeros elsewhere, and $\mathrm{\Omega }$ is a $K\times T_s$ matrix with the $k\text{,}n$th element being $e^{j{w}_{k}n{P}_s}$. By calculating the covariance matrix of $\mathbf{G}$ and then vectorizing it, we have$$\begin{array}{}\text{(9)}& {\mathbf{v}}_G={\overline{\mathbf{A}}}^H\odot {\overline{\mathbf{A}}}^T{\mathbf{q}}_s,\end{array}$$where $\overline{\mathbf{A}}={\mathbf{A}}^{\text{H}}\text{,\hspace{0.17em}}{\mathbf{A}}^{\text{T}}$ is the generated array manifold matrix and ${\mathbf{q}}_s$ can be seen as an equivalent impinging source vector $K\times 1$ with the kth element being ${E_k}^4$. Additionally, the $k\text{th}$ column of ${\overline{\mathbf{A}}}^H\odot {\overline{\mathbf{A}}}^T$ can be denoted as$$\begin{array}{}\text{(10)}& \tilde{\mathbf{a}}{\theta }_k=\begin{array}{c}\mathbf{a}{\theta }_k\otimes {\mathbf{a}}^{\ast }{\theta }_k\\ \mathbf{a}{\theta }_k\otimes \mathbf{a}{\theta }_k\\ {\mathbf{a}}^{\ast }{\theta }_k\otimes {\mathbf{a}}^{\ast }{\theta }_k\\ {\mathbf{a}}^{\ast }{\theta }_k\otimes \mathbf{a}{\theta }_k\end{array},\end{array}$$where the term $\tilde{\mathbf{a}}{\theta }_k$ in (10) behaves like a longer virtual steering vector that can provide more DOF for DOA estimation. More specifically, the union of $\mathbf{a}{\theta }_k\otimes {\mathbf{a}}^{\ast }{\theta }_k$ and ${\mathbf{a}}^{\ast }{\theta }_k\otimes \mathbf{a}{\theta }_k$, respectively, are corresponding to the DCA and its mirror version and the union of $\mathbf{a}{\theta }_k\otimes \mathbf{a}{\theta }_k$ and ${\mathbf{a}}^{\ast }{\theta }_k\otimes {\mathbf{a}}^{\ast }{\theta }_k$, respectively, are corresponding to the SCA and its mirror version. Then, the spatial smoothing MUSIC method or sparse construction techniques can be performed for DOA estimation. To conclude, the main steps of the proposed method are given in Table 1.

Table 1

Main steps of the proposed method.

(1) Calculate the time average function $R_{x_i^{\ast }{x}_j}\tau$ using (3)

(2) Perform sliding window data compression processing on xi(t) and xj(t) via (4) and rewrite (3) using (5)

(3) Construct the conjugate augmented vector g(τ)using (6)

(4) Construct the pseudodata matrix G using (8) and calculate its covariance matrix

(5) Vectorize the covariance matrix of G using (9)

(6) Perform spatial smoothing MUSIC method or sparse construction techniques on qs

4. Array Robustness Analysis

The array robustness to sensor failure directly affects the DOA estimation performance in the practical direction-finding system and the relevant analysis is discussed in detail in this section. For a sparse array with known array configuration, if the distribution of SDCA changes when one or more sensors are deleted from the physical array (PA), then these sensors are termed as essential sensors. Mathematically, for the sparse array $\mathcal{A}$ and the corresponding SDCA $\mathcal{S}$, when the lth sensor is deleted from $\mathcal{A}$, the remaining array becomes ${\mathcal{A}}_{-l}=\mathcal{A}\backslash l$ and the corresponding SDCA becomes ${\mathcal{S}}_{-l}$. If $\mathcal{S}\ne {\mathcal{S}}_{-l}$ holds, then the lth sensor is essential for the sparse array $\mathcal{A}$, but not vice versa.

An example of detecting essential sensor with ${\mathcal{A}}_1=\mathrm{0,1,2,3,4,6}$ is given in Figure 3, where the red circles and black circles, respectively, denote the locations of PA and SDCA. As can be seen from Figure 3, after removing the sensor 1, the PA becomes ${\mathcal{A}}_2=\mathrm{0,2,3,4,6}$; then, the corresponding SDCA has holes at $\pm 1$, which implies that sensor 1 is essential. On the contrary, after removing the sensor 2, the PA becomes ${\mathcal{A}}_3=\mathrm{0,1,3,4,6}$; then, the corresponding SDCA is the same as the original one; hence, the sensor 2 of PA is inessential.

[figure omitted; refer to PDF]

Then, the evaluation function of sparse array robustness is calculated as$$\begin{array}{}\text{(11)}& {ℱ}_k={{\rm E}}_k/\mathbb{A},\end{array}$$where ${\mathrm{{\rm E}}}_k$ is the number of essential sensors for sparse array $\mathbb{A}$ and $\mathbb{A}$ is the number of the whole sparse array $\mathbb{A}$. According to (10), the values of ${ℱ}_k$ ranges from 0 to 1. More specifically, if ${ℱ}_k⟶0$, then the sparse array has strong robustness, but the number of essential sensors is small; if ${ℱ}_k⟶1$, almost all the sensors in sparse array are essential, which is economic, but the array robustness is low. Therefore, several strategies, such as reducing the failure probability of essential sensors and introducing a certain number of inessential sensors, can be adopted to improve the robustness of sparse array and meanwhile ensure the economic benefits.

5. Numerical Simulations

In this section, numerical simulations are performed to evaluate the performance of the proposed method. Consider K = 20 narrowband sources uniformly distributed between $-{60}^{\circ }$ and ${60}^{\circ }$ and impinge on an 8-sensor CPA with SNR being 0 dB and pseudo snapshot number $T_s=400$, where the sensor locations of CPA is ${\mathbf{\mathbb{P}}}_{CPA}=\mathrm{0,4,5,8,10,12,15,16}$. To demonstrate the computational efficiency of the proposed method, the averaged CPU times of the proposed sliding window data compression (SWDC) and VCAM [21] versus snapshot number $T_P$ over 200 independent Monte Carlo trials are compared in Table 2, where the software used for implementation is MATLAB R2014a (version 8.3) and executed in PC Intel(R) Core(TM) i7-8550U processor with 8.0 GB RAM. It can be observed from Table 2 that the operation time of the proposed SWDC method is significantly reduced in comparison to VCAM, which is mainly attributed to reason that sliding window data compression can transform the antenna received matrix with a dimension of $L\times T_s{T}_P$ into an $L\times T_P+2T_s$ equivalent one, and the computational load is reduced accordingly.

Table 2

Averaged CPU times.

Time(sec)	Tp = 200	Tp = 300	Tp = 400	Tp = 500	Tp = 600
SWDC	0.0544	0.0666	0.0761	0.0852	0.0962
VCAM	0.0995	0.1368	0.1747	0.1950	0.1990

Then, array robustness to sensor failure is discussed in this simulation. For the 8-element CPA with ${\mathbf{\mathbb{P}}}_{CPA}=\mathrm{0,4,5,8,10,12,15,16}$, we model the sensor failure by deleting the sensors of CPA one by one; then, the remaining array location, holes of SDCA, and the consecutive DOF are listed in Table 3. As can be seen from Table 3 that if sensor failure occurs to sensor 0, the distribution of SDCA remains unchanged, which implies that sensor “0” is an inessential sensor for the 8-element CPA. Apart from sensor “0,” if sensor failure occurs to other sensors, the corresponding SDCA has more holes and less consecutive DOF. Thus, all the sensors in the 8-element CPA are essential, except for sensor “0.” Moreover, the importance of each essential sensor is different, e.g., the failure of sensor “5”, “10”, “12,” or “15” would cause more “holes” than the other essential sensors.

Table 3

The array robustness to sensor failure.

Sensor failure	Remaining array	Holes in SDCA	Consecutive DOF
Null	{0, 4, 5, 8, 10, 12, 15, 16}	{−29, 29}	57
“0”	{4, 5, 8, 10, 12, 15, 16}	{−29, 29}	57
“4”	{0, 5, 8, 10, 12, 15, 16}	{−29, −19, −14, 14, 19, 29}	27
“5”	{0, 4, 8, 10, 12, 15, 16}	{−29, −21, −17, −13, −9, 9, 13, 17, 21, 29}	17
“8”	{0, 4, 5, 10, 12, 15, 16}	{−29, −23, −18, −13, 13, 18, 23, 29}	25
“10”	{0, 4, 5, 8, 12, 15, 16}	{−29, −26, −25, −22, −18, −14, −6, −2, 2, 6, 14, 18, 22, 25, 26, 29}	9
“12”	{0, 4, 5, 8, 10, 15, 16}	{−29, −28, −27, −22, −17, 17, 22, 27, 28, 29}	33
“15”	{0, 4, 5, 8, 10, 12, 16}	{−29, −27, −25, −23, −19, 19, 23, 25, 27, 29}	37
“16”	{0, 4, 5, 8, 10, 12, 15}	{−29, −28, −26, −21, 21, 26, 28, 29}	41

The third simulation investigates the DOA estimation performance via MUSIC spectrum. All the conditions are the same as the first simulation except that the snapshot number $T_P$ is set to be 300. In addition, the searching step of MUSIC spectrum is set to be 0.5°. Figure 4 depicts the MUSIC spectrum of the proposed method, where the blue solid line represents the angle estimates and the red dotted line represents incident sources. As can be seen from Figure 4, the proposed method can resolve 20 impinging sources with only 8 sensors and the MUSIC spectrum has sharp and high power peaks in the vicinity of the true impinging sources.

[figure omitted; refer to PDF]

In the last simulation, the DOA estimation performance of the proposed SWDC, VCAM [21], and CPA-MUSIC [11] versus SNR and the number of snapshots are compared via 200 independent Monte Carlo trials, where the root mean square error (RMSE) is chosen for evaluating the DOA estimation performance:$$\begin{array}{}\text{(12)}& \text{RMSE}=\sqrt{\frac{1}{200K}{\displaystyle \sum _{i=1}^{200}{\displaystyle \sum _{k=1}^K{{\widehat{\theta }}_{k,i}-{\theta }_k}^2}}},\end{array}$$where ${\widehat{\theta }}_{k,i}$ is the estimate of ${\theta }_k$ for the ith Monte Carlo trial. The source distribution in this simulation is the same as that in the third simulation. The results from Figures 5 and 6 show that the DOA estimation accuracy of the proposed SWDC method is similar to that of VCAM, which is superior to the CPA-MUSIC. The reason is that the data information is not lost for the sliding window data compression processing, which implies that the proposed SWDC method and the VCAM method exploit the same data for DOA estimation, and the estimation accuracy is almost the same accordingly. By contrast, the CPA-MUSIC is performed based on DCA, where the available DOF and virtual array aperture are reduced.

[figure omitted; refer to PDF][figure omitted; refer to PDF]

6. Conclusions

In this paper, we have proposed a sliding window data compression method to reduce the computational burden of high-dimensional data processing for the spatial-time DOA estimation. By jointly using the temporal and spatial information of the impinging sources, the signal model is firstly formulated based on CPA from the perspective of SDCA. Then, a sliding window data compression processing is applied to the array output vector. Afterwards, we resort to the VCAM approach for DOA estimation. In addition, the sparse array robustness to sensor failure has been evaluated by introducing the concept of essential sensors. Simulation results have confirmed that the proposed method can resolve more sources than twice of physical sensors and has notable performance advantages in terms of computational load and DOA estimation accuracy.

Acknowledgments

This work was supported by the National Natural Science Foundation of China, under Grant no. 62101223, and Natural Science Foundation of the Jiangsu Higher Education Institutions of China, under Grant nos. 20KJB510027 and 20KJA510008.

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