The Analysis of Using Spatial Smoothing for DOA Estimation of Coherent Signals in Sparse Arrays

Xu, Haiyun; Cui, Weijia; Du, Yuxi; Mei, Fengtong; Ba, Bin

doi:https://doi.org/10.1155/2021/4447562

International Journal of Antennas and Propagation

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Abstract Introduction Conclusions Appendix Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2021 | Article ID 4447562 | https://doi.org/10.1155/2021/4447562

The Analysis of Using Spatial Smoothing for DOA Estimation of Coherent Signals in Sparse Arrays

Haiyun Xu,¹Weijia Cui,¹Yuxi Du,¹Fengtong Mei,¹and Bin Ba¹

Academic Editor: Francisco Falcone

Received14 May 2021

Revised25 Jul 2021

Accepted28 Aug 2021

Published17 Sept 2021

Abstract

When there is coexistence of uncorrelated and coherent signals in sparse arrays, the conventional algorithms for direction-of-arrival (DOA) estimation using difference coarray fail. In order to solve the problems, this paper analyzes the feasibility of using spatial smoothing in sparse arrays. Firstly, we summarize the two types of sparse arrays, one consisting of identical sparse subarrays and the other consisting of several uniform linear subarrays. Then, we give the feasibility analysis and the processes of applying spatial smoothing. At last, we discuss the performance of the number of detectable coherent signals in different sparse arrays. Numerical experiments prove the conclusions proposed by the paper.

1. Introduction

Direction of arrival (DOA) is the one of the key parameters of the wireless positioning technique. It has been widely used in the fifth generation communication systems, military early warning, radar monitoring, sonar targets positioning, and so on [1, 2]. Traditionally, the super-resolution DOA estimation methods, such as subspace methods, mainly use uniform nonsparse arrays like uniform linear arrays (ULAs). In recent years, many experts and scholars focus on nonuniform sparse arrays, which can provide larger array aperture with the same number of sensors. The typical sparse arrays are minimum redundancy arrays (MRAs) [3], coprime linear arrays (CLAs) [4], and nested linear arrays (NLAs) [5]. Through transforming the sparse array to a virtual ULA based on difference coarray, spatial smoothing multiple signal classification (SS-MUSIC) [6] and direct augmentation approach (DAA) [7] are proposed to solve DOAs. Moreover, compressed sensing can be directly applied to difference coarray to estimate DOAs [8–10].

Based on the DOA estimation methods mentioned above, many researchers proposed the improved array design methods. One design thought is based on the model of CLA. Coprime array with compressed interelement spacing (CACIS), coprime array with displaced subarrays (CADiS) [11, 12], coprime array with multiperiod subarrays (CAMpS) [13], shifted coprime array (SCA) [14], generalized nested array (GNA) [15], and novel sparse arrays with two uniform arrays (NSA-U2) [16] were proposed, where they all consist of two ULA-subarrays and have larger array aperture than that of CLA. CACIS and CADiS show that setting one subarray with a smaller interelement spacing can have larger aperture of virtual ULA. CAMpS demonstrates that in order to expand the aperture of virtual ULA, only one subarray can have a compressed interelement spacing. SCA reveals that the displacement between two subarrays is the main factor to the aperture of virtual ULA. Although GNA has the same degree of freedom as NLA, it owns a sparser array structure. NSA-U2 presents the solution to have the maximum degree of freedom for sparse arrays with two uniform arrays. In order to further improve the array aperture, the sparse arrays with multiple ULA-subarrays are proposed. Super-nested arrays (SNAs) [17, 18], the augmented nested array (ANA) [19], and the maximum interelement spacing constraint (MISC) [20] divide the dense subarray of CLA into several sparse ULA-subarrays. Another design thought is combining several identical sparse arrays, such as nested MRA (NMRA) [21, 22], generalized nested subarray (GNSA) [23], and displaced multistage cascade subarrays (MSC-DiSA) [24]. The subarrays can be any sparse array and the displacement between the subarrays depends on sensors’ location of another chosen sparse array. Although the aperture is smaller, the latter design method is less complex than that of the former method.

Unfortunately, the structure design of sparse arrays and DOA estimation algorithms are both based on the assumption that the impinging signals are uncorrelated to each other. But there exist coherent signals in real environment, such as in multipath channel. Many decoherence methods, such as spatial smoothing [25], forward/backward spatial smoothing (FBSS) [26], and Toeplitz reconstruction [27], are only applicable to the uniform structure arrays. Hence, the DOA estimation of coherent signals in sparse arrays has been a focus of interest. There have been some algorithms only for CLAs. Signal separation and Toeplitz reconstruction (SSTR) [28] and spatial smoothing using fourth-order cumulant (SS-FOC) [29] are all utilized in the uniform sparse subarrays and combined with common peak finding [30] to resolve the real values. Then, a method [31] combining spatial smoothing and matrix completion theory was proposed, which was applied to the data of physical sensors, but it also has strict restriction about array structure.

In order to solve coherent signals, we aim to give the process of using spatial smoothing in sparse arrays. We first summarize that the sparse array can be seen as two types: one is using several identical sparse arrays and the other consists of several ULA-subarrays. We then, respectively, apply the spatial smoothing and analyze the feasibility of decoherence methods. Next, we discuss the performance of the existing sparse arrays about the number of detectable coherent signals. At last, the simulation experiments are presented to prove the effectiveness of the proposed method.

The rest of this paper is organized as follows. Section 2 presents the model of received data including coherent and uncorrelated signals. Section 3 introduces the spatial smoothing process in sparse arrays. Section 4 gives the performance analysis and simulation experiments. Section 5 summarizes the paper. Throughout the paper, we make use of the notations shown in Table 1.

2. Signal Model

Suppose that there are far-field narrow-band signals impinging on a sparse array with sensors. Define the unit interelement spacing as , where is the wavelength of signals, and a integer set corresponding to the sensors location is given by (generally assuming ). Assume that there are P coherent signal groups, where the th group has signals. The coherent signal coming from is corresponding to the th multipath propagation of with power . The signals within each group are coherent to each other and uncorrelated to those in different groups. The total number of coherent signals is . In addition, the remaining signals, coming from with the power (), are uncorrelated to each other. The number of those signals is . Thus, the received signals iswhere the manifold matrix is denoted asand the steering vector can be given byand is the complex fading coefficient of the th coherent signal in the th group. The signal data vector iswhere , and is the number of snapshots. The noise vector is usually a Gaussian random variable with zero mean and variance .

From (1), the covariance matrix is denoted aswhere can be written as a block-diagonal matrix given by

Because , and . Thus, the conventional methods for DOA estimation fail. Spatial smoothing can let to satisfy the requirement of subspace methods, but it requires that the array can be divided into several same subarrays. Thus, in the next section, we will discuss the decomposition of sparse arrays.

3. Spatial Smoothing in Sparse Arrays

In this section, we try to divide any sparse array into several subarrays and summarize two situations. The first is that the sparse array is composed of several same sparse subarrays. The second is that the sparse array can be divided into several uniform sparse linear arrays, where although the structure of ULA is different, the process of spatial smoothing is applied to each ULA, and the ULAs with different interelement spacings are applied to remove the ambiguous values.

3.1. Sparse Arrays with Identical Sparse Subarrays

There exists a type of sparse arrays, which are made up of several same arrays. The sparse subarray can be CLAs, NLAs, MRAs, and so on. If the cascade number of array is , we have and each stage contains physical sensors, whose location set can be . MSC-DiSA and GNSA, which are two typical arrays, have different structure rules, but they are the same when the subarray is MRA with four sensors and the displacement is the function to the location of MRA with three sensors. The structures are shown in Figure 1(a). Thus, the received data in (1) can be rewritten aswhere

(a)

(b)

(c)

Each subarray meets the feature of rotational invariance, which can be expressed bywherewhere is displacement between the th subarray and first subarray and . Thus, the spatial smoothing covariance matrix is defined aswhere . In order to let , we give the following theorem.

Theorem 1. If , , and , .

Proof. See Appendix A.

Considering the requirement for setting the displacement between the subarrays [15], we have known that . So, when and , we can estimate all DOAs by applying subspace methods [32] to .

3.2. Sparse Arrays with ULA-Subarrays

The sparse arrays, which consist of ULA-subarrays, are capable to use spatial smoothing algorithm to solve coherent signals. We first give the general model of sensors location denoted aswhere is the displacement between the th ULA-subarray and the first ULA-subarray, is the sensor number of th ULA-subarray, and is the interelement spacing of th ULA-subarray.

3.2.1. Sparse Arrays with Two ULA-Subarrays

Figure 1(b) shows four arrays with 12 sensors using two ULAs. Thus, two subarrays, respectively, have and sensors. The location of subarrays can be denoted as , , where are coprime integers, and generally . The parameters of arrays are defined in Table 2. Then, the received data in (1) can be rewritten aswhere

Then, the spatial smoothing covariance matrix with size of th subarray is given bywhere . is given bywhere . Also, we define , wherewhere , and

Moreover,where and .

Based on the model above, the following statement gives the applicability of using spatial smoothing in this type of sparse array. Firstly, we introduce Theorem 2 in [30] and propose Theorem 3.

Theorem 2. Let and denote the interelement spacing of two ULAs, respectively. Let and denote the estimated DOA values of the th signal of two subarrays, respectively, where each set has multiple ambiguous values and one real value. If , .

Theorem 3. Let the directions of sources have a random distribution. When and , .

Proof. See Appendix B.

When and , and the requirement of using subspace methods is satisfied. Also, . Then, we know that are coprime integers, so we use the subspace method to and obtain the estimated values of the th signals, defined as and . From Theorem 2, common peak finding [30] tells that the real value .

3.2.2. Sparse Arrays with ULA-Subarrays

When it comes to the sparse arrays with ULAs-subarrays, we need to point out that the sensor location of existing sparse arrays satisfies (13). We just give the examples about ANAI-1, MISC, ANAII-1, and MRA with in Table 3.

Based on the analysis in sparse arrays with two subarrays, we just need to select two subarrays, whose interelement spacings are coprime integers, to solve the DOA estimation. Besides the requirement for setting the inter-element spacing, another two criteria for subarray parameters selection are the number of sensors that are as large as possible and the subarray apertures that are as large as possible. The former is to estimate as many sources as possible, and the latter is to ensure the accuracy of estimation. Thus, we can obtain the receiving data and of two chosen subarrays from . Then, we can use (16) to calculate the spatial smoothing covariance matrix and apply common peak finding [30] to find the DOAs of coherent signals.

4. Performance Analysis and Simulation Experiments

4.1. Performance Analysis

We discuss of each sparse array, that is, the maximum number of detectable coherent signals in one group. For GNSA, , so when is the maximum integer no more than , achieving maximum is equal to . For any sparse array consisting of ULAs, we arrange the number of subarray sensors in descending order and have the set , where . Based on the theorem that the paper proposes, we can have and . Thus, when and , . Due to subarrays, achieves maximum in theory. So, the more the number of ULA-subarrays, the smaller the value of .

Next, we compare the value of of GNSA with that of CLA, NLA, ANAI-1, MISC, ANAII-1, and MRA. We vary from 8 to 20 with 2 intervals, and the results are shown in Figure 2. becomes bigger with the increase of . The CLA and NLA with two ULA-subarrays have the biggest value of . But the ANAII-1 and MRA with 5 or more ULA-subarrays have the smallest value, and only when , they can use spatial smoothing to estimate coherent signals. Hence, with a fixed number of sensors, the conclusion that the less subarrays can have the bigger is corrected.

Moreover, based on the papers, where the sparse arrays are proposed, the maximum array aperture of consecutive virtual ULAs defined as meets that , and is generally between and . When they apply spatial smoothing, the main factor affecting the accuracy is the aperture of smoothing array, defined as . Then, we assume that and and compare and --ratio of different sparse arrays. Figure 3 presents that the more number of ULA-subarrays can achieve bigger , but for spatial smoothing, it is opposite that the less number of ULA-subarrays can achieve bigger .

(a)

(b)

4.2. Simulation Experiments

We use root mean square error (RMSE) to quantify the accuracy of DOA estimation, given bywhere is the Monte Carlo number, is the number of target signals, and is the DOA of the estimated source by the Monte Carlo experiment. The simulation conditions are shown in Table 4.

Simulation 1. feasibility of estimate coherent signals in different sparse arrays.
In first simulation, we show the feasibility of estimate coherent signals with maximum number in one group. Thus, assume that , , and of each sparse arrays can be seen in Figure 2. Set and . The estimation values of 100 times experiments are shown in Figure 4. The figure demonstrates that the sparse arrays can use spatial smoothing to estimate DOAs of coherent signals. Moreover, in the condition of and low SNR, all estimated values are still close to the real values, which means a favorable performance.

(a)

(b)

(c)

(d)

(e)

(f)

Simulation 2. RMSE performance comparison of different SNRs.
In this simulation, we compare the RMSEs of different sparse arrays and ULAs, when there are both uncorrelated and coherent signals. We set , where , and . Because ANAII-1 and MRA cannot estimate 3 signals after spatial smoothing, we only compare the other arrays, and the results are shown in Figure 5. Obviously, the RMSEs decrease with the increase of SNR. GNSA has the highest RMSE due to the smallest . ULA has the second smallest as 19, so the RMSE of it is just lower than that of GNSA. The other three sparse arrays have the close RMSE due to their close values of .

Simulation 3. RMSE performance comparison of different number of snapshots.
Similar to the simulation 2, we present the RMSEs of different sparse arrays under different snapshots with . From Figure 6, the RMSEs decrease with the increase of snapshot numbers, but when , the downtrend of RMSEs become slow. Other conclusions are the same as those in simulation 2.

Simulation 4. RMSE performance comparison using FBSS.
In this simulation, we use FBSS to replace spatial smoothing and do simulations 2 and 3 again. FBSS can be seen as an improvement method of spatial smoothing, where can be 1.5 times that of spatial smoothing [26]. Thus, we do not need to present the analysis in Part 1 of this section about using FBSS. But we should note that FBSS is not applicable to GNSA because it needs that the subarrays have uniform structure, where we still use spatial smoothing in GNSA. The results are shown in Figures 7 and 8. Compared with the results in simulations 2 and 3, FBSS has improved the accuracy. Thus, we can use FBSS in sparse arrays to find more coherent signals and obtain higher accuracy.

5. Conclusions

In this paper, the DOA estimation methods using spatial smoothing for coherent signals in sparse arrays are proposed. We divide the sparse arrays into two parts. The first type consists of several identical sparse arrays. The second type can be decomposed of several ULA-subarrays. In view of subarrays, spatial smoothing can be applied in sparse arrays. Based on the analysis of the maximum number of detectable coherent signals in one group, the sparse arrays with less subarrays are capable to estimate more signals and own bigger smoothing array aperture. Also, the simulation experiments prove that CLA and NLA have better performance than other arrays.

Appendix

A

Proof of Theorem 1

The matrix can be rewritten aswhere is a block-diagonal matrix and

Define , and (20) can be denoted as

If , . Thus, we rewrite as

Based on (25), it can also be simplified to give , wherewith denoting the Hermitian square root of .

Because , we need to prove that . We take column permutations to , which cannot change the rank of a matrix, and havewhere represents element in the th row and th column of and is expressed as

When each row of has at least one nonzero element and vectors are linearly independent, . Because every signal has the positive energy, every row of cannot be all zeros. The matrix can be seen as the manifold matrix of a sparse array and are denoted as the location of sensors.

Hence, we introduce the theorem in [32, 33], which tells that when , the array manifold is invertible. Invertibility means that if , then . Then, we can obtain the conclusion that if , the rank of is . So, we prove that if and , , and for any , then . Hence, when , .

B

Proof of Theorem 3

In this situation, is also a block-diagonal matrix, given byand is changed to

Similar to the Proof of Theorem 1, , wherewith denoting the Hermitian square root of . We just need to prove that because . Considering the analysis in Appendix A, we should prove that each row of has at least one nonzero element and vectors are linearly independent, where

It is easy to obtain that each row of has at least one nonzero element. is a Vandermonde matrix and are denoted as the location of sensors. Because , is not invertible. For example, if , , and , then and , while if , . In general, assuming that , iffor any integer , and . Considering that the directions of sources generally have a random distribution in real environment, the parameters satisfy (33) with probability one. In other words, if and , and in this situation. Hence, when , .

Data Availability

The data used in this article are provided by our simulations and the data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This study was supported by the National Natural Science Foundation of China (grant no. 61401513).

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Copyright © 2021 Haiyun Xu et al. 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.

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