# Simple Direction-of-Arrival Estimation under Nonuniform Noise Scenarios

Authors: Fenggang Sun; Jiefang Wu; Hao Fan; Lizhen Chen; Peng Lan
Journal: Wireless Communications and Mobile Computing
 (2022)

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## Abstract

The uniform white noise assumption is widely applied for most of the existing direction-of-arrival (DOA) estimation methods, which however may degrade the estimation performance. In practical scenarios, the nonuniform white noise model is more adequate. In this paper, we propose a simple DOA estimation method in the presence of spatially nonuniform white noise with unknown noise covariance matrix. The estimate of the unknown noise covariance matrix is firstly obtained by utilizing the shift property of identical subarrays. Then, the received signal covariance matrix is prewhitened by the estimated noise covariance, which can overcome the biased estimate of signal covariance. Finally, the performance improvement is verified by simulations in terms of resolution ability and estimation accuracy.


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## Body

## 1. Introduction

Direction-of-arrival (DOA) estimation is one of the most important research topics in the field of array signal processing and has been widely used in the radar, sonar, wireless communications, and acoustic tracking, to mention just a few [1–6]. Resolution ability and estimation accuracy, as the two main indicators for measuring the estimation performance, have been the consistent research focus. Over the past decades, there exist several high-resolution DOA estimation approaches. Among these approaches, the most popular approaches include minimum variance distortionless response beamformer (MVDR) [7], multiple signal classification (MUSIC) [8, 9], estimation of signal parameters via rotational invariance technique (ESPRIT) [10], and maximum likelihood (ML) [11, 12]. These approaches can provide high accuracy and high resolution for DOA estimation.For the abovementioned DOA methods, a fundamental assumption is that the DOA estimation methods are implemented in the presence of spatially uniform noise (Gaussian white noise). Under this assumption, the subspace-based approaches can successfully separate the orthogonally signal and noise subspaces, and the ML function with respect to the noise variance single parameter becomes possible [13]. In practical situations, the spatially uniform white noise can hardly be guaranteed. Except for uniform noise, the sensor noise can also be nonuniform [14], spatially correlated [15], or burst impulsive [16]. Since the array response and receive channels are usually nonideal and nonuniform, the practical noise powers across the array are nonidentical and usually behave as nonuniform [17]. The nonuniform noise usually leads to diagonal covariance matrix with nonidentical entries, which may degrade the estimation accuracy for traditional DOA estimation methods. To address this issue, the ML-based methods can deal with the nonuniform case in DOA estimation. In [18, 19], two ML-based estimators are proposed, respectively, based on iterative optimization. However, the iterative optimization procedure results in high complexity and is time-consuming. Moreover, the obtained result may be a local optimum rather than a global one [14]. For subspace-based approaches, there exist several methods to address the nonuniform noise issue [20, 21], which can avoid the iterative optimization procedure, and as a result, they are usually computationally efficient.In this paper, we propose a simple subspace-based DOA estimation method under the nonuniform noises scenarios, where the noise covariance matrix is estimated by utilizing the shift property of identical uniform subarrays, and then, some high-resolution approaches (such as MUSIC and root-MUSIC [22]) are adopted for the following DOA estimation. Simulations are provided to verify the performance improvement of the proposed method in terms of estimation accuracy and resolution ability.


## 2. System Model

In this section, we consider a uniform linear array (ULA) withM sensors, and the i-th sensor is located at the position li=i−1d,i=1,2,⋯,M, where d=λ/2 and λ is the signal wavelength. Assume K far-filed uncorrelated sources impinge on the array from distinct directions Θ=θ1,θ2,⋯,θK. Therefore, the received signal of the array at time t1≤t≤T is computed as
(1)xt=∑k=1Kaθkskt+nt=Ast+nt,where A=aθ1,aθ2,⋯,aθK is the M‐by‐K array manifold matrix with aθk being the M×1 steering vector for the k-th signal, and the i-th element in aθk is denoted as ejπi−1sinθk. st=s1t,s2t,⋯,sKtT denotes the source signal vector at time t with the subscript T being the transpose operation. nt is the M×1 noise vector received by the array, which is uncorrelated with the signal vector st. In general, the noise nt is usually assumed as the additive white Gaussian with zero mean and variance σ2. Under this assumption, the covariance matrix of xt can be obtained as
(2)Rxx=ExtxHt=ARssAH+σ2IM,where the subscript H denotes the conjugate transpose operation, E⋅ denotes expectation operation, Rss=EstsHt represents the signal covariance matrix of sources, and IM is the M‐by‐M unit matrix.In practice, the noisent is usually nonuniform; i.e., the noises at different sensors have different noise powers. We have
(3)Q=EntntH=diagq=diagσ12,σ22,⋯,σM2,where σi2,i=1,2,⋯,M is the sensor noise covariance at the i-th sensor with q=σ12,σ22,⋯,σM2T and diag⋅ denotes a diagonal matrix. In this case, the covariance matrix of xt becomes
(4)Rxx=ExtxHt=ARssAH+Q.From Equations (2) and (4), the uniform noise can be regarded as a special case of nonuniform noise when σ12=σ22=⋯=σM2=σ2, i.e., Q=σ2IM.Due to the effect of limited samples, the covariance matrixRxx is unavailable, but it can be estimated with L snapshots as
(5)R^xx=1L∑t=1LxtxHt,where the estimated covariance matrix R^xx converges to Rxx when the snapshot number L tends to be large enough. The eigenvalue decomposition can be written as
(6)R^xx=UsΛsUsH+UnΛnUnH,where Us and Un are the spanned signal and noise subspace and Λs and Λn are the corresponding eigenvalue matrices with respect to Us and Un, respectively. The pseudospectrum of MUSIC can be obtained as
(7)fθ=1aHθUnUnHaθ.Then, the DOAs can be estimated by finding the peaks offθ. However, as compared to the uniform noise scenarios, the nonuniform noise may interrupt the MUSIC spectrum and degrade the estimation performance. For clarity, we give an example of the MUSIC spectrum under both uniform and nonuniform noises in Figure 1. Here, we consider K=4 uncorrelated sources equally distributed between 10° and 50° impinge on the ULA with M=8, L=500, and the SNR being 0 dB. For the nonuniform noise case, the noise power vector is set as q=1,1,1,1,1,20,30,50T. As can be observed from Figure 1, the MUSIC spectrum performs well under the uniform noise case. Meanwhile, under the nonuniform noise scenarios, the MUSIC can only generate three peaks with larger error and missing one source, which indicates that the conventional MUSIC may become invalid for nonuniform noise cases.Figure 1
The MUSIC spectrum under uniform and nonuniform noises. Results with four signals arriving at angles equally distributed between10° and 50°, M=8, L=500, SNR=0dB, and q=1,1,1,1,1,20,30,50T.


## 3. Proposed Method

For anM-element ULA, it can be divided into three consecutive subarrays with the same aperture and no overlapping elements, as is shown in Figure 2. Each subarray is a uniform linear subarray with Ms elements, where Ms=M/3 with the symbol ⋅ being the round down to the nearest integral operation. The remaining elements can be regarded as the fourth subarray with Mr=M−3Ms elements and Mr can be 0, 1, or 2. Note that when Mr=0, the fourth subarray does not exist, i.e., the fourth subarray exists only when M≠3Ms.Figure 2
The ULA geometry and its subarrays.Without loss of generality, the covariance matrices of the four subarrays can be extracted directly fromRxxin Equation (4) as
(8)Rxx,1=Rxx1:Ms;1:Ms=A1RssA1H+Q1,Rxx,2=RxxMs+1:2Ms;Ms+1:2Ms=A2RssA2H+Q2,Rxx,3=Rxx2Ms+1:3Ms;2Ms+1:3Ms=A3RssA3H+Q3,Rxx,4=Rxx3Ms+1:M;3Ms+1:M=A4RssA4H+Q4,where A1, A2, and A3 are the Ms‐by‐K array manifold matrices of the former three subarrays extracted from A and satisfy Ai+1=FAi with F=diagejπ1+Mssinθ1,⋯,ejπ1+MssinθK for i=1,2. A4 is the Mr‐by‐K array manifold matrix for the fourth subarray. Q1=diagq1:Ms, Q2=diagq1+Ms:2Ms, Q3=diagq1+2Ms:3Ms, and Q4=diagq3Ms+1:M. From Equation (8), AiRssAiH (i=1,2,3,4) can be regarded as the noise-free covariance matrix.For the former three subarrays, they are identical subarrays with different origin shifts. SinceAi=FAi−1 holds for i=2,3 and F and Rss are both diagonal matrices, the noise-free covariance AiRssAiH can be simplified as AiRssAiH=A1Fi−1RssFi−1HA1H=A1RssA1H. Equivalently, the noise-free covariance matrices of the three former subarrays are approximately the same, which can be utilized to eliminate the effect of nonuniform noise.The cross-covariance matrices for the former three subarrays can be obtained in a similar approach as(9)C12=Rxx1:Ms;Ms+1:2Ms=A1RssA2H,C13=Rxx1:Ms;2Ms+1:3Ms=A1RssA3H,C23=RxxMs+1:2Ms;2Ms+1:3Ms=A2RssA3H,where Cij (i,j=1,2,3, i≠j) denotes the cross-covariance matrix between the i-th subarray and j-th subarray. According to the definition of cross-covariance matrix, Cij=CjiH holds. Note that the cross-covariance matrices do not contain the noise item since the noises are uncorrelated with the signal and are mutually uncorrelated at different sensors.We can define two new matrices as(10)T1=C12−1Rxx,1=A1RssA2H−1A1RssA1H+Q1,U1=C13−1C12=A1RssA3H−1A1RssA2H,where ⋅−1 denotes the matrix inverse operation. According to the relationship of A1, A2, and A3, we can simplify Equation (10) as T1=F+C12−1Q1 and U1=F, and thus, we have
(11)T1−U1=C12−1Q1.From Equation (11), Q1 can be estimated as
(12)Q^1=C12T1−U1=Rxx,1−C12C13−1C12.Similarly, by defining(13)T3=C23H−1Rxx,3=A3RssA2H−1A3RssA3H+Q3=FH+C23H−1Q3,U3=C13H−1C23H=A3RssA1H−1A3RssA2H=FH.AndQ3 can be estimated as
(14)Q^3=C23HT3−U3.With the estimatedQ^1 and Q^3, the corresponding noise-free estimated covariance matrices for Rxx,1 and Rxx,3 are
(15)R^xx,1=Rxx,1−Q^1≈A1RssA1H,R^xx,3=Rxx,3−Q^3≈A3RssA3H.Since the noise-free covariance matrices of the former three subarrays are approximately the same, then we can estimateQ2 as
(16)Q^2=Rxx,2−R^xx,1+R^xx,32.In the following, we will estimate the noise power matrix for the fourth subarray, which can be categorised into two cases.Case 1.
WhenMr=0, the fourth subarray does not exist.
With the estimatedQ^1, Q^2, and Q^3, the noise-free covariance matrix can be obtained correspondingly, i.e.,
(17)R^xx,nf1=R^xx−diagq^13,where q^13=diagQ^1,diagQ^2,diagQ^3.Case 2.
WhenMr≠0, the fourth subarray exists.The fourth subarray containsMr elements, and its covariance matrix is Rxx,4 in Equation (8). As the noise-free covariance matrices of identical ULAs with different shifts are the same, we can divide R^xx,nf1 into N=3Ms/Mr parts, which are identical with the fourth subarray with different shifts and no overlapping. And the covariance matrix of the n-th n=1,2,⋯,N part is
(18)R^n,free=R^xx,nf1n−1Mr+1:nMr;n−1Mr+1:nMr.According to the shift property of identical subarrays, the noise-free covariance matrix of the fourth subarray andR^n,free is the same. Q4 can be estimated as
(19)Q^4=Rxx,4−1N∑n=1NR^n,free.Then, we can obtain the noise-free covariance matrix of the entire array for Case2 as
(20)R^xx,nf2=R^xx−diagq^14,where q^14=diagQ^1,diagQ^2,diagQ^3,diagQ^4.Therefore, from Equations (12), (14), (16), and (19), an estimation of the noise-covariance matrix is obtained by the estimation of Q1, Q2, Q3, and Q4. Using the estimated noise-covariance matrix, we can prewhiten the received array data and obtain the noise-free covariance matrix, in Equations (17) and (20) for different cases. Then, DOAs can be estimated by several classic approaches, such as MUSIC and root-MUSIC.As depicted above, the proposed approach firstly estimates the noise covariance matrix to obtain the noise-free covariance matrix and then uses some classical approaches for the following DOA estimation. Therefore, the complexity mainly includes two parts: the estimation of noise-covariance matrix and the estimation of DOAs by classical approaches. For the estimation of noise-covariance matrix, it mainly involves the calculation of covariance matrixRxx and the estimation of noise covariance matrix; the complexities of which are OM2L and O6Ms3, respectively. For the classical DOA estimation approaches, such as MUSIC and root-MUSIC, their complexities can be approximately denoted as OM2S [23] and O20M2K [24], where S≫L denotes the number of spectral points of the total angular field of view. In general, by applying the proposed approach to MUSIC and root-MUSIC, the complexities become OM2S+M2L+6Ms3 and O20M2K+M2L+6Ms3, respectively. As compared to the original estimation approach, the complexities become slightly larger. However, the performance can be greatly improved, which is verified by the following numerical results.


## 4. Results and Discussion

In this section, we evaluate the performance of the proposed method via simulations. The worst noise power ratio (WNPR) is defined asWNPR=σmax2/σmin2 and the signal-to-noise ratio (SNR) is defined as SNR=σs2/∑k=1Kσk2/K, where σmax2 and σmin2 denote the maximum and minimum noise powers, respectively, and σs2 is the signal power. The root mean squared error (RMSE), defined as RMSE=1/KL∑k=1K∑j=1Jθ∧k,j−θk2, is adopted as the performance criteria, where θ^k,j denotes the estimate of the k-th DOA in the j-th trial.The effectiveness and robustness of the proposed approach are firstly verified. Here, we considerK=4 sources equally distributed between 10° and50°. The ULA is modelled with M=8 sensors, and the snapshot number is set as L=500. Figure 3 shows the normalized spectra of the proposed approach and MUSIC under different SNRs with q=1,1,1,1,1,20,30,50T. Figure 4 shows the normalized spectra of the proposed approach and MUSIC under different WNPRs with SNR=0dB. As is observed, the proposed approach can generate four obvious spectral peaks under different SNRs and WNPRs, all of which are coincident with the true DOAs. Meanwhile, the MUSIC approach may generate less peaks, especially for low SNRs. And the generated peaks deviate the positions of true DOAs. With the increase of SNR, the MUSIC approach can generate four peaks, but the proposed approach still provides sharper spectra than that of the MUSIC approach. Therefore, the proposed approach can achieve effective and robust DOA estimation under different SNRs and WNPRs.Figure 3
The normalized spectrum under different SNRs with (a)SNR=−5dB, (b) 0 dB, and (c) 5 dB. Results with four signals arriving at angles equally distributed between 10° and 50°, M=8, and L=500.
(a)
SNR=−5dB(b)
SNR=0dB(c)
SNR=5dBFigure 4
The normalized spectrum under different WNPRs with (a)WNPR=10, (b) 30, and (c) 50. Results with four signals arriving at angles equally distributed between 10° and 50°, M=8, and SNR=0dB.
(a)
WNPR=10(b)
WNPR=30(c)
WNPR=50To verify the resolution performance of the proposed approach, we test our approach and MUSIC at a small angular difference. Two closely located sources are assumed to impinge on the array from DOAs30°and35°, and other experimental conditions are kept unchanged. The spectra with different SNRs and WNPRs are shown in Figures 5 and 6, respectively. As is shown, the traditional MUSIC can generate only one spectral peak at low SNR and thus fail to distinguish the two sources. Despite the increase of SNR, the MUSIC can successfully detect the two closely located sources, but they may deviate the positions of true DOAs. In contrast, the spectrum of the proposed approach has two obvious peaks both under different SNRs and WNPRs. The DOAs corresponding to the peaks are coincident with the true DOAs, indicating that the proposed approach can achieve effective DOA estimation of closely located DOAs under different conditions.Figure 5
The normalized spectrum under different SNRs with (a)SNR=0dB, (b) 5 dB, and (c) 10 dB. Results with two closely located signals arriving at angles 30° and 35°, M=8, and L=500.
(a)
SNR=0dB(b)
SNR=5dB(c)
SNR=10dBFigure 6
The normalized spectrum under different WNPRs with (a)WNPR=10, (b) 30, and (c) 50. Results with two closely located signals arriving at angles 30° and 35°, M=8, and SNR=10dB.
(a)
WNPR =10(b)
WNPR = 30(c)
WNPR = 50Subsequently, the results of resolution probability to detect two closely located sources under different SNRs, snapshot number, and angular differences are shown in Figure7. Here, we consider the ULA with M=8, 9, and 10 sensors, and two closely located sources are assumed to impinge on the array from DOAs 30° and 35° for Figures 7(a) and 7(b). Other experimental conditions are kept unchanged. It is observed that both the proposed approach and MUSIC exhibit improved resolution probability with the increase of SNR, snapshot number, angular differences, and sensor number. At the same time, the proposed approach occupies higher resolution probability in all the considered cases and reaches 100% resolution probability much faster than that of the MUSIC method. The results in Figure 7 demonstrate the superiority of the proposed approach in terms of the resolution probability.Figure 7
The resolution probability of different approaches underM=8, 9, and 10 sensors with respect to (a) SNR, (b) snapshot number, and angular difference Δθ. Results with two closely located signals arriving at angles 30° and 35°. Other conditions include (a) L=500, (b) SNR=6dB, and (c) SNR=6dB, L=500, and θ1=30°.
(a)
Resolution probability vs. SNR(b)
Resolution probability vs. snapshot number(c)
Resolution probability vs. angular differenceΔθFurthermore, the RMSE performance under different SNRs and snapshot numbers is shown in Figure8. Here, we consider an ULA with M=8 sensors. There are K=4 incident DOAs equally distributed between 10° and50°. The MUSIC and root-MUSIC approaches are adopted for comparisons, and the proposed approaches are also performed for the two traditional methods (denoted as MUSIC_pre and root-MUSIC_pre), respectively. As can be observed, the RMSE performance is improved with the increase of SNR and snapshot number. By performing the proposed prewhitening procedure, the improved approach can provide lower RMSE, especially at low SNR and larger snapshot number. The reasons are given as follows. When the SNR is low, the nonuniform noises become the dominant factor to degrade the estimation accuracy, and the proposed approach can eliminate the effect; with the increase of SNR, the nonuniform noises become less important, and the estimation performance will converge to be similar. On the other hand, with the increase of snapshot number, the estimated noise power matrix becomes more accurate, which can enhance the estimation performance of DOAs. The results in Figure 8 effectively verify the superiority of the proposed approach under different SNRS and snapshot numbers.Figure 8
The RMSE performance of different approaches with respect to (a) SNR and (b) snapshot number. Results with four signals arriving at angles equally distributed between10° and 50°, M=8, and L=1000 (a), SNR=2dB (b).
(a)
RMSE vs. SNR(b)
RMSE vs. snapshot number


## 5. Conclusions

In this paper, we propose a simple DOA estimation method in the presence of spatially nonuniform white noise with unknown noise covariance matrix. Firstly, the unknown noise covariance matrix is estimated according to the shift property of identical subarrays. Then, the received signal covariance matrix is prewhitened by the estimated noise covariance to overcome the biased estimate of signal covariance. Therefore, the performance of the proposed method can be improved. Finally, simulation results are provided to verify the effectiveness of the proposed approach in terms of resolution ability and estimation accuracy.



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*Source: 3892542-2022-05-26.xml*