# Sparse Spatial Spectral Fitting with Nonuniform Noise Covariance Matrix Estimation Based on Semidefinite Optimization

Authors: Tuo Guo; Yang Bi; Xian Feng; Luoheng Yan
Journal: Wireless Communications and Mobile Computing
 (2022)

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## Abstract

In general, the azimuth estimation in array signal processing is derived under the assumption of uniform white noise, whose covariance matrix is a scaled identity matrix. However, in practice, the noise can be nonuniform with an arbitrary unknown diagonal covariance matrix. In this paper, the estimation of the noise covariance matrix is formulated into a solution to the semidefinite optimization problem which can obtain a more accurate sensor noise covariance matrix. In the proposed algorithm, the estimated nonuniform noise is subtracted from the sample covariance matrix. The simulation results show that the proposed algorithm can significantly improve the performance of the sparse spectrum fitting algorithm in nonuniform noise case, while the classic SpSF algorithm is used under uniform white noise assumption. The water pool experiments show that there are indeed significant differences in the noise covariance of the sensors.


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## Body

## 1. Introduction

Determining the direction-of-arrival (DOA) of incoming signals is an important research problem in array signal processing, which is widely used in radar, sonar, navigation, wireless communications, and acoustics [1–7]. Under the uniform white noise assumption, several high resolution DOA estimation approaches are well known to provide high accuracy and excellent asymptotic performance, such as multiple signal classification (MUSIC) [8] and estimation of signal parameters via rotational invariance technique (ESPRIT) [9]. According to the assumption, sensor noises are presumed to be a zero-mean Gaussian process with the covariance matrix σ2I, which is the unknown uniform noise variance, and I is the identity matrix [10].The sparse spectrum fitting (SpSF) [11] algorithm is another high resolution DOA estimation algorithm, which is based on the sparse signal model of the array and compressive sensing theory. SpSF is formulated by applying L1-norm penalization to the spatial sparse model of the sample covariance matrix. It is similar to Lasso-type algorithms [12–15] which utilize compressive sensing theory with L1-norm replacing L0-norm. However, the SpSF algorithm still assumes the sensors noise is uniform Gaussian white noise. In fact, the sensor noise may be nonuniform [16–18], spatially correlated [19–24], or block-correlated [25–28]. In particular, in the field of underwater acoustic signal processing, the noise of different hydrophones tends to be nonuniform due to inaccuracy in the manufacture of hydrophones.In this paper, as usual, we still assume that the noise entering the sensor is Gaussian noise, but the noise variances of sensor are nonidentical across the array, and then we propose a new DOA estimation algorithm with sparse spatial spectral fitting based on nonuniform noise estimation (NN-SpSF). The NN-SpSF algorithm significantly improves the performance of DOA estimation under nonuniform noise compared with the SpSF algorithm.The paper is organized as follows. The signal model is formulated in Section2. The SpSF algorithm based on nonuniform noise estimation is developed in Section 3. Simulation results are presented in Section 4. The results of the water pool experiment are shown in Section 5. Conclusions are provided in Section 6.


## 2. Signal Model

Consider a uniform linear hydrophone array ofM elements with its steering vector denoted by aθ. Suppose that KK<M far-field narrow-band signals impinge on the array from the unknown DOAs θ=θ1,⋯,θK, then the signal observed of the array at time t is given as
(1)xt=Aθst+et,t=1,⋯,L,where Aθ=aθ1,⋯,aθK is the steering matrix of the array, and amθ=1,ejφ⋯,ejM−1φT is the steering vector [29]. In a uniform linear array (ULA) with half-wavelength interelement spacing, the mth entry of amθ is given by
(2)φ=2πΔλsinθ,where st=s11,⋯,sKn and et are the signal and noise vectors, respectively. They are assumed to be uncorrelated. L is the number of snapshots. Then, the array output covariance matrix can be expressed as
(3)R=ExtxHt=APAH+Q,where E⋅ and ⋅ represent the mathematical expectation and Hermitian transpose, respectively. P=EstsHt is the signal covariance matrix, and Q=EeteHt is the noise covariance matrix. For uncorrelated sources, P is a diagonal matrix. In this paper, the sensor noise is considered to be nonuniform and can be modeled as a spatially and temporally uncorrelated zero-mean random process, and then Q is also a diagonal matrix having the form
(4)Q=diagσ12,⋯,σM2,where σ12,m=1,⋯,M are the sensor noise variances, and diag⋅ denotes a diagonal matrix. In practical applications, the array covariance matrix is usually estimated as R^=1/L∑t=1LxtxHt.


## 3. The Proposed SpSF Algorithm with Nonuniform Noise Estimation (NN-SpSF)

The SpSF algorithm is formulated by applyingl1-norm penalization of fitting the source covariance model to the estimated spatial covariance, which applies the vectorization operation [30–32] to R^, and it can obtain the following relation:
(5)r=ANθp∘+σ2vecI,where r=vecR^, vec, is the vectorization operation; ANθ=avθ1avθ2⋯avθNθ is the array manifold matrix, in which avθ=vecaθaHθ, and Nθ is the number of search angles for θ. According to the Formula (5), the noise entering the array is assumed to be uniform Gaussian white noise. However, if the noise is nonuniform, (5) can be rewritten as
(6)r=ANθp∘+vecQ,where Q is a diagonal matrix, but the elements are not equal; therefore, the bearing estimation performance of the SpSF algorithm will be significantly reduced. In the case of nonuniform Gaussian white noise, we need to remove the nonuniform noise information from Q by the following operations:
(7)Q‐QN⟶λ⋅I.To get the nonuniform diagonal matrixQN=diagσ′12,⋯,σ′M2, and σ′12≠σ′22≠⋯≠σ′M2, we can convert it to the following semipositive definite optimization problem, as shown in Formula (8):
(8)max∑m=1Mσ′m2,s.t.R^−QN≥η⋅I,QN≥0,η=mindiagR^.WhenQN is obtained, the DOA estimator of the SpSF algorithm in the nonuniform noise situation (written as NN-SpSF) can be given as
(9)p∗=argminp,μvecR^−QN‐ANθp22+μp1,s.t.pi≥0,i=1,⋯,Nθ.Formula (8) and (9) can be solved with the convex optimization tool like CVX [33], which is used to solve the semipositive definite optimization problem. Here in formula (9), the variable μ takes the value of 0.8.


## 4. Simulations

In this section, a series of numerical experiment results under different conditions are conducted to validate the performance of the proposed NN-SpSF algorithm. The experiments are performed with a uniform linear array (ULA) withM=12 sensors and half-wavelength space. Three equally powered independent narrowband signals impinge on the array from directions -8°, 0°, and 8° respectively. The noise is assumed spatially nonuniform and independent, which has the following covariance matrix:
(10)Q=diag10.25.68.511.27.89.584.896.17.22.The signal-to-noise ratio (SNR) is defined as(11)SNR=10log10σn2σs2∑i=1Mσi2,where σs2 denotes the power of source signal.In the first simulation, the number of snapshots is 500, and 10 independent experiment runs forSNR=0dB are performed. The comparison performance of the proposed NN-SpSF algorithm with the SpSF algorithm is shown in Figure 1.Figure 1
Comparisons of SpSF and NN-SpSF for DOA estimation.As can be seen from Figure1, the NN-SpSF algorithm has higher accuracy of azimuth estimation and lower sidelobe in nonuniform noise case.Next, we set the number of snapshots to 800 and evaluate the performances of the proposed algorithms at different SNR levels. The root-mean-square error (RMSE) of the estimated DOA of the sources is defined as(12)RMSE=1K∗Nm∑p=1Nm∑k=1Kθ∧kp−θk2,where θ^kp is the estimate of θk for the pth Monte Carlo trial, K is the number of sources, Nm is the number of the Monte Carlo trials, and Nm=200 in all the following simulations. The RMSE and success probability of DOA estimation versus SNR are shown in Figures 2 and 3. The SNR varies from -12 dB to 12 dB with 2 dB step size.Figure 2
Comparison of the DOA estimation RMSEs versus SNR.Figure 3
Comparison of the success probability of DOA estimation versus SNR.It can be seen from Figures2 and 3 that the NN-SpSF algorithm has lower RMSE and higher success probability than the SpSF algorithms when the signal-to-noise ratio changes. Figures 4 and 5 show the RMSE and success probability of DOA estimation versus snapshots, which varies from 200 to 3200 with 200 step sizes.Figure 4
Comparison of the DOA estimation RMSEs versus the number of snapshots.Figure 5
Success probability of DOA estimation versus the number of snapshots.It can be found that, as the snapshots increasing, the NN-SpSF algorithm has the similar performance with Figures2 and 3. Besides the performance of DOA estimation, we also evaluate the performance of the proposed NN-SpSF and SpSF algorithm for noise variance estimation. Figure 6 depicts the estimated noise variances averaged from 500 Monte Carlo trials at SNR=−8dB. Figure 7 shows the RMSE of noise variance estimation of all sensors versus SNR; the SNR varies from -12 dB to 12 dB with 2 dB step size.Figure 6
Averaged noise variances of sensors.Figure 7
RMSE versus SNR for noise variance estimation.It can be seen that the RMSE of noise variance decreases rapidly with the increase of the SNR, and the proposed algorithm can still estimate the nonuniform noise with a lower deviation at a lower SNR, such as the -8 dB case in Figure6.


## 5. Water Pool Experiment

The algorithm in this paper was verified by pool experiments in an anechoic pool. The water pool is 20 m long, 8 m wide, and 7 m deep. The receiving array is a vertical uniform linear array with 10 array elements, and the first hydrophone was placed at a depth of 0.7 m. The transmitting transducer is 7 m away from the receiving array, and its transmitting signal is a CW pulse with a frequency of 3 kHz. The CW pulse signal has a length of 400 ms and a period of 1 s.The results of the DOA estimation of the water pool experiment for the NN-SpSF algorithm is shown in Figure8, and it can be observed that the NN-SpSF algorithm has lower sidelobe. The noise variance estimation of all sensors of the array is shown in Figure 9. In the actual situation, there are indeed some differences in the noise background of the sensors in the array due to manufacturing, installation, and other reasons.Figure 8
Comparison of SpSF and NN-SpSF algorithm for DOA estimation of pool experiment.Figure 9
Noise variance estimation of the all sensors.


## 6. Conclusion

In this paper, a new noise variance estimation algorithm of all sensors in an array in nonuniform noise for DOA estimation is proposed. The estimation of the noise covariance matrix is formulated into a solution to the semidefinite optimization problem which can obtain a more accurate sensor noise covariance matrix. Simulation results show that subtracting the estimated nonuniform noise from the sample covariance matrix can significantly improve the performance of the SpSF algorithm. The water pool experiments show that due to manufacturing and other reasons, nonuniform noise of the sensor does exist, but independent nonuniform noise in the sensor array is the simplest assumption. In future, we will study the noise covariance matrix estimation algorithm closer to the real situation to improve the performance of azimuth estimation.



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