# An Improved Spatial Difference Smoothing Method Based on Multistage Wiener Filtering

Authors: Jian Gong; Yiduo Guo; Qun Wan
Journal: Mathematical Problems in Engineering
 (2019)

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## Abstract

In order to solve the angle estimation problem of coherent sources in the colored background noise, an improved forward and backward spatial difference smoothing algorithm is proposed by combining the improved spatial smoothing algorithm with the spatial difference algorithm. By the algorithm we can not only decoherent the coherent source but also suppress the influence of the color noise. In order to further reduce the computational complexity of the IFBSDS algorithm, an improved forward and backward spatial difference smoothing algorithm based on Wiener filtering is also proposed. Thus, the eigenvalue decomposition operation of subspace class algorithm can be avoided, and at the same time, the same performance with the IFBSDS algorithm can be obtained, which is more consistent with the real demand of MIMO radar signal real-time processing.


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## Body

## 1. Introduction

At present, the study of MIMO radar target angle estimation generally assumes that the environmental noise is white noise obeying Gauss distribution [1–3]. However, in practical applications, there are many factors such as the electronic interference caused by the enemy, the random scattering caused by distributed source, and the mutual coupling characteristics between the receiving channels and so on, and ambient noise often appears as a nonideal colored noise with unknown statistical characteristics. Besides, electronic jamming and low-altitude multipath effect make MIMO radar need angle estimation for coherent sources [4–6]. Under the condition of colored noise background and coherent source, the performance of the classical subspace algorithm such as multiple signal classification (MUSIC) algorithm and estimating signal parameter via rotational invariance techniques (ESPRITs) algorithm will become very poor [7–9]. The four-order cumulant matrix of echo data is constructed in [10, 11], and the angle of the target can be estimated effectively, under the background of colored noise and white noise, but the influence of coherent sources is not eliminated. In [12], an extended rotation invariance factor is obtained for the implementation of the four-order cumulative data, which has high aperture utilization and no phase ambiguity, but the computational complexity is high. In [13, 14], the covariance matrix is reconstructed through the Toeplitz submatrix of a set of received data, but the Gauss color noise cannot be suppressed. In [15], an improved spatial differencing method (ISD) is proposed by constructing a spatial differencing matrix with neighboring subarrays, and the proposed method performs better under the coexistence of both uncorrelated and coherent signals. In [16], by extracting all the data information of each subarray, both the forward only ISD (FO-ISD) and forward backward ISD (FB-ISD) methods are proposed using the reconstructed submatrices.In this paper, an improved spatial difference smoothing algorithm based on multistage Wiener filtering is proposed. The remainder of this paper is organized as follows. In Section2, the echo model is established and its characteristics under color noise are analyzed. Then, an improved spatial difference smoothing (ISDS) algorithm based on multistage Wiener filtering is proposed in Section 3, followed by performance analysis of ISDS algorithm based on multistage Wiener filtering (MWF-ISDS) in Section 4. Some simulations are conducted to verify the performance of the proposed method in Section 5. Finally, we conclude the paper in Section 6.Notation:.∗ denotes the conjugate operator; (.)H denotes the matrix conjugate-transpose operator; (.)T denotes the matrix transpose operator; ⊗ denotes the Kronecker product operator; Σ denotes the sum operator; Π denotes the multiplication operator.


## 2. Echo Model and Its Characteristics under Color Noise

Considering the single base MIMO radar as shown in Figure1, the transmitting and receiving antenna adopt M and N elements, respectively. The spacing of array elements is d=λ/2, and λ is the carrier wavelength. Assuming that the direction of arrival (DOA) and the direction of departure (DOD) both are θp, the echo signal is processed by matching filtering as below:(1)y=Aβt+nt,where A=arθ1⊗atθ1,arθ2⊗atθ2,…,arθP⊗atθP, the transmit steering vector of the pth target is atθp=1,exp−jπsinθp,…,exp−jπM−1sinθpT, the receive steering vector is arθp=1,exp−jπsinθp,…,exp−jπN−1sinθpT, β=ξ1ej2πfd1t,ξ2ej2πfd2t,…,ξPej2πfdPtT, ξp is the reflection coefficient of the pth target, fdp is the normalized Doppler frequency of the pth target, and nt representation of noisy column vectors.Figure 1
The structure of a monostatic MIMO radar [17].If defined,(2)arθp⊗atθp≜Fbθp.In the form,bθp=1,exp−jπsinθp,…,exp−jπM+N−1sinθpT, F∈CMN×M+N−1 is a dimensionality reduction matrix, which can be expressed as [18](3)F=10⋯00⋯001⋯00⋯0⋮⋮⋯⋮⋮⋯⋮00⋯10⋯0M01⋯00⋯000100⋯0⋮⋮⋯⋮⋮⋯⋮00⋯01⋯0M⋮0⋯010⋯00⋯001⋯0⋮⋮⋯⋮⋮⋯⋮0⋯000⋯1M∈CMN×M+N−1.Then, the matrixA can be expressed as follows:(4)A=FB,where B=bθ1,bθ2,…,bθP∈CM+N−1P, and it is a Vandermonde matrix.According to formula (3), if the formula W≜FHF is defined, there are(5)W=diag1,2,…,minM,N,…,minM,N︸M−N+1,…,2,1.By using the reduced dimension transformationW−1FH, we can get(6)y′tl=W−1FHytl=W−1FHFBβtl+W−1FHntl=Bβtl+n′tl,where n′tl=W−1FHntl is a M+N−1 dimension additive noise vector.When there are independent sources and coherent sources in theP target, it may be possible to set up G as coherent source number, and then U=P−G is the number of independent source. The array manifold composed of coherent sources can be expressed as Bc=bθ1,...,bθG, and the array manifold composed of independent sources can be represented as Bnc=bθG+1,...,bθP. Incoherent sources can be expressed as βnct=βG+1t,...,βPtT. If ρi is used to represent the generation factor of the ith coherent source, the coherent source can be expressed as βct=ρβ0t, the echo signal of the matched filter can be expressed as follows:(7)y=∑i=1Gbθiρiβ0t+∑j=G+1Pbθjβjt+n′t=Bcβct+Bncβnct+n′t.So, the covariance matrixR is(8)R=BcPcBcH+BncPncBncH+φ=Rc+Rnc+φ,where φ represents the color noise covariance matrix and Rc and Rnc are the covariance matrix of the coherent source and the independent source, respectively. Among them, Rnc is Hermite, Toeplitz matrix, Rc is Hermite, non-Toeplitz matrix, and φ is Hermite, Toeplitz matrix when space stationary color noise.Under the color noise, there is a large angle estimation error in the classical DOA algorithm. The angle estimation loss caused by the theoretical noise than the real noise can be found in the literature [19].


## 3. ISDS Angle Estimation Algorithm Based on Multistage Wiener Filtering

### 3.1. Improvement of Spatial Smoothing Algorithm

Assuming that the number of virtual elements after the dimension reduction transformation isI, then I=M+N−1. The forward space smoothing algorithm divides the array into q(q≥G+1) subarrays. The number of array elements of each subarray is m=I−q+1. By using the translation invariance property between subarrays, the covariance matrix of q subarrays can be expressed as follows:(9)Rmii=BmDi−1PDi−1HBmH+σ2Im,i=1,…,q,where Rmii=Ri:I+i−q,i:I+i−q, Bm is an array manifold of the initial subarray, σ2 indicates noise power, and D=diage−jπsinθ1,e−jπsinθ2,…,e−jπsinθG.The equivalent covariance matrixRf of the conventional forward space smoothing algorithm can be obtained as follows:(10)Rf=1q∑i=1qRmii.And the covariance matrix of the forward and backward spatial smoothing subarray can be expressed as follows:(11)Rfb=12q∑i=1qRmii+R¯mii=12q∑i=1qRmii+JRmii∗J,where J represents the m order permutation matrix and ∗ represents the conjugate operation.In order to further excavate the useful information of the received data, this paper improves the conventional spatial smoothing algorithm from two directions: strengthening the influence of the main diagonal elements and adding intercorrelation information between the subarrays. The new smoothing matrix performs cross-correlation operations on theq subarray autocorrelation matrix, and then the cross-correlation matrix of the symmetric subarray is cross-correlation to obtain the average as follows:(12)R˜f=12q∑i=1q∑j=1qRmiiRmjj+RmijRmji,(13)R˜fb=14q∑i=1q∑j=1qRmiiRmjj+RmijRmji+R¯miiR¯mjj+R¯mijR¯mji=14q∑i=1q∑j=1qRmiiRmjj+RmijRmji+JRmiiRmjj+RmijRmji∗J.


### 3.2. Modified Spatial Difference Smoothing Algorithm

The spatial difference smoothing algorithm is proposed by [20] using the Toeplitz decomposition characteristics and differential smoothing operations of the uniform linear array covariance matrix to estimate the target angle. The expression of the spatial difference matrix is as follows:(14)ΔR≜R−JR∗J.If formula (7) is replaced by formula (14), we can obtain(15)ΔR=Rc−Rnc+φ−JRc−Rnc+φ∗J,where Rnc and φ are Hermite, Toeplitz matrix, conforming to conjugate reverse order invariance:(16)Rnc=JRnc∗J,φ=Jφ∗J.So, formula (15) can be written as follows:(17)ΔR≜Rc−JRc∗J.Formula (17) shows that there is neither the noise item nor the Toeplitz term of the independent source, which contains only the information of coherent sources.From the construction process of matrixΔR, it can be seen that ΔR is a negative antisymmetric matrix, and its eigenvalues are the combination of positive and negative. Since the trace of a matrix equals the sum of eigenvalues, when the order K of ΔR is an odd number, it must have a 0 eigenvalue as follows:(18)rankΔR=K−1.It can be seen that the negative antisymmetry ofΔR will lead to rank deficit in some cases, and ΔR needs to be corrected and decoherent.In this paper, an improved spatial difference smoothing (ISDS) algorithm is proposed, which preprocessesΔR by Formulae (12) and (13):(19)ΔR˜f=12q∑i=1q∑j=1qΔRmiiΔRmjj+ΔRmijΔRmji,(20)ΔR˜fb=14q∑i=1q∑j=1qΔRmiiΔRmjj+ΔRmijΔRmji+JΔRmiiΔRmjj+ΔRmijΔRmji∗J,where ΔRmii=ΔRi:I+i−q,i:I+i−q and ΔRmij=ΔRi:I+i−q,j:I+j−q.In order to further distinguish the proposed ISDS algorithm, the operation of formula (18) is called the improved forward spatial difference smoothing (IFSDS) algorithm in this paper, and the operation of formula (19) is called the improved forward and backward spatial difference smoothing (IFBSDS) algorithm.


### 3.3. The Principle of MWF-ISDS Algorithm

When the number of virtual elements is large, the computation of signal subspace directly through eigenvalue decomposition is large. In this paper, an improved spatial difference smoothing algorithm based on multistage Wiener filtering is proposed. The signal subspace is obtained by the decomposition process of multistage Wiener filtering, and the computational complexity of the algorithm is further reduced. The processing flow is shown in Figure2.Figure 2
Flow-process diagram of multistage Wiener filter.In Figure2, x0 is the initial array antenna receiving data vector; x1, … is each stage residual vector after the recursive projection of the array antenna receiving data vector; h1, h2, … is the next recursive normalized projection vector; C1, C2, … is each stage forward recursive vector of the array antenna received data vector; and d=d1,d2,…,dKT is oriented vector space, which can be expressed in the following way:(21)d=d1,d2,…,dKT=h1B1h2⋮∏r=1K−2BrhK−1∏r=1K−1BrhK⋅x0t,where K represents the number of filter recursion times.If vectorw1,w2,…,wI=h1,B1h2,…,∏r=1K−2BrhK−1,∏r=1K−1BrhK is used to represent the forward recursive matching filter of multistage Wiener filter, it can be proved that [18](22)spanw1,w2,…,wI=spanh1,Rrh1,Rr2h1,…,RrI−1h1,where Rr represents the received data covariance matrix of MIMO radar and I is the number of virtual array elements.The proposed ISDS algorithm is used to decompose the matrixΔR˜f or ΔR˜fb to get the subspace US=u1,u2,…,uG. The relationship between the signal subspace VS=v1,v2,…,vG and US of MWF algorithm satisfies [21](23)spanv1,v2,…,vG=spanu1,u2,…,uG.The initialization reference signal of the MWF algorithm is the row mean ofΔR˜ as follows:(24)d0t=1I∑i=1IΔR˜i,:.ByG recursions, the subspace of MIMO radar echo data can be obtained as(25)US=w1,w2,…,wG.The noise subspace of MIMO radar echo data is(26)UNUNH=I−USUSH.According to the MUSIC algorithm, the DOAs forG coherent sources are(27)θ^=argminθbHθUNUNHbθ.Based on the above analysis, the operational steps of angle estimation by using the MWF-ISDS algorithm can be summarized as follows:Step 1.
According to formula (6), the received signal of matched filtering is dimensionality reducedStep 2.
Use the MWF algorithm to estimate the angle of independent sourcesStep 3.
According to formula (15), the array difference matrix ΔR of MIMO radar is calculatedStep 4.
CalculateΔR˜f or ΔR˜fb according to formula (19) or (20)Step 5.
Use the MWF algorithm to estimate the signal subspace ofΔR˜f or ΔR˜fbStep 6.
Calculate the noise subspace according to formula (26)Step 7.
Estimate the angle of the coherent sources using formula (27)


## 3.1. Improvement of Spatial Smoothing Algorithm

Assuming that the number of virtual elements after the dimension reduction transformation isI, then I=M+N−1. The forward space smoothing algorithm divides the array into q(q≥G+1) subarrays. The number of array elements of each subarray is m=I−q+1. By using the translation invariance property between subarrays, the covariance matrix of q subarrays can be expressed as follows:(9)Rmii=BmDi−1PDi−1HBmH+σ2Im,i=1,…,q,where Rmii=Ri:I+i−q,i:I+i−q, Bm is an array manifold of the initial subarray, σ2 indicates noise power, and D=diage−jπsinθ1,e−jπsinθ2,…,e−jπsinθG.The equivalent covariance matrixRf of the conventional forward space smoothing algorithm can be obtained as follows:(10)Rf=1q∑i=1qRmii.And the covariance matrix of the forward and backward spatial smoothing subarray can be expressed as follows:(11)Rfb=12q∑i=1qRmii+R¯mii=12q∑i=1qRmii+JRmii∗J,where J represents the m order permutation matrix and ∗ represents the conjugate operation.In order to further excavate the useful information of the received data, this paper improves the conventional spatial smoothing algorithm from two directions: strengthening the influence of the main diagonal elements and adding intercorrelation information between the subarrays. The new smoothing matrix performs cross-correlation operations on theq subarray autocorrelation matrix, and then the cross-correlation matrix of the symmetric subarray is cross-correlation to obtain the average as follows:(12)R˜f=12q∑i=1q∑j=1qRmiiRmjj+RmijRmji,(13)R˜fb=14q∑i=1q∑j=1qRmiiRmjj+RmijRmji+R¯miiR¯mjj+R¯mijR¯mji=14q∑i=1q∑j=1qRmiiRmjj+RmijRmji+JRmiiRmjj+RmijRmji∗J.


## 3.2. Modified Spatial Difference Smoothing Algorithm

The spatial difference smoothing algorithm is proposed by [20] using the Toeplitz decomposition characteristics and differential smoothing operations of the uniform linear array covariance matrix to estimate the target angle. The expression of the spatial difference matrix is as follows:(14)ΔR≜R−JR∗J.If formula (7) is replaced by formula (14), we can obtain(15)ΔR=Rc−Rnc+φ−JRc−Rnc+φ∗J,where Rnc and φ are Hermite, Toeplitz matrix, conforming to conjugate reverse order invariance:(16)Rnc=JRnc∗J,φ=Jφ∗J.So, formula (15) can be written as follows:(17)ΔR≜Rc−JRc∗J.Formula (17) shows that there is neither the noise item nor the Toeplitz term of the independent source, which contains only the information of coherent sources.From the construction process of matrixΔR, it can be seen that ΔR is a negative antisymmetric matrix, and its eigenvalues are the combination of positive and negative. Since the trace of a matrix equals the sum of eigenvalues, when the order K of ΔR is an odd number, it must have a 0 eigenvalue as follows:(18)rankΔR=K−1.It can be seen that the negative antisymmetry ofΔR will lead to rank deficit in some cases, and ΔR needs to be corrected and decoherent.In this paper, an improved spatial difference smoothing (ISDS) algorithm is proposed, which preprocessesΔR by Formulae (12) and (13):(19)ΔR˜f=12q∑i=1q∑j=1qΔRmiiΔRmjj+ΔRmijΔRmji,(20)ΔR˜fb=14q∑i=1q∑j=1qΔRmiiΔRmjj+ΔRmijΔRmji+JΔRmiiΔRmjj+ΔRmijΔRmji∗J,where ΔRmii=ΔRi:I+i−q,i:I+i−q and ΔRmij=ΔRi:I+i−q,j:I+j−q.In order to further distinguish the proposed ISDS algorithm, the operation of formula (18) is called the improved forward spatial difference smoothing (IFSDS) algorithm in this paper, and the operation of formula (19) is called the improved forward and backward spatial difference smoothing (IFBSDS) algorithm.


## 3.3. The Principle of MWF-ISDS Algorithm

When the number of virtual elements is large, the computation of signal subspace directly through eigenvalue decomposition is large. In this paper, an improved spatial difference smoothing algorithm based on multistage Wiener filtering is proposed. The signal subspace is obtained by the decomposition process of multistage Wiener filtering, and the computational complexity of the algorithm is further reduced. The processing flow is shown in Figure2.Figure 2
Flow-process diagram of multistage Wiener filter.In Figure2, x0 is the initial array antenna receiving data vector; x1, … is each stage residual vector after the recursive projection of the array antenna receiving data vector; h1, h2, … is the next recursive normalized projection vector; C1, C2, … is each stage forward recursive vector of the array antenna received data vector; and d=d1,d2,…,dKT is oriented vector space, which can be expressed in the following way:(21)d=d1,d2,…,dKT=h1B1h2⋮∏r=1K−2BrhK−1∏r=1K−1BrhK⋅x0t,where K represents the number of filter recursion times.If vectorw1,w2,…,wI=h1,B1h2,…,∏r=1K−2BrhK−1,∏r=1K−1BrhK is used to represent the forward recursive matching filter of multistage Wiener filter, it can be proved that [18](22)spanw1,w2,…,wI=spanh1,Rrh1,Rr2h1,…,RrI−1h1,where Rr represents the received data covariance matrix of MIMO radar and I is the number of virtual array elements.The proposed ISDS algorithm is used to decompose the matrixΔR˜f or ΔR˜fb to get the subspace US=u1,u2,…,uG. The relationship between the signal subspace VS=v1,v2,…,vG and US of MWF algorithm satisfies [21](23)spanv1,v2,…,vG=spanu1,u2,…,uG.The initialization reference signal of the MWF algorithm is the row mean ofΔR˜ as follows:(24)d0t=1I∑i=1IΔR˜i,:.ByG recursions, the subspace of MIMO radar echo data can be obtained as(25)US=w1,w2,…,wG.The noise subspace of MIMO radar echo data is(26)UNUNH=I−USUSH.According to the MUSIC algorithm, the DOAs forG coherent sources are(27)θ^=argminθbHθUNUNHbθ.Based on the above analysis, the operational steps of angle estimation by using the MWF-ISDS algorithm can be summarized as follows:Step 1.
According to formula (6), the received signal of matched filtering is dimensionality reducedStep 2.
Use the MWF algorithm to estimate the angle of independent sourcesStep 3.
According to formula (15), the array difference matrix ΔR of MIMO radar is calculatedStep 4.
CalculateΔR˜f or ΔR˜fb according to formula (19) or (20)Step 5.
Use the MWF algorithm to estimate the signal subspace ofΔR˜f or ΔR˜fbStep 6.
Calculate the noise subspace according to formula (26)Step 7.
Estimate the angle of the coherent sources using formula (27)


## 4. Performance Analysis of MWF-ISDS Algorithm

### 4.1. Improved Performance of the Algorithm

In order to compare the algorithm performance, the improvement of the signal-to-noise ratio can be expressed by the ratio of covariance matrix trace of source to covariance matrix trace of noise. Then, the SNR improvement factorQsn is defined as follows:(28)Qsn=trPtrφ,where trP is the trace of the source covariance matrix and trφ is the trace of the noise covariance matrix. The larger the value Qsn is, the smaller the noise perturbation is, and the better the estimation performance of the algorithm is.If the classic forward smoothing, there is(29)Rf=1q∑i=1qRmii=Bm1q∑i=1qDi−1PDi−1HBmH+σ2Im=BmPfBmH+σ2Im,where Pf is the echo covariance matrix of the classical forward smoothing algorithm.By formula (29), the signal-to-noise ratio improvement factor of the classical forward smoothing algorithm can be obtained as follows:(30)Qsn−f=trPftrσ2Im.If the improved forward smoothing, there is(31)R˜f=12qBm∑i,j=1i≠jqwij+2σ2∑i,j=1i≠jqDi−1PDj−1H+2∑i=1qwi+σ2∑i=1qDi−1PDi−1HBmH+σ4Im,where wij=βHDi−1HBmHBmDj−1β and wj=βHDj−1HBmHBmDj−1β.By formula (31), the improvement factor of the signal to noise ratio of the improved forward smoothing algorithm can be obtained as follows:(32)Q˜sn−f=tr1/2q∑i,j=1,i≠jqwij+2σ2∑i,j=1,i≠jqDi−1PDj−1Htrσ4I+1σ2∑j=1qwj+1Qsn−f.It is clear that the proposed forward smoothing algorithm is better than the classical forward smoothing algorithm.


### 4.2. Comparison of Computational Complexity

If do not reduce the covariance matrix dimension, the multiplication times of the subspace algorithms as MUSIC approximate toOM2N2L+OM3N3. The number of multiplication times required by the multistage Wiener filtering method is approximately [22] OGMNL. If the dimension is reduced, the subspace algorithms such as MUSIC need to decompose the M+N−1 covariance matrix, and the multiplication times is OM+N−12L+OM+N−13. The number of multiplication times required by the multistage Wiener filtering method is approximately OGM+N−1L. It can be seen that the MWF-ISDS algorithm proposed in this paper reduces the dimension of the matrix operation by reducing the dimension of the dimension, and further reduces the computational complexity by using the multistage Wiener filtering method.


## 4.1. Improved Performance of the Algorithm

In order to compare the algorithm performance, the improvement of the signal-to-noise ratio can be expressed by the ratio of covariance matrix trace of source to covariance matrix trace of noise. Then, the SNR improvement factorQsn is defined as follows:(28)Qsn=trPtrφ,where trP is the trace of the source covariance matrix and trφ is the trace of the noise covariance matrix. The larger the value Qsn is, the smaller the noise perturbation is, and the better the estimation performance of the algorithm is.If the classic forward smoothing, there is(29)Rf=1q∑i=1qRmii=Bm1q∑i=1qDi−1PDi−1HBmH+σ2Im=BmPfBmH+σ2Im,where Pf is the echo covariance matrix of the classical forward smoothing algorithm.By formula (29), the signal-to-noise ratio improvement factor of the classical forward smoothing algorithm can be obtained as follows:(30)Qsn−f=trPftrσ2Im.If the improved forward smoothing, there is(31)R˜f=12qBm∑i,j=1i≠jqwij+2σ2∑i,j=1i≠jqDi−1PDj−1H+2∑i=1qwi+σ2∑i=1qDi−1PDi−1HBmH+σ4Im,where wij=βHDi−1HBmHBmDj−1β and wj=βHDj−1HBmHBmDj−1β.By formula (31), the improvement factor of the signal to noise ratio of the improved forward smoothing algorithm can be obtained as follows:(32)Q˜sn−f=tr1/2q∑i,j=1,i≠jqwij+2σ2∑i,j=1,i≠jqDi−1PDj−1Htrσ4I+1σ2∑j=1qwj+1Qsn−f.It is clear that the proposed forward smoothing algorithm is better than the classical forward smoothing algorithm.


## 4.2. Comparison of Computational Complexity

If do not reduce the covariance matrix dimension, the multiplication times of the subspace algorithms as MUSIC approximate toOM2N2L+OM3N3. The number of multiplication times required by the multistage Wiener filtering method is approximately [22] OGMNL. If the dimension is reduced, the subspace algorithms such as MUSIC need to decompose the M+N−1 covariance matrix, and the multiplication times is OM+N−12L+OM+N−13. The number of multiplication times required by the multistage Wiener filtering method is approximately OGM+N−1L. It can be seen that the MWF-ISDS algorithm proposed in this paper reduces the dimension of the matrix operation by reducing the dimension of the dimension, and further reduces the computational complexity by using the multistage Wiener filtering method.


## 5. Computer Simulation Results

It is assumed that the number of transmitter elements and the number of receiving elements of the monostatic MIMO radar are 6 and 8, respectively, and the number of virtual elements is 13 after dimensionality reduction. It may be possible to establish a spatial stationary color covariance matrix ofφ, and its kth line lth column element is φk,l=σn20.9k−lexpjπk−l/16. The signal-to-noise ratio is S/N=10lgσs2/σn2, where σs2 indicates signal power and σn2 indicates noise power.Experiment 1 (comparison of spatial spectrum curves of different algorithms under colored noise mixed source background).
In the background of colored noise, angle estimation of 6 equal power mixed sources is carried out. The angles are −40°, −30°, −10°, 5°, 25°, and 35°, respectively. The first 4 are coherent sources and the other 2 are incoherent sources. The number of pulses is 200, the number of smooth subarrays is 4, and the signal-to-noise ratio is fixed to 0 dB. Figure3 simulates the normalized spatial spectral curves of six different algorithms of FSS, FBSS, MUSIC, SDS, IFBSDS, and MWF-IFBSDS.
The simulation results from Figure3 show that, under the background of color noise, the FSS and FBSS algorithms only use the smooth processing without differential processing; the angle estimation performance is low, and the estimated results have a large error with the true direction of arrival. The MUSIC algorithm cannot estimate the angle of the coherent source and can effectively estimate the angle of the independent source. But because of the inability to overcome the influence of colored noise, the normalized spatial spectral curve still has a large pseudo peak. The SDS, IFBSDS, and MWF-IFBSDS algorithms use both the smoothing and the differential processing, so it is good to estimate the angle of the coherent source in the background of the color noise. Because of the improved smoothing algorithm in ISD, IFBSDS, and MWF-IFBSDS algorithms, the estimation performance of the angle is better than that of the SDS algorithm. Compared with the ISD algorithm, the proposed MWF-IFBSDS algorithm still maintains the angle estimation performance which is not much different from the ISD algorithm with reducing the computational complexity.Figure 3
Angle estimation performance of mixed noise sources using different algorithms. (a) FSS and FBSS algorithm. (b) MUSIC algorithm. (c) SDS, IFBSDS, ISD, and MWF-IFBSDS algorithms.
(a)
(b)
(c)Experiment 2 (statistical performance comparison of angle estimation algorithms under colored noise coherent sources).
In the background of colored noise, the angle estimation of 2 equal power coherent sources is 0° and 10°, respectively. The number of pulsing pulses is 200 and the number of smooth subarrays is 2. Figure4 simulates angle estimation performance of SDS, IFBSDS, and MWF-IFBSDS algorithms in 200 Monte–Carlo experiments.
The simulation results in Figure4 show that when the SNR is low, the IFBSDS and MWF-IFBSDS algorithms outperform the SDS algorithm in terms of probability of success and root mean square error of estimation. In the case of low SNR, the angle estimation performance of the IFBSDS and MWF-IFBSDS algorithms is better than that of the SDS algorithm. With the increase of SNR, the statistical performance of the three algorithms is basically the same.Figure 4
Comparison of angle estimation performance when SNR varies. (a) Success probability. (b) Root mean square error.
(a)
(b)


## 6. Conclusions

In this paper, the echo model under the background of colored noise is established and the characteristics of its covariance matrix are analyzed. An ISDS (MWF-ISDS) angle estimation algorithm based on multistage Wiener filtering is proposed. The improved performance and computational complexity of the algorithm are analyzed and simulated. Theoretical analysis and simulation experiments show that the MWF-IFBSDS algorithm has better angle estimation performance than the FSS, FBSS, MUSIC, and ISDS algorithms under low SNR and small angle interval, and the proposed algorithm has stronger array element saving ability and source overload capability.



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