# Weighted Direct Position Determination via the Dimension Reduction Method for Noncircular Signals

Authors: Xinlei Shi; Xiaofei Zhang
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

This work studies the direct position determination (DPD) of noncircular (NC) signals with multiple arrays. Existing DPD algorithms of NC sources ignore the impact of path propagation loss on the performance of the algorithms. In practice, the signal-to-noise ratios (SNRs) of different observation stations are often different and unstable when the NC signal of the same radiation target strikes different observation locations. Besides, NC features of the target signals are applied not only to extend the virtual array manifold but also to bring high-dimensional search. For the sake of addressing the above problems, this study develops a DPD method of NC sources for multiple arrays combing weighted subspace data fusion (SDF) and dimension reduction (RD) search. First, NC features of the target signals are applied to extend the virtual array manifold. Second, we assign a weight to balance the error and obtain higher location accuracy with better robustness. Then, the RD method is used to eliminate the high computational complexity caused by the NC phase search dimension. Finally, the weighted fusion cost function is constructed by using the eigenvalues of the received signal covariance matrixes. It is verified by simulation that the proposed algorithm can effectively improve the location performance, get better robustness, and distinguish more targets compared with two-step location technology and SDF technology. In addition, without losing the estimation performance, the proposed algorithm can significantly reduce the complexity caused by the NC phase search dimension.


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## Body

## 1. Introduction

As we all know, wireless position determination is a significant research work about signal processing, which has developed rapidly in recent years. It has been widely used in many fields including communication, radar, military, astronomy, and so on [1–4]. Research on wireless location can be divided into two categories: traditional two-step location technology and direct position determination technology (DPD). Traditional two-step location technology: this kind of technology first needs to extract the intermediate parameters including the target position information from the original data, such as direction of arrival. Then, target positions can be calculated from the spatial geometric relationship [5–7]. DPD can directly estimate target positions from the original data without estimating intermediate parameters [8–10].Theoretically, DPD avoids the transmission of intermediate parameter error and can obtain more accurate estimation results than two-step location technology, so it has been widely concerned. In the modern communication system, according to whether the ellipse covariance is zero or not, the signal could be split into circular signal and noncircular signals [11]. To date, DPD for circular signal has achieved many research results. Weiss proposed a DPD method based on maximum likelihood (ML) in 2007 [12]. Then, Oispuu proposed a DPD technology combining maximum likelihood and Capon in 2010 to improve the estimation accuracy [9]. A DPD method based on time-varying delay is developed by Shangyu, which can effectively improve the location performance [13]. T Zhou proposed an iterative adaptive DPD method, which can deal with the case of coherent sources [14]. In modern communication systems, quadrature phase shift keying (QPSK) signals and amplitude modulation (AM) signals all belong to NC signals, so it is of great significance for the research of the DPD algorithm with NC sources. However, so far, there are a few research studies on DPD methods for NC signals. The features of NC signals are applied to ameliorate the location accuracy in [15–17]. Then, Doppler shifts have been applied to extend the array manifold for higher estimation accuracy [18, 19]. Recently, the sparse array and spatial smoothing method have been used to realize the DPD of underdetermined conditions [20], [21].However, the existing methods do not solve the high complexity problem caused by noncircular phase. In addition, the existing DPD methods of NC sources have not considered the impact of path propagation loss, so performances of the relevant algorithms are unstable and the location accuracy has some limitations [22–25]. According to the principle of power allocation in reference [26], combined with the idea of SDF and NC characteristics, this study derives the weighted SDF algorithm for NC sources (NC-WSDF). Due to the high-dimensional search caused by NC phase, this study introduces the idea of dimension reduction search [27, 28] and develops a DPD technology of NC signals with multiple arrays by weighted the SDF and RD method (NRD-WSDF). It is verified by simulation that the proposed method is more robust and has better location performance compared with the traditional two-step technology, SDF technology, and NC-SDF technology. Besides, the proposed DPD technology significantly reduces the computational complexity through the RD method without losing the estimation performance.The main contributions are summarized as follows:(1)
When the NC signal of the same radiation target strikes different observation locations, the SNRs of different observation stations are often different and unstable because of the path propagation loss. Therefore, a method based on SNR weighting is applied to balance the error and obtain higher location accuracy and higher level of robustness.(2)
Combined with the idea of expanding virtual array aperture based on NC characteristics and SDF algorithm, the NC-WSDF algorithm is derived, which has more degrees of freedom and can distinguish more targets(3)
The RD method is introduced to reduce the high computational complexity caused by NC phase search, and an NRD-WSDF algorithm is proposed, which effectively reduces the computational complexity without losing the estimation performance.(4)
Complexity analysis and simulation results are explained to prove the excellent performance of the proposed algorithmThe rest of the study is given as follows: Section2 introduces the DPD model with multiple arrays and NC sources model. In Section 3, we develop a DPD method of NC sources for multiple arrays combing weighted subspace data fusion (SDF) and dimension reduction (RD) search. Sections 4 and 5 give the performance analysis and conclusions, respectively.In this study, vector and matrix are lower-case bold and upper-case bold, respectively;· represents the l2 norm; ·∗, ·T, ·H, and ·−1 represent conjugate, transposition, conjugate transpose, and the inverse of matrix, respectively; 0N denotes the N dimensional zero array; IN denotes the N dimensional unit array; diag· represents the diagonal matrix.


## 2. Model Formulation

### 2.1. DPD Model with Multiple Arrays

Consider a two-dimensional localization scene as shown in Figure1. Suppose that there are K uncorrelated narrow-band noncircular signals incident on L base stations in the far-field, each base station is outfitted with a uniform linear array (ULA), and the interval between arrays is λ/2, where λ denotes the wavelength. The target positions are pk=xk,ykTk=1,2,…,K. The locations of the observation stations are ul=xl,ylTl=1,2,…,L. The number of snapshots per observation station is T.Figure 1
Geometry of multiple arrays.On the basis of the free space propagation loss model [29–31], the SNRs of different observation stations are different when the signal of the same radiation target strikes different observation locations. The path propagation loss coefficient can be stated as(1)αl,k=Pl,kPk,where Pk represents the power of the emitter signal, and Pl,k stands for the power of the signal from the kth target received at the lth base location.Suppose thetth snapshot for the kth target at the lth observation station is sl,kt. The signal at the tth snapshot received by the lth observation station can be stated as [31](2)rlt=∑k=1Kαl,kaθl,ksl,kt+nlt,where the steering vector is(3)aθl,k=1,e−jπcosθl,k,…,e−jπM−1cosθl,kT∈ℂM×1,where θl,k represents azimuth of the kth target received by the lth observation station, and nlt denotes the Gaussian white noise vector. As the targets are far from the observation stations, the observation stations can be considered as points. Therefore, equation (3) can be stated as(4)alpk=1,…,e−jπM−1ul1−pk1/ul−pkT∈ℂM×1,where |·| represents the l2 norm.Thus, equation (2) can be simplified as(5)rlt=AlpΛslt+nlt∈ℂM×1,where(6)Alp=alp1,alp2,…,alpK∈ℂM×K,Λ=αl,10…00αl,2⋱⋮⋮⋱⋱00…0αl,K∈ℝK×K,slt=sl,1t,…,sl,KtT∈ℂK×1,p=p1T,p2T,…,pKTT,nlt=nl,1t,nl,2t,…,nl,MtT∈ℂM×1,where ·T denotes the transposition.


### 2.2. Noncircular Sources Model

According to [11], we can get the following equation:(7)EsltslHt=ρejφEsltslTt,where φ represents the noncircular phase, ρρ∈0,1 denotes the noncircular ratio, and ·H stands for the conjugate transpose. For the sake of simplicity, this study only considers the case that the noncircular ratio is 1. From [15], NC signals can be written as(8)slt=ΦslRt,where(9)Φ=e−jφ10…00e−jφ2⋱⋮⋮⋱⋱00…e−jφK∈ℂK×K,slRt=sl,1Rt,sl,2Rt,…,sl,KRtT∈ℝK×1,where φk stands for the NC phase of the kth signal, and sl,kRt denotes the signal amplitude.


## 2.1. DPD Model with Multiple Arrays

Consider a two-dimensional localization scene as shown in Figure1. Suppose that there are K uncorrelated narrow-band noncircular signals incident on L base stations in the far-field, each base station is outfitted with a uniform linear array (ULA), and the interval between arrays is λ/2, where λ denotes the wavelength. The target positions are pk=xk,ykTk=1,2,…,K. The locations of the observation stations are ul=xl,ylTl=1,2,…,L. The number of snapshots per observation station is T.Figure 1
Geometry of multiple arrays.On the basis of the free space propagation loss model [29–31], the SNRs of different observation stations are different when the signal of the same radiation target strikes different observation locations. The path propagation loss coefficient can be stated as(1)αl,k=Pl,kPk,where Pk represents the power of the emitter signal, and Pl,k stands for the power of the signal from the kth target received at the lth base location.Suppose thetth snapshot for the kth target at the lth observation station is sl,kt. The signal at the tth snapshot received by the lth observation station can be stated as [31](2)rlt=∑k=1Kαl,kaθl,ksl,kt+nlt,where the steering vector is(3)aθl,k=1,e−jπcosθl,k,…,e−jπM−1cosθl,kT∈ℂM×1,where θl,k represents azimuth of the kth target received by the lth observation station, and nlt denotes the Gaussian white noise vector. As the targets are far from the observation stations, the observation stations can be considered as points. Therefore, equation (3) can be stated as(4)alpk=1,…,e−jπM−1ul1−pk1/ul−pkT∈ℂM×1,where |·| represents the l2 norm.Thus, equation (2) can be simplified as(5)rlt=AlpΛslt+nlt∈ℂM×1,where(6)Alp=alp1,alp2,…,alpK∈ℂM×K,Λ=αl,10…00αl,2⋱⋮⋮⋱⋱00…0αl,K∈ℝK×K,slt=sl,1t,…,sl,KtT∈ℂK×1,p=p1T,p2T,…,pKTT,nlt=nl,1t,nl,2t,…,nl,MtT∈ℂM×1,where ·T denotes the transposition.


## 2.2. Noncircular Sources Model

According to [11], we can get the following equation:(7)EsltslHt=ρejφEsltslTt,where φ represents the noncircular phase, ρρ∈0,1 denotes the noncircular ratio, and ·H stands for the conjugate transpose. For the sake of simplicity, this study only considers the case that the noncircular ratio is 1. From [15], NC signals can be written as(8)slt=ΦslRt,where(9)Φ=e−jφ10…00e−jφ2⋱⋮⋮⋱⋱00…e−jφK∈ℂK×K,slRt=sl,1Rt,sl,2Rt,…,sl,KRtT∈ℝK×1,where φk stands for the NC phase of the kth signal, and sl,kRt denotes the signal amplitude.


## 3. The Proposed Algorithm

### 3.1. NC-WSDF

According to equation (8), equation (5) can be rewritten as(10)rlt=AlpΛΦslRt+nlt∈ℂM×1.On the basis of the noncircular characteristics in [21], the received signal vector can be extended to(11)zlt=rltrl∗t=AltΛsltAl∗tΛ∗sl∗t+nltnl∗t∈ℂ2M×1,where ·∗ denotes the conjugate.As(12)sl∗t=Φ∗slR∗t=Φ∗Φ−1slt=Φ∗2slt,where ·−1 stands for the inverse of matrix.Thus, equation (11) can be expressed as(13)zlt=AltAl∗tΦ∗Φ∗Λslt+nltnl∗t=BlpΛslt+nltnl∗t∈ℂ2M×1,where(14)Blp=AltAl∗tΦ∗Φ∗=blp1,φ1,blp2,φ2,…,blpK,φK∈ℂ2M×K,blpk,φk=alpkal∗pke−j2φk∈ℂ2M×1.Then, we can get the covariance matrix of signals received by thelth observation location:(15)Rl=1T∑t=1TzltzlHt.The eigenvalue decomposition (EVD) ofRl is carried out [28].(16)Rl=Uls,UlnΣlUls,UlnH.If we useλl,mm=1,2,…,2M to represent the eigenvalues of the lth observation location, which is sorted from large to small, the corresponding eigenvectors are represented by el,mm=1,2,…,2M. Then, in equation (16), signal subspace and noise subspace can be expressed as Uls=el,1,…,el,K and Uln=el,K+1,…,el,2M, respectively, and Σl=diagλl,1,λl,2,…,λl,2M, where diag⋅ represents the diagonal matrix.Then, we can get the cost function according to the idea of(17)fNC−SDFp,φ=argminp,φ∑l=1LUlnHblp,φ2.However, the cost function in equation (17) does not consider the influence of path propagation loss matrix Λ. In practice, the path propagation loss cannot be ignored [29]. As long as there is one with poor performance, the cost function is easily disturbed. Therefore, in order to balance the projection error, obtain higher location accuracy and higher level of robustness, and we assign weight as(18)fNC−WSDFp,φ=argminp,φ∑l=1LwlUlnHblp,φ2,where wl is the weight of the lth observation station.As higher SNR will lead to smaller error, we can use SNR to weight the projection result of each observation location. Therefore, we rewrite equation (15) by substituting equation (13) into equation (15):(19)Rl=1T∑t=1T∑k=1Kαl,k2Pkblp,φblHp,φ+σn2I2M×2M,where IN denotes the N dimensional unit array.Under the assumption that the noise power is stationary during the observation, it can be seen from equation (19) that the SNR of different base stations is proportional to αl,k2Pk, i.e., Pl,k, which is unknown in practice. However, the covariance matrix can be decomposed into SDH:(20)Rl=Rs+Rn=Blp,φdiagPl,1,…,Pl,KBlHp,φ+σn,l2I2M×2M.Under the same assumption, the eigenvalues of covariance matrix can be expressed as(21)λl,m=σs,m2+σn,l2,1≤m≤K,σn,l2,K+1≤m≤2M,where σs,m21≤m≤K are the larger nonzero eigenvalues of Rs, which represent the power of the received signal. Then, the noise power can be expressed as(22)σ^n,l2=12M−K∑m=K+12Mλl,m.According to the estimated noise power, the estimated signal power of thelth observation location can be obtained as follows:(23)P^l=∑m=K+12Mλl,m−σ^n,l2.Then, we can get the weighted cost function:(24)fNC−WSDFp,φ=argminp,φ∑l=1LP^lσ^n,l2UlnHblp,φ2.


### 3.2. NRD-WSDF

To reduce the complexity and improve the practicability of the algorithm, the idea of dimension reduction in [32] and [33] is introduced to remove the noncircular phase search dimension. Since slRt is a real vector, we can get(25)slRt=slR∗t.Thus, equation (11) can be rewritten as(26)zlt=rltrl∗t=AltΦAl∗tΦ∗Λslt+nltnl∗t=Hlp,φΛslt+nlct,where(27)Hlp,φ=AltΦAl∗tΦ∗=hlp1,φ1,…,hlpK,φK∈ℂ2M×K,(28)nlct=nltnl∗t∈ℂ2M×1,(29)hlpk,φk=alpke−jφkal∗pkejφk∈ℂ2M×1.Equation (29) is an extended steering vector, which contains the position information and noncircular phase information of the kth target. In order to separate the position information from the noncircular phase information, matrix conversion is performed as(30)hlpk,φk=alpke−jφkal∗pkejφk=alpk0M×10M×1al∗pke−jφkejφk=Cpkηφk,where 0N denotes the N dimensional zero array.(31)Cpk=alpk0M×10M×1al∗pk,ηφk=e−jφkejφk.Then, we can get the loss function of thelth observation location:(32)flp,φ=argminp,φP^lσ^n,l2ηHφJlpηφ.It is obvious that(33)flp,φ=e−jφflp,φejφ,=argminp,φe−jφP^lσ^n,l2ηHφJlpηφejφ=argminp,φP^lσ^n,l2gHφJlpgφ,where(34)gHφ=ηφejφ=1ej2φ,Jlp=CHpUlnUlnHCp.Equation (33) is a quadratic optimization problem for unknown parameter φ. Thus, we can reconstruct the optimization problem as [28](35)minp,φgHφJlpgφ,s.t.η¯Hgφ=1,where(36)η¯=10.In order to solve the above optimization problem, the Lagrange method is adopted, so the following functions are constructed:(37)Lp,φ=gHφJlpgφ−μη¯Hgφ−1,where μ in equation (37) is the Lagrange multiplier. Let the derivative of Lp,φ over ηφ be zero.(38)∂Lp,φ∂ηφ=2Jlpgφ+μη¯=0.Then, we can get(39)gφ=κJlp−1η¯.Sinceη¯Hgφ=1,(40)κ=1η¯HJlp−1η¯,gφ=Jlp−1η¯η¯HJlp−1η¯.Thus, equation (32) can be reduced to(41)flp=argminpP^lσ^n,l21η¯HJlp−1η¯.Finally, we can get the cost function of NRD-WSDF:(42)fNRD−WSDFp=minp∑l=1LP^lσ^n,l21η¯HCHpUlnUlnHCp−1η¯.Through spectral peak search, theK minimum points correspond to the target positions.


### 3.3. Main Steps of the Proposed Algorithm

The main steps of NRD-WSDF are summarized as follows:1.
Construct the NC-DPD model according to equation (10)(2)
Expand the received signal vector according to equation (11) and calculate the covariance matrix Rl considering equation (15). Then, we can obtain the noise subspace Uln by equation (16).(3)
Getσ^n,l2 and P^l from equation (22) and equation (23)(4)
Use equation (36) and equation (42) to get fNRD−WSDFp(5)
Through spectral peak search, theK minimum points correspond to the target positions


### 3.4. Summary of Algorithm Advantages

Advantages of NRD-WSDF are as follows.(1)
The estimation accuracy and the robustness of the proposed NRD-WSDF algorithm are better than that of two-step location technology, SDF technology, and NC-SDF method(2)
The proposed NRD-WSDF method has more degrees of freedom and can distinguish more targets compared with the two-step location technology and SDF algorithm(3)
The proposed NRD-WSDF algorithm can significantly reduce the complexity caused by the NC phase without losing the estimation performance


## 3.1. NC-WSDF

According to equation (8), equation (5) can be rewritten as(10)rlt=AlpΛΦslRt+nlt∈ℂM×1.On the basis of the noncircular characteristics in [21], the received signal vector can be extended to(11)zlt=rltrl∗t=AltΛsltAl∗tΛ∗sl∗t+nltnl∗t∈ℂ2M×1,where ·∗ denotes the conjugate.As(12)sl∗t=Φ∗slR∗t=Φ∗Φ−1slt=Φ∗2slt,where ·−1 stands for the inverse of matrix.Thus, equation (11) can be expressed as(13)zlt=AltAl∗tΦ∗Φ∗Λslt+nltnl∗t=BlpΛslt+nltnl∗t∈ℂ2M×1,where(14)Blp=AltAl∗tΦ∗Φ∗=blp1,φ1,blp2,φ2,…,blpK,φK∈ℂ2M×K,blpk,φk=alpkal∗pke−j2φk∈ℂ2M×1.Then, we can get the covariance matrix of signals received by thelth observation location:(15)Rl=1T∑t=1TzltzlHt.The eigenvalue decomposition (EVD) ofRl is carried out [28].(16)Rl=Uls,UlnΣlUls,UlnH.If we useλl,mm=1,2,…,2M to represent the eigenvalues of the lth observation location, which is sorted from large to small, the corresponding eigenvectors are represented by el,mm=1,2,…,2M. Then, in equation (16), signal subspace and noise subspace can be expressed as Uls=el,1,…,el,K and Uln=el,K+1,…,el,2M, respectively, and Σl=diagλl,1,λl,2,…,λl,2M, where diag⋅ represents the diagonal matrix.Then, we can get the cost function according to the idea of(17)fNC−SDFp,φ=argminp,φ∑l=1LUlnHblp,φ2.However, the cost function in equation (17) does not consider the influence of path propagation loss matrix Λ. In practice, the path propagation loss cannot be ignored [29]. As long as there is one with poor performance, the cost function is easily disturbed. Therefore, in order to balance the projection error, obtain higher location accuracy and higher level of robustness, and we assign weight as(18)fNC−WSDFp,φ=argminp,φ∑l=1LwlUlnHblp,φ2,where wl is the weight of the lth observation station.As higher SNR will lead to smaller error, we can use SNR to weight the projection result of each observation location. Therefore, we rewrite equation (15) by substituting equation (13) into equation (15):(19)Rl=1T∑t=1T∑k=1Kαl,k2Pkblp,φblHp,φ+σn2I2M×2M,where IN denotes the N dimensional unit array.Under the assumption that the noise power is stationary during the observation, it can be seen from equation (19) that the SNR of different base stations is proportional to αl,k2Pk, i.e., Pl,k, which is unknown in practice. However, the covariance matrix can be decomposed into SDH:(20)Rl=Rs+Rn=Blp,φdiagPl,1,…,Pl,KBlHp,φ+σn,l2I2M×2M.Under the same assumption, the eigenvalues of covariance matrix can be expressed as(21)λl,m=σs,m2+σn,l2,1≤m≤K,σn,l2,K+1≤m≤2M,where σs,m21≤m≤K are the larger nonzero eigenvalues of Rs, which represent the power of the received signal. Then, the noise power can be expressed as(22)σ^n,l2=12M−K∑m=K+12Mλl,m.According to the estimated noise power, the estimated signal power of thelth observation location can be obtained as follows:(23)P^l=∑m=K+12Mλl,m−σ^n,l2.Then, we can get the weighted cost function:(24)fNC−WSDFp,φ=argminp,φ∑l=1LP^lσ^n,l2UlnHblp,φ2.


## 3.2. NRD-WSDF

To reduce the complexity and improve the practicability of the algorithm, the idea of dimension reduction in [32] and [33] is introduced to remove the noncircular phase search dimension. Since slRt is a real vector, we can get(25)slRt=slR∗t.Thus, equation (11) can be rewritten as(26)zlt=rltrl∗t=AltΦAl∗tΦ∗Λslt+nltnl∗t=Hlp,φΛslt+nlct,where(27)Hlp,φ=AltΦAl∗tΦ∗=hlp1,φ1,…,hlpK,φK∈ℂ2M×K,(28)nlct=nltnl∗t∈ℂ2M×1,(29)hlpk,φk=alpke−jφkal∗pkejφk∈ℂ2M×1.Equation (29) is an extended steering vector, which contains the position information and noncircular phase information of the kth target. In order to separate the position information from the noncircular phase information, matrix conversion is performed as(30)hlpk,φk=alpke−jφkal∗pkejφk=alpk0M×10M×1al∗pke−jφkejφk=Cpkηφk,where 0N denotes the N dimensional zero array.(31)Cpk=alpk0M×10M×1al∗pk,ηφk=e−jφkejφk.Then, we can get the loss function of thelth observation location:(32)flp,φ=argminp,φP^lσ^n,l2ηHφJlpηφ.It is obvious that(33)flp,φ=e−jφflp,φejφ,=argminp,φe−jφP^lσ^n,l2ηHφJlpηφejφ=argminp,φP^lσ^n,l2gHφJlpgφ,where(34)gHφ=ηφejφ=1ej2φ,Jlp=CHpUlnUlnHCp.Equation (33) is a quadratic optimization problem for unknown parameter φ. Thus, we can reconstruct the optimization problem as [28](35)minp,φgHφJlpgφ,s.t.η¯Hgφ=1,where(36)η¯=10.In order to solve the above optimization problem, the Lagrange method is adopted, so the following functions are constructed:(37)Lp,φ=gHφJlpgφ−μη¯Hgφ−1,where μ in equation (37) is the Lagrange multiplier. Let the derivative of Lp,φ over ηφ be zero.(38)∂Lp,φ∂ηφ=2Jlpgφ+μη¯=0.Then, we can get(39)gφ=κJlp−1η¯.Sinceη¯Hgφ=1,(40)κ=1η¯HJlp−1η¯,gφ=Jlp−1η¯η¯HJlp−1η¯.Thus, equation (32) can be reduced to(41)flp=argminpP^lσ^n,l21η¯HJlp−1η¯.Finally, we can get the cost function of NRD-WSDF:(42)fNRD−WSDFp=minp∑l=1LP^lσ^n,l21η¯HCHpUlnUlnHCp−1η¯.Through spectral peak search, theK minimum points correspond to the target positions.


## 3.3. Main Steps of the Proposed Algorithm

The main steps of NRD-WSDF are summarized as follows:1.
Construct the NC-DPD model according to equation (10)(2)
Expand the received signal vector according to equation (11) and calculate the covariance matrix Rl considering equation (15). Then, we can obtain the noise subspace Uln by equation (16).(3)
Getσ^n,l2 and P^l from equation (22) and equation (23)(4)
Use equation (36) and equation (42) to get fNRD−WSDFp(5)
Through spectral peak search, theK minimum points correspond to the target positions


## 3.4. Summary of Algorithm Advantages

Advantages of NRD-WSDF are as follows.(1)
The estimation accuracy and the robustness of the proposed NRD-WSDF algorithm are better than that of two-step location technology, SDF technology, and NC-SDF method(2)
The proposed NRD-WSDF method has more degrees of freedom and can distinguish more targets compared with the two-step location technology and SDF algorithm(3)
The proposed NRD-WSDF algorithm can significantly reduce the complexity caused by the NC phase without losing the estimation performance


## 4. Performance Analysis

### 4.1. Complexity Analysis

The complexity of the algorithms is analyzed as follows: the number of array elements isM, the number of targets is K, and the number of snapshots is T. L represents the number of observation stations. Lx, Ly, Lθ, and Lφ stand for the number of search grids of x coordinates, y coordinates, angles, and NC phase. The NC-WSDF algorithm in this study is mainly composed of three steps: the calculation of covariance matrix, EVD, and the calculation of searching spectral function. The corresponding computational complexity is 4LTM2, 8LM3, and LLxLyLφ8M3−4M2K+4M2+2M. In contrast, the complexity of spectral function search for the proposed NRD-WSDF algorithm is LLxLy8M3−4M2K+8M2+8M+14, which only contains the x coordinates and the y coordinates. The weight calculation does not need complex multiplication. The complexity of different algorithms is given in Table 1.Table 1
Complexity of different algorithms.
AlgorithmComplexityTwo-stepOLM3+LT+LθM2−LθKM+LKLxLySDFOLTM2+LM3+LLxLyM2M−K+M2+MNC-SDFO4LTM2+8LM3+LLxLyLφ8M3−4M2K+4M2+2MNC-WSDFO4LTM2+8LM3+LLxLyLφ8M3−4M2K+4M2+2MNRD-WSDFO4LTM2+8LM3+LLxLy8M3−4M2K+8M2+8M+14The computational complexity of the algorithms in Table1 is shown in Figure 2 when Lx, Ly, Lθ, and Lφ take the same value and change from 1000 to 9000, K=3, L=5, T=200, M=10.Figure 2
Comparison of different algorithms in complexity.It can be seen from Figure2 that the proposed NRD-WSDF algorithm has higher computational complexity than the SDF algorithm and the two-step technology. However, since the search dimension of NC phase is removed, the complexity of the proposed NRD-WSDF method is significantly reduced compared with the NC-SDF technology and the NC-WSDF method.


### 4.2. Simulation and Discussion

We applied Monte Carlo experiments to verify the estimation performance of the proposed NRD-WSDF algorithm. The estimation performance of this algorithm is checked by root mean square error (RMSE) as(43)RMSE=1K∑k=1K1Mon∑mn=1Monx^k,mn−xk2+y^k,mn−yk2,where K denotes the number of targets, and Mon represents the number of Monte Carlo experiments. All the simulation parameters are given in Table 2. Besides, to express the heteroscedasticity of the observation station conveniently, we define(44)σerror2=1L∑l=1Lαl−α¯l2,where(45)αl=αl,1,αl,2,…,αl,KT∈ℝK×1,α¯l=α¯l,α¯l,…,α¯lT∈ℝK×1,α¯l=1K∑k=1Kαl,k.Table 2
Simulation parameters.
ParameterValueMnumber of antennas4Knumber of targets5Tnumber of snapshots10–500Lnumber of observation stations6Monnumber of Monte Carlo500SNRsignal−to−noise ratio0–20 dBσerror2heteroscedasticity10ptarget locations(−2900, −2900); (−2900, 2900); (0, −1000); (2900, -2900); (2900, 2900)φNC phase(10, 20, 30, 40, 50)uobservation locations(−6000, −9000); (−3600, −7000); (−1200, −10000); (1200, −8000); (3600, −11000); (6000, −12000)Simulation 1.
This section simulates the proposed algorithm in the underdetermined condition with different SNRs. There are 5 targets incident on the observation stations (M=4, σerror2=10T=200). The SNRs in Figure 3 and 4 are 0 dB and 20 dB, respectively. From the figures, the proposed NRD-WSDF algorithm can estimate the targets successfully at low SNR and the estimation performance becomes better with the increase of SNR.Figure 3
Location performance of NRD-WSDF (SNR=0 dB).Figure 4
Location performance of NRD-WSDF (SNR=20dB).Simulation 2.
This section simulates the proposed algorithm in the underdetermined condition with different snapshots. There are 5 targets incident on the observation stations (M=4, σerror2=10, SNR = 15 dB). The snapshots in Figure 5 and 6 are 10 and 200, respectively. From the figures, the proposed NRD-WSDF algorithm can estimate the targets successfully with small snapshot, and the estimation accuracy becomes higher with the increase of snapshot.Figure 5
Location performance of NRD-WSDF (snapshot=10).Figure 6
Location performance of NRD-WSDF (snapshot=200).Simulation 3.
Theoretically, the minimum number of array elementsM needed to distinguish K targets is M=K+2/2. This section simulates the proposed algorithm in the underdetermined condition with different array elements. There are 4 targets incident on the observation stations (T=200, σerror2=10, SNR = 25 dB). Their NC phase and positions are (10, 20, 30, 40) and ((−2900, −2900), (−2900, 2900), (2900, −2900), (2900, 2900)), respectively. The number of array elements in Figures 7 and 8 is 3 and 4, respectively. From the figures, the proposed NRD-WSDF algorithm can estimate the targets successfully with fewer array elements, and the estimation accuracy becomes higher with the increase of array elements.Figure 7
Spectral peak of NRD-WSDF (M=3).Figure 8
Spectral peak of NRD-WSDF (M=4).Simulation 4.
This section compares the RMSE performance of different algorithms with different SNRs under differentσerror2. There are 3 targets incident on the observation stations. Their NC phase and positions are (10, 20, 30) and ((−300, −300), (100, 100), (900, 900)), respectively. The other simulation parameters are given in Table 2. The estimation is simulated when σerror2 is 5 (Figure 9) and 50 (Figure 10). The simulation results show that the performance of the proposed NRD-WSDF algorithm is superior than two-step location technology (using ESPRIT technology and clustering algorithm [34]), SDF technology, and NC-SDF method. Compared with the algorithm before dimensionality reduction, the estimation accuracy of the proposed algorithm is close to it. Besides, the excellent performance of the NRD-WSDF algorithm is more prominent when the heteroscedasticity of observation stations increases.Figure 9
RMSE of different algorithms with different SNRs. (snapshot=100,σerror2=5).Figure 10
RMSE of different algorithms with different SNRs. (snapshot=100,σerror2=50).Simulation 5.
This section compares the RMSE performance of different algorithms with different snapshots under differentσerror2. The number of snapshots changed from 10 to 460. The estimation is simulated when σerror2 is 2 (Figure 11) and 50 (Figure 12). The SNR is 10 dB and the other parameters are the same as simulation 3. The simulation results show that the performance of the proposed NRD-WSDF algorithm is better than two-step location technology, SDF method, and NC-SDF technology. Compared with the algorithm before dimensionality reduction, the performance of the proposed algorithm is close to it. Besides, the excellent performance of the NRD-WSDF algorithm is more prominent when the heteroscedasticity of observation stations increases.Figure 11
RMSE of different algorithms with different snapshots (SNR=10dB,σerror2=2).Figure 12
RMSE of different algorithms with different snapshots (SNR=10dB,σerror2=50).


## 4.1. Complexity Analysis

The complexity of the algorithms is analyzed as follows: the number of array elements isM, the number of targets is K, and the number of snapshots is T. L represents the number of observation stations. Lx, Ly, Lθ, and Lφ stand for the number of search grids of x coordinates, y coordinates, angles, and NC phase. The NC-WSDF algorithm in this study is mainly composed of three steps: the calculation of covariance matrix, EVD, and the calculation of searching spectral function. The corresponding computational complexity is 4LTM2, 8LM3, and LLxLyLφ8M3−4M2K+4M2+2M. In contrast, the complexity of spectral function search for the proposed NRD-WSDF algorithm is LLxLy8M3−4M2K+8M2+8M+14, which only contains the x coordinates and the y coordinates. The weight calculation does not need complex multiplication. The complexity of different algorithms is given in Table 1.Table 1
Complexity of different algorithms.
AlgorithmComplexityTwo-stepOLM3+LT+LθM2−LθKM+LKLxLySDFOLTM2+LM3+LLxLyM2M−K+M2+MNC-SDFO4LTM2+8LM3+LLxLyLφ8M3−4M2K+4M2+2MNC-WSDFO4LTM2+8LM3+LLxLyLφ8M3−4M2K+4M2+2MNRD-WSDFO4LTM2+8LM3+LLxLy8M3−4M2K+8M2+8M+14The computational complexity of the algorithms in Table1 is shown in Figure 2 when Lx, Ly, Lθ, and Lφ take the same value and change from 1000 to 9000, K=3, L=5, T=200, M=10.Figure 2
Comparison of different algorithms in complexity.It can be seen from Figure2 that the proposed NRD-WSDF algorithm has higher computational complexity than the SDF algorithm and the two-step technology. However, since the search dimension of NC phase is removed, the complexity of the proposed NRD-WSDF method is significantly reduced compared with the NC-SDF technology and the NC-WSDF method.


## 4.2. Simulation and Discussion

We applied Monte Carlo experiments to verify the estimation performance of the proposed NRD-WSDF algorithm. The estimation performance of this algorithm is checked by root mean square error (RMSE) as(43)RMSE=1K∑k=1K1Mon∑mn=1Monx^k,mn−xk2+y^k,mn−yk2,where K denotes the number of targets, and Mon represents the number of Monte Carlo experiments. All the simulation parameters are given in Table 2. Besides, to express the heteroscedasticity of the observation station conveniently, we define(44)σerror2=1L∑l=1Lαl−α¯l2,where(45)αl=αl,1,αl,2,…,αl,KT∈ℝK×1,α¯l=α¯l,α¯l,…,α¯lT∈ℝK×1,α¯l=1K∑k=1Kαl,k.Table 2
Simulation parameters.
ParameterValueMnumber of antennas4Knumber of targets5Tnumber of snapshots10–500Lnumber of observation stations6Monnumber of Monte Carlo500SNRsignal−to−noise ratio0–20 dBσerror2heteroscedasticity10ptarget locations(−2900, −2900); (−2900, 2900); (0, −1000); (2900, -2900); (2900, 2900)φNC phase(10, 20, 30, 40, 50)uobservation locations(−6000, −9000); (−3600, −7000); (−1200, −10000); (1200, −8000); (3600, −11000); (6000, −12000)Simulation 1.
This section simulates the proposed algorithm in the underdetermined condition with different SNRs. There are 5 targets incident on the observation stations (M=4, σerror2=10T=200). The SNRs in Figure 3 and 4 are 0 dB and 20 dB, respectively. From the figures, the proposed NRD-WSDF algorithm can estimate the targets successfully at low SNR and the estimation performance becomes better with the increase of SNR.Figure 3
Location performance of NRD-WSDF (SNR=0 dB).Figure 4
Location performance of NRD-WSDF (SNR=20dB).Simulation 2.
This section simulates the proposed algorithm in the underdetermined condition with different snapshots. There are 5 targets incident on the observation stations (M=4, σerror2=10, SNR = 15 dB). The snapshots in Figure 5 and 6 are 10 and 200, respectively. From the figures, the proposed NRD-WSDF algorithm can estimate the targets successfully with small snapshot, and the estimation accuracy becomes higher with the increase of snapshot.Figure 5
Location performance of NRD-WSDF (snapshot=10).Figure 6
Location performance of NRD-WSDF (snapshot=200).Simulation 3.
Theoretically, the minimum number of array elementsM needed to distinguish K targets is M=K+2/2. This section simulates the proposed algorithm in the underdetermined condition with different array elements. There are 4 targets incident on the observation stations (T=200, σerror2=10, SNR = 25 dB). Their NC phase and positions are (10, 20, 30, 40) and ((−2900, −2900), (−2900, 2900), (2900, −2900), (2900, 2900)), respectively. The number of array elements in Figures 7 and 8 is 3 and 4, respectively. From the figures, the proposed NRD-WSDF algorithm can estimate the targets successfully with fewer array elements, and the estimation accuracy becomes higher with the increase of array elements.Figure 7
Spectral peak of NRD-WSDF (M=3).Figure 8
Spectral peak of NRD-WSDF (M=4).Simulation 4.
This section compares the RMSE performance of different algorithms with different SNRs under differentσerror2. There are 3 targets incident on the observation stations. Their NC phase and positions are (10, 20, 30) and ((−300, −300), (100, 100), (900, 900)), respectively. The other simulation parameters are given in Table 2. The estimation is simulated when σerror2 is 5 (Figure 9) and 50 (Figure 10). The simulation results show that the performance of the proposed NRD-WSDF algorithm is superior than two-step location technology (using ESPRIT technology and clustering algorithm [34]), SDF technology, and NC-SDF method. Compared with the algorithm before dimensionality reduction, the estimation accuracy of the proposed algorithm is close to it. Besides, the excellent performance of the NRD-WSDF algorithm is more prominent when the heteroscedasticity of observation stations increases.Figure 9
RMSE of different algorithms with different SNRs. (snapshot=100,σerror2=5).Figure 10
RMSE of different algorithms with different SNRs. (snapshot=100,σerror2=50).Simulation 5.
This section compares the RMSE performance of different algorithms with different snapshots under differentσerror2. The number of snapshots changed from 10 to 460. The estimation is simulated when σerror2 is 2 (Figure 11) and 50 (Figure 12). The SNR is 10 dB and the other parameters are the same as simulation 3. The simulation results show that the performance of the proposed NRD-WSDF algorithm is better than two-step location technology, SDF method, and NC-SDF technology. Compared with the algorithm before dimensionality reduction, the performance of the proposed algorithm is close to it. Besides, the excellent performance of the NRD-WSDF algorithm is more prominent when the heteroscedasticity of observation stations increases.Figure 11
RMSE of different algorithms with different snapshots (SNR=10dB,σerror2=2).Figure 12
RMSE of different algorithms with different snapshots (SNR=10dB,σerror2=50).


## 5. Conclusions

Existing DPD algorithms of NC signals with multiple arrays do not consider the impact of path propagation loss on the performance of the algorithms, which leads to unstable performance. Thus, we assign a weight to balance the error and obtain higher location accuracy with better robustness. Besides, the high computational complexity caused by NC phase reduces the practicability of the algorithm. So, we introduce the RD method to reduce the computational complexity. The performance analysis confirms that the NRD-WSDF method can effectively improve the location accuracy, get better robustness, and distinguish more targets compared with traditional two-step location technology and SDF technology. In addition, without losing the estimation performance, the proposed algorithm can significantly reduce the complexity caused by the NC phase search dimension.



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*Source: 8427575-2021-12-21.xml*