# Direct Position Determination of Noncircular Sources with Multiple Nested Arrays: Reduced Dimension Subspace Data Fusion

Authors: Yang Qian; Dalin Zhao; Haowei Zeng
Journal: Wireless Communications and Mobile Computing
 (2021)

---

## Abstract

Direct position determination (DPD) of noncircular (NC) sources with multiple nested arrays (NA) is investigated in this paper. Noncircular sources are used to expand the dimension of the received signal matrix, so the number of identifiable information sources and the accuracy of direct position determination are improved. Furthermore, nested array increases spatial degree of freedom. In this paper, the high-dimensional search problem of noncircular sources is investigated. Therefore, we propose algorithm dimension reduction subspace data fusion (RD-SDF) to reduce complexity and increase positioning accuracy. Simulation results show that the proposed RD-SDF algorithm for multiple nested arrays with noncircular sources has improved positioning accuracy with higher spatial degree of freedom than SDF, Capon, and two-step algorithms with uniform linear array and circular sources (CS).


---

## Body

## 1. Introduction

Nowadays, there are many research directions in wireless positioning technology. Traditional two-step location determination methods such as angle of arrival (AOA) and time difference of arrival (TDOA) are extensively used in navigation, seismic exploration, and industrial production [1–3]. Two-step location determination method estimates location parameters by the time, angle, and frequency difference [4]. In the process of positioning, the more intermediate processing steps are experienced, the more information is lost, and the subsequent positioning accuracy will be affected [5]. In order to solve the problem of two-step positioning, in recent years, direct position determination technology has been proposed [6]. In 2004, Weiss firstly proposed direct position determination [7]. This method obtains the target location from the observation data directly and need not any additional intermediate parameters [8–10]. Due to the direct use of observational data, the target information is fully utilized, and the parameter estimation steps are effectively reduced. So the positioning accuracy of the location can be greatly improved [11, 12].With the fast development of computer and communication technology in recent years, direct positioning technology is used more and more. At present, most of the research is using angle of the sources or time delay for DPD with single station or multiple stations [13]. The DPD algorithm for multiple arrays mostly uses uniform linear array, which leads to low positioning accuracy due to the restricted array aperture [14].Nowadays, there have been few reports about DPD methods making use of a nested array. In 2010, Pal and Vaidyanathan proposed a nested array structure, in which the more typical nested array consisted of two-stage subarrays [15]. Nested arrays increase spatial degree of freedom, so the number of identifiable information sources and the accuracy of direct position determination are improved [16–21]. Noncircular signal is a kind of signal that has been studied a lot now. Amplitude modulation signal, binary phase shift keying, and pulse amplitude modulation signal are indispensable signal forms in the modern communication system, and they all belong to noncircular source types. Therefore, the research of the NC source location algorithm has important practical significance [11, 22–25].In this paper, the NC characteristics of an emitter signal are used to expand the spatial information to obtain larger virtual array aperture and more recognizable sources. Nested arrays are introduced into observation stations to increase the spatial degree of freedom and more recognizable sources, and then, combined with spatial smoothing technology, the DPD algorithm in the form of SDF for NC sources is derived. However, the high-dimensional search caused by the NC phase greatly increases the complexity of the algorithm. Therefore, this paper introduces multiple nested arrays into observation stations and proposes reduced dimension subspace data fusion (RD-SDF) algorithm with noncircular sources [26, 27].We list the main contributions as follows:(1)
We introduce noncircular sources into the direct location algorithm model. We use noncircular sources to expand the dimension of the received signal matrix, so the number of identifiable information sources and the accuracy of direct position determination are improved.(2)
We introduce multiple nested arrays into multiple observation stations and combine with spatial smoothing technology. Furthermore, we smooth sum array and difference array and splice together to form a longer virtual array which increases spatial degree of freedom.(3)
We reduce the search dimension to lower the computational complexity. Furthermore, we lower the complexity of the algorithm to make sure of the estimation accuracy.The structure of this paper is as follows. In Section2, we introduce a few concepts of nested arrays and noncircular sources for DPD. In Section 3, we depict the proposed RD-SDF algorithm. In Section 4, we depict the performance analysis about the RD-SDF algorithm for nested arrays with noncircular sources. In Section 5, we simulate the proposed RD-SDF algorithm for nested arrays with noncircular sources. Finally, Section 6 summarizes this paper.Notations are as follows:•H denotes conjugate transpose, •∗ denotes the conjugate, and •T denotes transposition. The symbol vec• denotes the received covariance matrix virtualization, and symbol ⊗ denotes Kronecker product. E• denotes the mathematical expectation, and In denotes an n×n identity matrix.


## 2. Preliminaries

In this section, we introduce a few notions of nested array and the model of DPD. Then, we introduce a noncircular source model and characteristic.


### 2.1. Nested Array Model for Direct Positioning

We use a two-stage nested array withM=N1+N2 sensors for direct position determination. The intensive uniform liner subarray has N1 sensors in which interelement spacing is d1=d. Another sparse one has N2 sensors with d2=N1+1d, where d=λ/2, and λ is expressed as wavelength of the operating wave [20]. The array elements are set at L′=dL=l1,l2,⋯,lN, and L can be expressed as
(1)L=n10≤n1≤N1−1∪n2N1+1−11≤n2≤N2.Figure1 depicts a two-stage nested array, where N1=N2=3. Figure 2 describes the sum coarray and difference coarray. In Figure 2, continuous virtual elements are locating from −11d to 11d in the difference coarray with six physical sensors.Figure 1
Two-level nested array.Figure 2
Difference coarray and sum coarray.


### 2.2. Multiple Nested Array Positioning Model

We use the positioning scenario as shown in Figure3. Assuming that Q uncorrelated far-field narrow-band noncircular sources are in the two-dimensional X‐Y plane [10]. There are L two-stage nested arrays located at L stations. The target sources are located at pq=xq,yqTq=1,2,⋯Q and L stations are located at ul=xul,yulTl=1,2,⋯,L. Each observation station places M array elements.Figure 3
Multiple array combination positioning scene.Assuming that the signal envelope changes slowly, the envelope of each signal is approximately equal when it arrives at the observation. The complex envelope of thelthl=1,2,⋯,L observation corresponding to the kth (k=1,2,3.⋯K) sampling snapshot time can be expressed as [10].
(2)rlk=∑q=1Qalpqsl,qk+nlk,wheresl,qk denotes the signal waveform of the qth source at the kth sampling fast beat time of the lth observation station, alpq denotes the direction vector, and nlk denotes the noise vector of the antenna array of the lth observation station. It is determined by the arrival direction of the signal θlpq [10].
(3)θlpq=arctanxul1−pq1yul2−pq2,(4)alpq=1,e−j2πdsinθlpq,⋯,e−j2πM−1dsinθlpqT.Equation (2) can be expressed as [10]
(5)rlk=Alpslk+nlk,where
(6)Alp=alp1,alp2,⋯,alpQ,(7)slk=sl,1k,sl,2k,⋯,sl,QkT,(8)p=p1T,p2T,⋯,pQTT,(9)nlk=nl,1k,nl,2k,⋯,nl,MkT.


### 2.3. Noncircular Source Model

According to the second-order statistical properties, we can divide sources into circular sources and noncircular sources. For random vectorsslk of complex circular sources, covariance matrix EslkslHk≠0, but ellipse covariance matrix EslkslHk=0. But for noncircular sources, neither is zero [6]. For noncircular sources, the covariance matrix and elliptic covariance matrix satisfy
(10)EslkslHk=ρejφEslkslTk,where φ is the noncircular phase and ρ is the noncircular rate of the value in 0~1.The signal with a noncircular rate of 1 is called the maximum noncircular rate signal. For the sake of simplicity, this paper assumes that the noncircular rate is 1. According to reference [11], noncircular sources can be expressed as
(11)st=Φs0t,where
(12)Φ=e−jφ10⋯00e−jφ2⋯.....0.0e−jφQ,where s0t represents the real part of the signal.According to equation (11),equation (5) can be rewritten as
(13)rlk=AlpΦsl0k+nlk,where
(14)sl0k=sl,10k,sl,20k,⋯,sl,Q0kT.


## 2.1. Nested Array Model for Direct Positioning

We use a two-stage nested array withM=N1+N2 sensors for direct position determination. The intensive uniform liner subarray has N1 sensors in which interelement spacing is d1=d. Another sparse one has N2 sensors with d2=N1+1d, where d=λ/2, and λ is expressed as wavelength of the operating wave [20]. The array elements are set at L′=dL=l1,l2,⋯,lN, and L can be expressed as
(1)L=n10≤n1≤N1−1∪n2N1+1−11≤n2≤N2.Figure1 depicts a two-stage nested array, where N1=N2=3. Figure 2 describes the sum coarray and difference coarray. In Figure 2, continuous virtual elements are locating from −11d to 11d in the difference coarray with six physical sensors.Figure 1
Two-level nested array.Figure 2
Difference coarray and sum coarray.


## 2.2. Multiple Nested Array Positioning Model

We use the positioning scenario as shown in Figure3. Assuming that Q uncorrelated far-field narrow-band noncircular sources are in the two-dimensional X‐Y plane [10]. There are L two-stage nested arrays located at L stations. The target sources are located at pq=xq,yqTq=1,2,⋯Q and L stations are located at ul=xul,yulTl=1,2,⋯,L. Each observation station places M array elements.Figure 3
Multiple array combination positioning scene.Assuming that the signal envelope changes slowly, the envelope of each signal is approximately equal when it arrives at the observation. The complex envelope of thelthl=1,2,⋯,L observation corresponding to the kth (k=1,2,3.⋯K) sampling snapshot time can be expressed as [10].
(2)rlk=∑q=1Qalpqsl,qk+nlk,wheresl,qk denotes the signal waveform of the qth source at the kth sampling fast beat time of the lth observation station, alpq denotes the direction vector, and nlk denotes the noise vector of the antenna array of the lth observation station. It is determined by the arrival direction of the signal θlpq [10].
(3)θlpq=arctanxul1−pq1yul2−pq2,(4)alpq=1,e−j2πdsinθlpq,⋯,e−j2πM−1dsinθlpqT.Equation (2) can be expressed as [10]
(5)rlk=Alpslk+nlk,where
(6)Alp=alp1,alp2,⋯,alpQ,(7)slk=sl,1k,sl,2k,⋯,sl,QkT,(8)p=p1T,p2T,⋯,pQTT,(9)nlk=nl,1k,nl,2k,⋯,nl,MkT.


## 2.3. Noncircular Source Model

According to the second-order statistical properties, we can divide sources into circular sources and noncircular sources. For random vectorsslk of complex circular sources, covariance matrix EslkslHk≠0, but ellipse covariance matrix EslkslHk=0. But for noncircular sources, neither is zero [6]. For noncircular sources, the covariance matrix and elliptic covariance matrix satisfy
(10)EslkslHk=ρejφEslkslTk,where φ is the noncircular phase and ρ is the noncircular rate of the value in 0~1.The signal with a noncircular rate of 1 is called the maximum noncircular rate signal. For the sake of simplicity, this paper assumes that the noncircular rate is 1. According to reference [11], noncircular sources can be expressed as
(11)st=Φs0t,where
(12)Φ=e−jφ10⋯00e−jφ2⋯.....0.0e−jφQ,where s0t represents the real part of the signal.According to equation (11),equation (5) can be rewritten as
(13)rlk=AlpΦsl0k+nlk,where
(14)sl0k=sl,10k,sl,20k,⋯,sl,Q0kT.


## 3. The Proposed RD-SDF Algorithm

In this section, we describe the proposed RD-SDF algorithm steps and describe the covariance vectorization signal and spatial smoothing technology in detail.


### 3.1. Covariance Vectorization Signal

According to the noncircular feature of the signal, we construct the direct position determination model by nested arrays. Combined with the idea of SDF, the cost function of the subspace data fusion algorithm with noncircular sources is obtained. Considering that the algorithm needs to search for high spectral peaks, this paper introduces the idea of dimension reduction, removes the noncircular phase search dimension, and finally gets the result reduced dimension subspace data fusion (RD-SDF).The vector of the received signal can be extended by using the characteristic that the elliptic covariance of the noncircular sources is not zero [11]. We construct the data model
(15)zlk=r1kr1∗k=AlpslkAl∗psl∗k+nlknl∗k.It can be obtained from equation (11)
(16)sl∗k=Φ∗sl0k=Φ∗Φ−1slk=Φ∗2slk.Then, equation (15) can be expressed as
(17)zlk=AlpAl∗pΦ∗Φ∗slk+nlknl∗k=Blpslk+nlknl∗k,where
(18)Blp=AlpAlpΦ∗Φ∗=blp1,blp2,⋯blpQ,where
(19)blpq=alpalpej2φq.The covariance matrix of the received signal is(20)Rl=1K∑k=1KZlkZlHk=∑i=1qσl,q2blpqblHpq+σn2I,where σl,q2 represents the power of the qth transmitted source at the lth observation position and σn2 represents noise power. In order to take advantage of characteristics of a nested matrix, we vectorize the covariance matrix [22]
(21)zl=vecRl=vec∑iDσl,q2blpqblHpq+σn2I~=Hlpμ+σn2I~,where μ is the signal power vector and
(22)Hlp=bl∗p1⊗blp1,bl∗p2⊗blp2,⋯,bl∗pQ⊗blpQ,(23)I~=vecIM,where
(24)bl∗pl⊗blpl=alpqal∗pqej2φq∗⊗alpqal∗pqej2φq=al∗pq⊗alpqal∗pq⊗al∗pqej2φqalpq⊗alpqe−j2φqalpq⊗al∗pq=d1d2d3d4,where d1=al∗pq⊗alpq, d2=al∗pq⊗al∗pqej2φq, d3=alpq⊗alpqe−j2φq, and d4=alpq⊗al∗pq. In the above equation (24),al∗pq⊗alpq and alpq⊗al∗pq can be used to generate difference coarray (Diff I and Diff II). While al∗pq⊗al∗pqej2φq and alpq⊗alpqe−j2φq can be used to generate the sum coarray (SUM I and SUM II).The corresponding array structures of difference coarray 1, sum coarray 1, sum coarray 2, and difference coarray 2 are shown in Figure4. The array elements of Diff I and Diff II are distributed as −M1−1d,M1−1d, where M1=N1N2+N2. The continuous array elements (SCA I and SCA II) of SUM I and SUM II are distributed as follows −M2−1d,0 and 0,M2−1d, where M2=N1N2+N1+N2.Figure 4
Structure diagram of sum difference array.


### 3.2. Spatial Smoothing Technology and Noncircular Phase Removal

In Figure4, for continuous difference coarray DCA, we divide DCA into R1+1 subarrays with equal length and equal distance; each subarray has R1+1 elements.According to references [17, 22], R1=N1N2+N2−1, R2=0, andR3=N1N2+N1+N2−1. The received signal of the ithii=1,2,⋯,R1+1 subarray is ydi∈CR1+1×1 and can be expressed as
(25)ydi=H~dΨi−1γ+σn2u~i,whereu~i is the ith element and is 1, and the other elements are all vectors of 0. γ is single snapshot correlation signal. H~d=b~dp1,b~dp2,⋯,b~dpq is the direction matrix of the first smooth subarray. Its kth direction can be expressed as
(26)b∼dpq=1,e−jπsinθlpq,e−j2πsinθlpq,⋯,e−jR1πsinθlpqT,(27)Ψ=diagejπsinθlp1,ejπsinθlp2,⋯,ejπsinθlpQ.The received signals ofR1+1 subarrays are combined together, and it can get the received signal matrix after spatial smoothing Yd∈CR1+1×R1+1.
(28)Yd=yd1,yd2,⋯,ydR1+1=H~dγ,Ψγ,⋯,ΨR1γ+σn2IR1+1=H~dS~+σn2IR1+1,where S~=γ,Ψγ,⋯,ΨR1γ eliminates the equivalent signal matrix of correlation and Yd can be expressed as the received signals of the first spatial smoothing subarray SS-Diff. The array element position range is 0,R1.Then, the backward spatial smoothing is performed for SCA I and SCA II, respectively. In order to produce the same number of equivalent received signals, for continuous sum coarray, they are also divided into theR1+1 subarray. After division, the ithii=1,2,⋯,R1+1 received signals of the SCA I and SCA II, respectively, are
(29)ysi−=Hs−~Ψi−1γ,(30)ysi+=Hs+~Ψi−1γ,where Hs−~=bs−~p1,bs−~p2,⋯,bs−~pq is the direction matrix of the first spatial smoothing subarray of SCA I. The array element position of the SS-SCA I range is −R3−R1,0 and its qth direction vector can be expressed as
(31)bs−∼pq=ejR3−R1πsinθlpqej2φq,⋯,ejπsinθlpqej2φq,ej2φqT,where Hs+~=bs+~p1,bs+~p2,⋯,bs+~pq is the first spatial smoothing subarray of SCA II. The array element position of the SS-SCA II range is R1,R3 and its qth direction vector can be expressed as
(32)bs+∼pq=e−jR1πsinθlpqe−j2φq,e−jR1+1πsinθlpqe−j2φq,⋯,e−jR3πsinθlpqe−j2φqT.Combining the received signals of allR1+1 subarrays together, the equivalent received signal matrix of SS-SCA I and SS-SCA II can be obtained, respectively.
(33)Ys−=ys1−,ys2−,⋯,ysR1+1−=Hs−~S~,(34)Ys+=ys1+,ys2+,⋯,ysR1+1+=Hs+~S~.After smoothing, the received signal is composed ofYs−,Yd,Ys+,
(35)Y∼=Ys−YdYs+.After smoothing, we splice SS-DCA, SS-SCA I, and SS-SCA II together and connect three arrays end to end. A longer virtual array is formed. As shown in Figure5, a longer virtual array is formed.Figure 5
Array structure diagram corresponding to spatial smoothing.In order to lower the complexity and improve the practicability of the algorithm, this paper introduces the RD-SDF algorithm to reduce the search dimension and remove the noncircular phase. After smoothing, equation (36) denotes the qth direction
(36)b~pq,φq=ejR3−R1πsinθlpqej2φq..ejπsinθlpqej2φq−−−−−−−−−−1e−jπsinθlpq..e−jR1πsinθlpq−−−−−−−−−−e−jR1πsinθlpqe−j2φq...e−jR3πsinθlpqe−j2φq=ejR3−R1πsinθlpq⋯ejπsinθlpq11e−jπsinθlpq⋯e−jR1πsinθlpqe−jR1πsinθlpqe−jR1+1πsinθlpq⋯e−jR3πsinθlpqej2φq1e−j2φq=Θpqφφq.Therefore, separate the matrixφφq=ej2φq1e−j2φqT. Θpq is another matrix after decomposition. We construct matrix e=010T to reduce dimension and remove the noncircular phase.The cost function of noncircular direct location of nested array is(37)fRD‐SDFp=argmax∑l=1LeHΘHpElnElnHΘpe.


### 3.3. The Steps of the Proposed RD-SDF Algorithm

We list several steps about RD-SDF algorithm.Step1. Set up the noncircular sources and nested array model for DPD.Step2. Generate sum coarray and difference coarray and adopt vector and spatial smoothing technology.Step3. Calculate covariance matrix to obtain noise subspaceEln.Step4. The noncircular phase is removed by dimension reduction method and the cost function is constructedfRD‐SDFp.Step5. The target location search area is divided into several two-dimensional plane grids, and the cost function value corresponding to each grid point is calculated. The spectral peak searching value is the position estimation valuex∧q,y∧q.


## 3.1. Covariance Vectorization Signal

According to the noncircular feature of the signal, we construct the direct position determination model by nested arrays. Combined with the idea of SDF, the cost function of the subspace data fusion algorithm with noncircular sources is obtained. Considering that the algorithm needs to search for high spectral peaks, this paper introduces the idea of dimension reduction, removes the noncircular phase search dimension, and finally gets the result reduced dimension subspace data fusion (RD-SDF).The vector of the received signal can be extended by using the characteristic that the elliptic covariance of the noncircular sources is not zero [11]. We construct the data model
(15)zlk=r1kr1∗k=AlpslkAl∗psl∗k+nlknl∗k.It can be obtained from equation (11)
(16)sl∗k=Φ∗sl0k=Φ∗Φ−1slk=Φ∗2slk.Then, equation (15) can be expressed as
(17)zlk=AlpAl∗pΦ∗Φ∗slk+nlknl∗k=Blpslk+nlknl∗k,where
(18)Blp=AlpAlpΦ∗Φ∗=blp1,blp2,⋯blpQ,where
(19)blpq=alpalpej2φq.The covariance matrix of the received signal is(20)Rl=1K∑k=1KZlkZlHk=∑i=1qσl,q2blpqblHpq+σn2I,where σl,q2 represents the power of the qth transmitted source at the lth observation position and σn2 represents noise power. In order to take advantage of characteristics of a nested matrix, we vectorize the covariance matrix [22]
(21)zl=vecRl=vec∑iDσl,q2blpqblHpq+σn2I~=Hlpμ+σn2I~,where μ is the signal power vector and
(22)Hlp=bl∗p1⊗blp1,bl∗p2⊗blp2,⋯,bl∗pQ⊗blpQ,(23)I~=vecIM,where
(24)bl∗pl⊗blpl=alpqal∗pqej2φq∗⊗alpqal∗pqej2φq=al∗pq⊗alpqal∗pq⊗al∗pqej2φqalpq⊗alpqe−j2φqalpq⊗al∗pq=d1d2d3d4,where d1=al∗pq⊗alpq, d2=al∗pq⊗al∗pqej2φq, d3=alpq⊗alpqe−j2φq, and d4=alpq⊗al∗pq. In the above equation (24),al∗pq⊗alpq and alpq⊗al∗pq can be used to generate difference coarray (Diff I and Diff II). While al∗pq⊗al∗pqej2φq and alpq⊗alpqe−j2φq can be used to generate the sum coarray (SUM I and SUM II).The corresponding array structures of difference coarray 1, sum coarray 1, sum coarray 2, and difference coarray 2 are shown in Figure4. The array elements of Diff I and Diff II are distributed as −M1−1d,M1−1d, where M1=N1N2+N2. The continuous array elements (SCA I and SCA II) of SUM I and SUM II are distributed as follows −M2−1d,0 and 0,M2−1d, where M2=N1N2+N1+N2.Figure 4
Structure diagram of sum difference array.


## 3.2. Spatial Smoothing Technology and Noncircular Phase Removal

In Figure4, for continuous difference coarray DCA, we divide DCA into R1+1 subarrays with equal length and equal distance; each subarray has R1+1 elements.According to references [17, 22], R1=N1N2+N2−1, R2=0, andR3=N1N2+N1+N2−1. The received signal of the ithii=1,2,⋯,R1+1 subarray is ydi∈CR1+1×1 and can be expressed as
(25)ydi=H~dΨi−1γ+σn2u~i,whereu~i is the ith element and is 1, and the other elements are all vectors of 0. γ is single snapshot correlation signal. H~d=b~dp1,b~dp2,⋯,b~dpq is the direction matrix of the first smooth subarray. Its kth direction can be expressed as
(26)b∼dpq=1,e−jπsinθlpq,e−j2πsinθlpq,⋯,e−jR1πsinθlpqT,(27)Ψ=diagejπsinθlp1,ejπsinθlp2,⋯,ejπsinθlpQ.The received signals ofR1+1 subarrays are combined together, and it can get the received signal matrix after spatial smoothing Yd∈CR1+1×R1+1.
(28)Yd=yd1,yd2,⋯,ydR1+1=H~dγ,Ψγ,⋯,ΨR1γ+σn2IR1+1=H~dS~+σn2IR1+1,where S~=γ,Ψγ,⋯,ΨR1γ eliminates the equivalent signal matrix of correlation and Yd can be expressed as the received signals of the first spatial smoothing subarray SS-Diff. The array element position range is 0,R1.Then, the backward spatial smoothing is performed for SCA I and SCA II, respectively. In order to produce the same number of equivalent received signals, for continuous sum coarray, they are also divided into theR1+1 subarray. After division, the ithii=1,2,⋯,R1+1 received signals of the SCA I and SCA II, respectively, are
(29)ysi−=Hs−~Ψi−1γ,(30)ysi+=Hs+~Ψi−1γ,where Hs−~=bs−~p1,bs−~p2,⋯,bs−~pq is the direction matrix of the first spatial smoothing subarray of SCA I. The array element position of the SS-SCA I range is −R3−R1,0 and its qth direction vector can be expressed as
(31)bs−∼pq=ejR3−R1πsinθlpqej2φq,⋯,ejπsinθlpqej2φq,ej2φqT,where Hs+~=bs+~p1,bs+~p2,⋯,bs+~pq is the first spatial smoothing subarray of SCA II. The array element position of the SS-SCA II range is R1,R3 and its qth direction vector can be expressed as
(32)bs+∼pq=e−jR1πsinθlpqe−j2φq,e−jR1+1πsinθlpqe−j2φq,⋯,e−jR3πsinθlpqe−j2φqT.Combining the received signals of allR1+1 subarrays together, the equivalent received signal matrix of SS-SCA I and SS-SCA II can be obtained, respectively.
(33)Ys−=ys1−,ys2−,⋯,ysR1+1−=Hs−~S~,(34)Ys+=ys1+,ys2+,⋯,ysR1+1+=Hs+~S~.After smoothing, the received signal is composed ofYs−,Yd,Ys+,
(35)Y∼=Ys−YdYs+.After smoothing, we splice SS-DCA, SS-SCA I, and SS-SCA II together and connect three arrays end to end. A longer virtual array is formed. As shown in Figure5, a longer virtual array is formed.Figure 5
Array structure diagram corresponding to spatial smoothing.In order to lower the complexity and improve the practicability of the algorithm, this paper introduces the RD-SDF algorithm to reduce the search dimension and remove the noncircular phase. After smoothing, equation (36) denotes the qth direction
(36)b~pq,φq=ejR3−R1πsinθlpqej2φq..ejπsinθlpqej2φq−−−−−−−−−−1e−jπsinθlpq..e−jR1πsinθlpq−−−−−−−−−−e−jR1πsinθlpqe−j2φq...e−jR3πsinθlpqe−j2φq=ejR3−R1πsinθlpq⋯ejπsinθlpq11e−jπsinθlpq⋯e−jR1πsinθlpqe−jR1πsinθlpqe−jR1+1πsinθlpq⋯e−jR3πsinθlpqej2φq1e−j2φq=Θpqφφq.Therefore, separate the matrixφφq=ej2φq1e−j2φqT. Θpq is another matrix after decomposition. We construct matrix e=010T to reduce dimension and remove the noncircular phase.The cost function of noncircular direct location of nested array is(37)fRD‐SDFp=argmax∑l=1LeHΘHpElnElnHΘpe.


## 3.3. The Steps of the Proposed RD-SDF Algorithm

We list several steps about RD-SDF algorithm.Step1. Set up the noncircular sources and nested array model for DPD.Step2. Generate sum coarray and difference coarray and adopt vector and spatial smoothing technology.Step3. Calculate covariance matrix to obtain noise subspaceEln.Step4. The noncircular phase is removed by dimension reduction method and the cost function is constructedfRD‐SDFp.Step5. The target location search area is divided into several two-dimensional plane grids, and the cost function value corresponding to each grid point is calculated. The spectral peak searching value is the position estimation valuex∧q,y∧q.


## 4. Performance Analysis

In this section, we analyze the performance of the RD-SDF algorithm, the obtained spatial degree of freedom, and the computational complexity of the RD-SDF algorithm and depict the advantages of the proposed RD-SDF algorithm.


### 4.1. Achievable DOFs

We define thatM represents the number of array elements, N1 represents the number of intensive uniform liner subarray, and N2 represents the number of sparse liner subarray. After smoothing, three smoothing subarrays are spliced together to form a longer virtual array. The DOF of the proposed algorithm after spatial smoothing is N1N2+2N1+N2. Table 1 shows the DOF of the uniform linear array and nested array algorithm under circular and noncircular sources.Table 1
DOF of different arrays with different sources.
Different arrays with different sourcesDOFCircular sources for ULAMNC sources for ULA2MCircular sources for nested arrayN1N2+N2NC sources for nested arrayN1N2+2N1+N2This paper emulates the degree of freedom of algorithm with different arrays for different sources. It can be seen from the Figure6, the combination of noncircular sources and nested array improves the spatial degree of freedom and increases with the number of physical array elements. Therefore, it can identify more sources.Figure 6
Comparison of DOF of different signals and arrays.


### 4.2. Complexity Analysis

We define thatM denotes array element numbers, Q denotes source numbers, L denotes observation station numbers, and K denotes snapshot numbers. The X direction is divided into Lx equal parts, and Y direction is divided into Ly equal parts [10]. The noncircular phase is divided into Lφ. In this paper, the complexity of the algorithm before dimension reduction is mainly composed of covariance matrix calculation, eigenvalue decomposition, and search spectrum function calculation [12, 13].The computational complexity of the RD-SDF algorithm contains several aspects: the calculation of covariance matrix of receiving sourcesO4M2LK and the computational complexity of covariance after smoothing is ON1N2+N2+2N12N1N2+N2L. The eigenvalue decomposition of the covariance matrix of the receiving sources is ON1N2+N2+2N13L, and the computational complexity of spectral peak value after dimension reduction is OLLxLy(3N1N2+2N1+N22+9N1N2+2N1+N2+N1N2+2N1+N22N1N2+2N1+N2−Q+39.Complexity before dimension reduction is(38)O4M2LK+N1N2+N2+2N12N1N2+N2L+N1N2+N2+2N13L+LLxLyLφN1N2+2N1+N22+N1N2+2N1+N2+N1N2+2N1+N22N1N2+2N1+N2−Q.Complexity after dimension reduction is(39)O4M2LK+N1N2+N2+2N12N1N2+N2L+N1N2+N2+2N13L+LLxLy3N1N2+2N1+N22+9N1N2+2N1+N2+N1N2+2N1+N22N1N2+2N1+N2−Q+39.From the complexity, it can be seen that the complexity is significantly reduced after dimension reduction.


### 4.3. Advantages

We analyze advantages about the RD-SDF algorithm for nested arrays with noncircular sources.(1)
The proposed algorithm uses noncircular signal characteristics. Compared with the algorithm of circular sources and the uniform array, DOF has improved a lot.(2)
The high-dimensional search caused by the NC phase greatly increases the complexity of the algorithm. Based on this problem, we use the dimension reduction method to reduce complexity resulting from high-dimensional search. The computational complexity of the proposed RD-SDF algorithm is reduced.(3)
We compare the proposed algorithm with algorithms for uniform linear array and two-step positioning method, the positioning accuracy of the RD-SDF algorithm for a nested array with noncircular sources is greatly improved.(4)
We compare the proposed algorithm with algorithms for circular sources and two-step positioning method, the positioning accuracy of the proposed RD-SDF algorithm for nested arrays with noncircular sources is greatly improved.


## 4.1. Achievable DOFs

We define thatM represents the number of array elements, N1 represents the number of intensive uniform liner subarray, and N2 represents the number of sparse liner subarray. After smoothing, three smoothing subarrays are spliced together to form a longer virtual array. The DOF of the proposed algorithm after spatial smoothing is N1N2+2N1+N2. Table 1 shows the DOF of the uniform linear array and nested array algorithm under circular and noncircular sources.Table 1
DOF of different arrays with different sources.
Different arrays with different sourcesDOFCircular sources for ULAMNC sources for ULA2MCircular sources for nested arrayN1N2+N2NC sources for nested arrayN1N2+2N1+N2This paper emulates the degree of freedom of algorithm with different arrays for different sources. It can be seen from the Figure6, the combination of noncircular sources and nested array improves the spatial degree of freedom and increases with the number of physical array elements. Therefore, it can identify more sources.Figure 6
Comparison of DOF of different signals and arrays.


## 4.2. Complexity Analysis

We define thatM denotes array element numbers, Q denotes source numbers, L denotes observation station numbers, and K denotes snapshot numbers. The X direction is divided into Lx equal parts, and Y direction is divided into Ly equal parts [10]. The noncircular phase is divided into Lφ. In this paper, the complexity of the algorithm before dimension reduction is mainly composed of covariance matrix calculation, eigenvalue decomposition, and search spectrum function calculation [12, 13].The computational complexity of the RD-SDF algorithm contains several aspects: the calculation of covariance matrix of receiving sourcesO4M2LK and the computational complexity of covariance after smoothing is ON1N2+N2+2N12N1N2+N2L. The eigenvalue decomposition of the covariance matrix of the receiving sources is ON1N2+N2+2N13L, and the computational complexity of spectral peak value after dimension reduction is OLLxLy(3N1N2+2N1+N22+9N1N2+2N1+N2+N1N2+2N1+N22N1N2+2N1+N2−Q+39.Complexity before dimension reduction is(38)O4M2LK+N1N2+N2+2N12N1N2+N2L+N1N2+N2+2N13L+LLxLyLφN1N2+2N1+N22+N1N2+2N1+N2+N1N2+2N1+N22N1N2+2N1+N2−Q.Complexity after dimension reduction is(39)O4M2LK+N1N2+N2+2N12N1N2+N2L+N1N2+N2+2N13L+LLxLy3N1N2+2N1+N22+9N1N2+2N1+N2+N1N2+2N1+N22N1N2+2N1+N2−Q+39.From the complexity, it can be seen that the complexity is significantly reduced after dimension reduction.


## 4.3. Advantages

We analyze advantages about the RD-SDF algorithm for nested arrays with noncircular sources.(1)
The proposed algorithm uses noncircular signal characteristics. Compared with the algorithm of circular sources and the uniform array, DOF has improved a lot.(2)
The high-dimensional search caused by the NC phase greatly increases the complexity of the algorithm. Based on this problem, we use the dimension reduction method to reduce complexity resulting from high-dimensional search. The computational complexity of the proposed RD-SDF algorithm is reduced.(3)
We compare the proposed algorithm with algorithms for uniform linear array and two-step positioning method, the positioning accuracy of the RD-SDF algorithm for a nested array with noncircular sources is greatly improved.(4)
We compare the proposed algorithm with algorithms for circular sources and two-step positioning method, the positioning accuracy of the proposed RD-SDF algorithm for nested arrays with noncircular sources is greatly improved.


## 5. Simulation Results

In this section, we simulate the proposed RD-SDF algorithm for nested arrays with noncircular sources, obtain the spectral peak map and location map, and compare the performance of the RD-SDF algorithm with other algorithms for DPD.


### 5.1. Simulations Results versus Proposed RD-SDF Algorithm

Multiple targets areP1=−800m,800m,P2=0m,500m, and P3=800m,200m. The noncircular phase is π/6,π/4,π/3. There is a single target P4=500m,300m, and the noncircular phase is π/6 correspondingly. Multiple nested arrays are located at multiple observation stations and observation stations are U1=−1000m,−500m, U2=−500m,−500m, U3=0m,−500m, U4=500m,−500m, and U5=1000m,−500m. It can be seen from Figures 7 and 8 that the proposed algorithm can accurately locate a single target and multiple targets. Figure 9 shows the location map of the two-dimensional target.Figure 7
Single-target direct position.Figure 8
Multiple target direct position.Figure 9
Direct location of 2D target location.The estimation performance of the proposed RD-SDF algorithm is analyzed by calculating the root mean square error (RMSE), it can be denoted as [11].
(40)RMSE=1Q∑q=1Q1MC∑mc=1MCx∧q,mc−xq2+y∧q,mc−yq2,where MC is the number of Monte Carlo (MC) simulation test, Q is the number of target sources, x∧q,mc,y∧q,mc represents the estimated value of the location of the qth target in the mcth simulation experiment, and xq,yq is the real value of the qth target.


### 5.2. RMSE Results versus Comparison of Different Algorithms with Different Arrays

The nested array element number isN1,N2=3,3. The snapshot number is 300. Figure 10 shows the performance of the RD-SDF algorithm with different arrays for DPD. The performance of the RD-SDF algorithm for a nested array with noncircular sources is better than the algorithm of the RD-SDF algorithm for uniform array with noncircular sources. The performance of the RD-SDF algorithm for the nested array with noncircular sources is better than SDF, PM, and Capon algorithm [9–11] for uniform array with circular sources and the two-step localization algorithm. In Figure 10, CRB for the nested array with noncircular sources is better than CRB for a uniform array with noncircular sources.Figure 10
Comparison of different algorithms for the nested array and uniform array.


### 5.3. RMSE Results versus Comparison of Different Algorithms with Circular Sources and Noncircular Sources

The nested array element number isN1,N2=3,3. The snapshot number is 300. Figure 11 shows the comparison of algorithms with nested arrays under different signal types. The performance of the RD-SDF algorithm for noncircular sources is slightly better than the algorithm SDF for circular sources with nested array. Compared with the Capon algorithm and two-step positioning algorithm for circular sources with nested array, the proposed algorithm has better performance. CRB for nested array with noncircular sources is better than CRB for the nested array with circular sources.Figure 11
Algorithms comparison between circular sources and noncircular sources.


### 5.4. RMSE Results versus Comparison of Different Array Element Numbers

The snapshot number is 300. Figure12 shows the performance of two subarrays of nested array with the number of elements N1,N2=3,3,5,5,5,8. When we improve the array element numbers, the performance of RD-SDF for nested array with noncircular sources will be relatively good.Figure 12
Comparison of different element numbers with RD-SDF.


### 5.5. RMSE Results versus Comparison of Different Snapshot Numbers

This paper emulates the comparison of proposed algorithm by changing snapshot numbers. The number of snapshots was set to 100, 300, 500, and 800. The nested array element number isN1,N2=3,3. The performance of different snapshots is shown in Figure 13. With the increase of SNR, we can clearly see that to improve snapshot numbers, the performance of RD-SDF for nested arrays with noncircular sources will be better.Figure 13
Comparison of RD-SDF under different snapshot numbers.


## 5.1. Simulations Results versus Proposed RD-SDF Algorithm

Multiple targets areP1=−800m,800m,P2=0m,500m, and P3=800m,200m. The noncircular phase is π/6,π/4,π/3. There is a single target P4=500m,300m, and the noncircular phase is π/6 correspondingly. Multiple nested arrays are located at multiple observation stations and observation stations are U1=−1000m,−500m, U2=−500m,−500m, U3=0m,−500m, U4=500m,−500m, and U5=1000m,−500m. It can be seen from Figures 7 and 8 that the proposed algorithm can accurately locate a single target and multiple targets. Figure 9 shows the location map of the two-dimensional target.Figure 7
Single-target direct position.Figure 8
Multiple target direct position.Figure 9
Direct location of 2D target location.The estimation performance of the proposed RD-SDF algorithm is analyzed by calculating the root mean square error (RMSE), it can be denoted as [11].
(40)RMSE=1Q∑q=1Q1MC∑mc=1MCx∧q,mc−xq2+y∧q,mc−yq2,where MC is the number of Monte Carlo (MC) simulation test, Q is the number of target sources, x∧q,mc,y∧q,mc represents the estimated value of the location of the qth target in the mcth simulation experiment, and xq,yq is the real value of the qth target.


## 5.2. RMSE Results versus Comparison of Different Algorithms with Different Arrays

The nested array element number isN1,N2=3,3. The snapshot number is 300. Figure 10 shows the performance of the RD-SDF algorithm with different arrays for DPD. The performance of the RD-SDF algorithm for a nested array with noncircular sources is better than the algorithm of the RD-SDF algorithm for uniform array with noncircular sources. The performance of the RD-SDF algorithm for the nested array with noncircular sources is better than SDF, PM, and Capon algorithm [9–11] for uniform array with circular sources and the two-step localization algorithm. In Figure 10, CRB for the nested array with noncircular sources is better than CRB for a uniform array with noncircular sources.Figure 10
Comparison of different algorithms for the nested array and uniform array.


## 5.3. RMSE Results versus Comparison of Different Algorithms with Circular Sources and Noncircular Sources

The nested array element number isN1,N2=3,3. The snapshot number is 300. Figure 11 shows the comparison of algorithms with nested arrays under different signal types. The performance of the RD-SDF algorithm for noncircular sources is slightly better than the algorithm SDF for circular sources with nested array. Compared with the Capon algorithm and two-step positioning algorithm for circular sources with nested array, the proposed algorithm has better performance. CRB for nested array with noncircular sources is better than CRB for the nested array with circular sources.Figure 11
Algorithms comparison between circular sources and noncircular sources.


## 5.4. RMSE Results versus Comparison of Different Array Element Numbers

The snapshot number is 300. Figure12 shows the performance of two subarrays of nested array with the number of elements N1,N2=3,3,5,5,5,8. When we improve the array element numbers, the performance of RD-SDF for nested array with noncircular sources will be relatively good.Figure 12
Comparison of different element numbers with RD-SDF.


## 5.5. RMSE Results versus Comparison of Different Snapshot Numbers

This paper emulates the comparison of proposed algorithm by changing snapshot numbers. The number of snapshots was set to 100, 300, 500, and 800. The nested array element number isN1,N2=3,3. The performance of different snapshots is shown in Figure 13. With the increase of SNR, we can clearly see that to improve snapshot numbers, the performance of RD-SDF for nested arrays with noncircular sources will be better.Figure 13
Comparison of RD-SDF under different snapshot numbers.


## 6. Conclusion

In order to improve direct positioning accuracy and spatial degree of freedom with uniform linear array and circular sources for DPD. This paper uses multiple nested arrays with noncircular sources for DPD. We use the RD-SDF algorithm to lower the computational complexity and increase positioning accuracy. The complexity of high-dimensional search caused by NC phase is eliminated by data reconstruction. Simulation results show that the proposed RD-SDF algorithm for nested array with noncircular sources has a prominent promotion in positioning accuracy with higher spatial degree of freedom than the SDF, Capon, and two-step algorithms with uniform linear array and circular sources.



---

*Source: 9950518-2021-08-29.xml*