# Joint 2D-DOA and Frequency Estimation for L-Shaped Array Using Iterative Least Squares Method

Authors: Ling-yun Xu; Xiao-fei Zhang; Zong-ze Xu; Miao Yu
Journal: International Journal of Antennas and Propagation
 (2012)

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## Abstract

We introduce an iterative least squares method (ILS) for estimating the 2D-DOA and frequency based on L-shaped array. The ILS iteratively finds direction matrix and delay matrix, then 2D-DOA and frequency can be obtained by the least squares method. Without spectral peak searching and pairing, this algorithm works well and pairs the parameters automatically. Moreover, our algorithm has better performance than conventional ESPRIT algorithm and propagator method. The useful behavior of the proposed algorithm is verified by simulations.


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## Body

## 1.  Introduction

Antenna arrays have been used in many fields such as radar, sonar, and mobile communications, and so forth, [1–6]. The direction of arrival and frequency estimation of signals impinging on an array of sensors have received considerable attention in the field of array signal processing. For example, these parameters can be applied to locate the mobiles and allocate pilot tones in space division multiple access (SDMA) systems. Furthermore, a precise estimation of these parameters is helpful to attain a better channel estimate and thus enhances the system performance [7]. Uniform linear arrays for estimation of wave arrival have been studied extensively, and they contain maximum likelihood (ML) [8], multiple signal classification (MUSIC) algorithm [9, 10], estimation of signal parameters via rotational invariance techniques (ESPRIT) [11, 12], propagator method (PM) [13], and so forth.The ML method is often applicable but might be computationally prohibitive. ESPRIT and MUSIC algorithms are based on signal subspace and have better parameter estimation performance. The main advantages of MUSIC/ESPRIT are the high-resolution estimates of direction of arrivals (DOAs) and frequencies, while the computational effort compared to ML method is significantly reduced. MUSIC requires multiple dimensional spectral peak searching, and it is the search that is still computationally expensive. The primary computational advantage of ESPRIT is that it eliminates the search procedure inherent. ESPRIT method requires eigenvalue decomposition (EVD) to the cross spectral matrix or singular value decomposition (SVD) to the received data. Reference [13] presented propagator method to estimate the angle and frequency with uniform linear array. Propagator method has low complexity, but its parameters’ estimation performance is less than ESPRIT algorithm. Compared with uniform linear array, L-shaped array can identify 2D-DOA and is very close to actual situation [14, 15]. And thus we propose a novel iterative-based angle and frequency estimation algorithm with L-shaped array which can achieve better performance than ESPRIT [11] and propagator method [13]. The proposed algorithm can obtain automatically paired 2D-DOA and frequency estimation. This method is an iterative algorithm, which does not need EVD or SVD, and only requires fewer iterations for convergence. The useful behavior of the proposed algorithm is verified by simulations.The remainder of this paper is structured as follows. Section2 develops data model. Section 3 deals with algorithmic issues. Section 4 presents simulation results, and Section 5 summarizes our conclusions.Denote
We denote by(·)* the complex conjugation, by (·)T the matrix transpose, and by (·)H the matrix conjugate transpose. diag{·} is to construct a diagonal matrix.


## 2. Data Model

We consider an L-shaped array with sensors at2M-1 different locations as shown in Figure 1. A uniform linear array containing M elements is located in Y-axis, and the other uniform linear array containing M elements is located in X-axis. We suppose that there are K narrowband uncorrelated signal sources impinge on the L-shaped array with (ϕk,θk),k=1,2,…,K, where ϕk,θk are the elevation angle and the azimuth angle of the kth source, respectively.Figure 1
The structure of L-shaped array.The received signal ofM elements in X-axis is shown
(1)x1(t)=Axs(t)+w1(t),
where s(t)=[s1(t),s2(t),…,sK(t)]T is the source matrix, w1(t) is the received additive white Gaussian noise, and Ax=[ax(θ1,ϕ1,f1),ax(θ2,ϕ2,f2),…, ax(θK,ϕK,fK)]∈ℂM×K is the X-axis direction matrix, and ax(θk,ϕk,fk)=[1,u(θk,ϕk,fk),…,uM(θk,ϕk,fk)]T,u(θk,ϕk,fk)=exp[-j2πdcosθksinϕkfk/c].The received signal ofM elements in Y-axis is denoted as
(2)x2(t)=Ays(t)+w2(t),
where Ay=[ay(θ1,ϕ1,f1),ay(θ2,ϕ2,f2),…,ay(θK,ϕK,fK)]∈ℂ(M-1)×K is the Y-axis direction matrix and ay(θk,ϕk,fk)=[v(θk,ϕk,fk),v2(θk,ϕk,fk),…, vM-1(θk,ϕk,fk)]T,  v(θk,ϕk,fk)=exp[-j2πdsinθksinϕkfk/c],  w2(t) is the received white Gaussian noise.The received signal of the L-shaped array antennas can be denoted as(3)x(t)=[x1x2]=[AxAy]s(t)+w(t)=As(t)+w(t),
where the direction matrix A and noise matrix w(t) are shown as follows:
(4)A=[AxAy],w(t)=[w1(t)w2(t)].Ax and Ay are Vandermonde matrices. The delayed signal for (3) with τp can be denoted as
(5)x(t-τp)=As(t-τp)+w(t-τp)=AΦps(t)+w(t-τp),
where Φp=diag[e-j2πf1τp,e-j2πf2τp,…,e-j2πfKτp].We assume that channel state information is constant forN symbols. The received signal of array antennas can be denoted as
(6)Xp=AΦpS+Wp,
where Wp=[w(t1-τp),w(t2-τp),…,w(tN-τp)],S=[s(t1),s(t2),…,s(tN)].Define the delay matrix as(7)Φ=[11⋯1e-j2πf1τ1e-j2πf2τ1⋯e-j2πfKτ1⋮⋮⋱⋮e-j2πf1τPe-j2πf2τP⋯e-j2πfKτP]∈ℂ(P+1)×K.
Equation (6) is also denoted as
(8)Xp=ADp(Φ)S+Wp,p=1,2,…,P+1,
where Dm(·) is to extract the mth row of its matrix argument and constructs a diagonal matrix out of it.


## 3. Iterative Least Squares Method

It can be seen from (8) that if the direction matrices A and Φ are obtained, then the estimates of 2D-DOA and frequency follow immediately based on L-shaped array geometries. Hence our main focus in this section is on estimating A and Φ.


### 3.1. ILS Cost Function

The cost function can be constructed via the least squares criterion and given by(9)f(A,D1(Φ),…,DP+1(Φ),S)=∑p=1P+1∥Xp-ADp(Φ)S∥F2,
where ∥∥F2 denotes the Frobenius norm. Interestingly, if any two subsets of A,{Dp(Φ)}p=1P+1 and S are fixed, the remaining parameter subset can be easily obtained by minimizing (9).Firstly, fixA and S. Recall that A=[a1,a2,…,aK], S=[s1,s2,…,sK]T, and Dp(Φ)=diag{β1(p),β2(p),…βK(p)}, where βk(p)=e-j2πfkτp-1,k=1,2,…,K. Then (9) can be rewritten as
(10)f=∑p=1P+1∥Xp-∑k=1Kakβk(p)sk∥F2=∑p=1P+1tr{[Xp-∑k=1Kakβk(p)sk][XpH-∑k=1KskHβk*(p)akH]}.
Let the gradient of f with respect to βk*(p), be equal to zero, one obtains
(11)akH[∑k=1Kakβk(p)sk]skH-akHXpskH=0,
for p=1,2,…,P+1 and k=1,2,…,K. The above equation yields
(12)akHADp(Φ)SskH=akHXpskH,p=1,2,…,P+1,k=1,2,…,K.
Let β(p)=[β1(p),β2(p),…βK(p)]T,E=(AHA)⊕(SSH)*, and d(p)=[a1HXps1H,…,aKHXpsKH]T, where ⊕ denotes the Hadamard product. Then (12) can be rewritten in matrix form
(13)Eβ(p)=d(p),p=1,2,…,P+1.
Hence, the estimate of Dp(Φ) can be easily computed as
(14)Dp(Φ)=diag{β(p)}=diag{E-1d(p)},p=1,2,…,P+1.
Secondly, let A and {Dp(Φ)}p=1P+1 be fixed. Differentiation of (9) with respect to S yields the estimate of S^(15)S^H=[∑p=1P+1XpHADpH(Φ)][∑p=1P+1Dp(Φ)AHADpH(Φ)]-1.
Thirdly fix {Dp(Φ)}p=1P+1, and S. Let the gradient of (9) with respect to A be equal to zero, we have
(16)∑p=1P+1[ADp(Φ)S-Xp]SHDpH(Φ)=0,
which gives the estimate of A(17)A^=[∑p=1P+1XpSHDpH(Φ)][∑p=1P+1Dp(Φ)SSHDpH(Φ)]-1.
Based on the above analysis, the ILS algorithm is summarized as follows.Let the initial valuesA^0=A-∈ℂ(2M-1)×K and S^0=S-∈ℂK×N, the elements of A- and S- are equal to 1. For iteration index i=1,2,…, do the following three steps.Step 1.
Calculate{Dp(Φ)i} in (14) using A^i-1 and S^i-1, for p=1,2,…,P+1.Step 2.
CalculateS^i in (15) using {Dp(Φ)i}p=1P+1 and A^i-1.Step 3.
CalculateA^i in (17) using S^i and {Dp(Φ)i}p=1P+1.δ
=
∥
A
^
i
-
A
^
i
-
1
∥
F
2 with a threshold ξ(0<ξ≤1); if δi<ξ, then stop; otherwise, go to Step 1.The basic idea of the iterative least square algorithm is to minimize the cost functionf(A,D1(Φ),…,DP+1(Φ),S) with respect to A, {Dp(Φ)}p=1P+1, and S by using the cyclic minimization (CM) technique [16], which monotonically decreases the cost function. Furthermore, we know ∑p=1P+1∥Xp-ADp(Φ)S∥F2≥0. Hence, the proposed algorithm is convergent, only 20 iterations are required to achieve convergence for this iterative algorithm (also shown in Figure 2).Figure 2
Performance of algorithmic convergence.We obtain the estimated matricesA^=AΠΔ1, Φ^=ΦΠΔ2, and S^=SΠΔ3, where Π is a permutation matrix and Δ1,Δ2,Δ3 stand for diagonal scaling matrices satisfying Δ1Δ2Δ3=IK. That is to say, the ith column of A^ corresponds to ith column of Φ^ and S^. So our algorithm can estimate 2D-DOA and frequency estimation without extra pairing.


### 3.2. Frequency and 2D-DOA Estimation

We use iterative least squares method to attain the direction matrixA and the delay matrix Φ, and then angle and frequency are estimated according to the least squares principle and Vandermonde characteristic. Define the delay vector g(fi) as g(fi)=[1,ej2πfiτ1,…,ej2π(M-1)fiτP]T. g(fi) is the ith column of the delay matrix. We get h=angle(g(fi))=[0,2πfiτ1,…,2πfi(M-1)τp]T, where angle(·) is  to get phase part of a complex number. The estimated frequency vector g^(fi) (the ith column of the estimated delay matrix Φ) is processed through normalization, which also resolves the scale ambiguity, and then normalized sequence is processed to attain h^ according to above processing. Finally, we use leaset squares principle to estimatefi. Least squares fitting is PC=h^, where
(18)P=[11⋮102πτ1⋮2π(M-1)τP],C=[c0fi].
The least squares solution for C is
(19)C^=(PTP)-1PTh^.
Since Ax,Ay, and the delay matrix Φ are with Vandermonde characteristic, the direction matrix A has a quasi-Vandermonde structure, so we can use the same method with frequency estimation method to estimate DOAs. The estimated receive array steer vector a^(θk,ϕk,fk) (the kth column of the estimated matrix A^) is processed through normalization, which also resolves the scale ambiguity, then direction estimation matrix is divided into two submatrices, the first M rows form submatrices A^1.The last M-1 rows form sub-matrix A^2, the kth column of the sub-matrix A^1 and A^2 is a^1(θk,ϕk,fk) and a^2(θk,ϕk,fk), respectively. Then normalized sequence is processed to attain g^1,g^2(20)g^1=-angle(a^1(θk,ϕk,fk))=[0,2πdcosϕksinθkfkc,…,2πd(M-1)cosϕksinθkfkc]T=[0,mfkc,…,(M-1)mfkc]T,g^2=-angle(a^2(θk,ϕk,fk))=[2πdsinϕksinθkfkc,…,2πd(M-1)sinϕksinθkfkc]T=[nfkc,…,(M-1)nfkc]T,
where m=2πdsinθkcosϕk, n=2πdsinθksinϕk. Least squares fitting is P1C1=g^1, where
(21)P1=[101f^kc⋮⋮1(M-1)f^kc],C1=[m0m^],
where f^k is the estimated frequency through (19) and m^ is the estimated value of 2πdsinθkcosϕk. The LS solution to m is
(22)[m0m^]=(P1TP1)-1P1Tg^1.
Similarly, P2C2=g^2, where
(23)P2=[1f^kc12f^kc⋮⋮1(M-1)f^kc],C2=[n0n^],
in which n^ is the estimated value of 2πdsinϕksinθk. The LS solution to n is
(24)[n0n^]=(P2TP2)-1P2Tg^2.
The 2D-DOAs are estimated via
(25)θ^i=sin-1(m^2+n^22πd),ϕ^i=tan-1(n^m^).


### 3.3. Complexity Analysis and CRB

In contrast to ESPRIT algorithms in [11] and PM algorithm in [13], our algorithm has a heavier computational load, in our method, computing {Dp(Φ)}p=1P+1 in (14) takes a complexity of O((P+1)[2MK2+NK2+K3+2MN]), the complexity for computing S^ in (15) is O((P+1)[(2M-1)NK+NK2+2K2(2M-1)]+(P+2)K3+NK2]); the complexity for computing A^ in (17) is O((P+1)[(2M-1)NK+2NK2+(2M-1)K2+K3]+K3+(2M-1)K2). Therefore, the complexity per iteration in the ILS algorithm is O((P+1)[2(2M-1)NK+4NK2+(8M-3)K2+3K3+2MN]+2K3+(2M+N-1)K2); when the ILS algorithm converge within 20 steps, the total computational complexity of the proposed algorithm is about O(20((P+1)[2(2M-1)NK+4NK2+(8M-3)K2+3K3+2MN]+2K3+(2M+N-1)K2)), and ESPRIT algorithm in [11] needs O[(2M-1)2(P+1)2N+(2M-1)3(P+1)3+(2M-1)PK2+2(M-1)K2+6K3], and PM algorithm in [13] needs O(NK[(2M-1)(P+1)-K]+(4M-K-3)K2+4K3).According to [17], we derive the CRB for 2D-DOA and frequency estimation with L-shaped array
(26)CRB=σ22N{Re[(DHΠA⊥D)⊕PT]}-1,
where ⊕ stands for Hadamard product. ΠA⊥=I-A(AHA)-1AH, P=(1/N)∑n=1Ns(tn)sH(tn), D=[d1,d2,…,dK,f1,f2,…,fK,g1,g2,…,gK], dk=∂ak/∂ϕk, fk=∂ak/∂θk, gk=∂ak/∂fk.The advantages of the proposed algorithm can be summarized as follows. (1) The 2D-DOA and frequency can be paired automatically. (2) The proposed algorithm has better angle and frequency estimation performance than ESPRIT algorithm and PM algorithm.


## 3.1. ILS Cost Function

The cost function can be constructed via the least squares criterion and given by(9)f(A,D1(Φ),…,DP+1(Φ),S)=∑p=1P+1∥Xp-ADp(Φ)S∥F2,
where ∥∥F2 denotes the Frobenius norm. Interestingly, if any two subsets of A,{Dp(Φ)}p=1P+1 and S are fixed, the remaining parameter subset can be easily obtained by minimizing (9).Firstly, fixA and S. Recall that A=[a1,a2,…,aK], S=[s1,s2,…,sK]T, and Dp(Φ)=diag{β1(p),β2(p),…βK(p)}, where βk(p)=e-j2πfkτp-1,k=1,2,…,K. Then (9) can be rewritten as
(10)f=∑p=1P+1∥Xp-∑k=1Kakβk(p)sk∥F2=∑p=1P+1tr{[Xp-∑k=1Kakβk(p)sk][XpH-∑k=1KskHβk*(p)akH]}.
Let the gradient of f with respect to βk*(p), be equal to zero, one obtains
(11)akH[∑k=1Kakβk(p)sk]skH-akHXpskH=0,
for p=1,2,…,P+1 and k=1,2,…,K. The above equation yields
(12)akHADp(Φ)SskH=akHXpskH,p=1,2,…,P+1,k=1,2,…,K.
Let β(p)=[β1(p),β2(p),…βK(p)]T,E=(AHA)⊕(SSH)*, and d(p)=[a1HXps1H,…,aKHXpsKH]T, where ⊕ denotes the Hadamard product. Then (12) can be rewritten in matrix form
(13)Eβ(p)=d(p),p=1,2,…,P+1.
Hence, the estimate of Dp(Φ) can be easily computed as
(14)Dp(Φ)=diag{β(p)}=diag{E-1d(p)},p=1,2,…,P+1.
Secondly, let A and {Dp(Φ)}p=1P+1 be fixed. Differentiation of (9) with respect to S yields the estimate of S^(15)S^H=[∑p=1P+1XpHADpH(Φ)][∑p=1P+1Dp(Φ)AHADpH(Φ)]-1.
Thirdly fix {Dp(Φ)}p=1P+1, and S. Let the gradient of (9) with respect to A be equal to zero, we have
(16)∑p=1P+1[ADp(Φ)S-Xp]SHDpH(Φ)=0,
which gives the estimate of A(17)A^=[∑p=1P+1XpSHDpH(Φ)][∑p=1P+1Dp(Φ)SSHDpH(Φ)]-1.
Based on the above analysis, the ILS algorithm is summarized as follows.Let the initial valuesA^0=A-∈ℂ(2M-1)×K and S^0=S-∈ℂK×N, the elements of A- and S- are equal to 1. For iteration index i=1,2,…, do the following three steps.Step 1.
Calculate{Dp(Φ)i} in (14) using A^i-1 and S^i-1, for p=1,2,…,P+1.Step 2.
CalculateS^i in (15) using {Dp(Φ)i}p=1P+1 and A^i-1.Step 3.
CalculateA^i in (17) using S^i and {Dp(Φ)i}p=1P+1.δ
=
∥
A
^
i
-
A
^
i
-
1
∥
F
2 with a threshold ξ(0<ξ≤1); if δi<ξ, then stop; otherwise, go to Step 1.The basic idea of the iterative least square algorithm is to minimize the cost functionf(A,D1(Φ),…,DP+1(Φ),S) with respect to A, {Dp(Φ)}p=1P+1, and S by using the cyclic minimization (CM) technique [16], which monotonically decreases the cost function. Furthermore, we know ∑p=1P+1∥Xp-ADp(Φ)S∥F2≥0. Hence, the proposed algorithm is convergent, only 20 iterations are required to achieve convergence for this iterative algorithm (also shown in Figure 2).Figure 2
Performance of algorithmic convergence.We obtain the estimated matricesA^=AΠΔ1, Φ^=ΦΠΔ2, and S^=SΠΔ3, where Π is a permutation matrix and Δ1,Δ2,Δ3 stand for diagonal scaling matrices satisfying Δ1Δ2Δ3=IK. That is to say, the ith column of A^ corresponds to ith column of Φ^ and S^. So our algorithm can estimate 2D-DOA and frequency estimation without extra pairing.


## 3.2. Frequency and 2D-DOA Estimation

We use iterative least squares method to attain the direction matrixA and the delay matrix Φ, and then angle and frequency are estimated according to the least squares principle and Vandermonde characteristic. Define the delay vector g(fi) as g(fi)=[1,ej2πfiτ1,…,ej2π(M-1)fiτP]T. g(fi) is the ith column of the delay matrix. We get h=angle(g(fi))=[0,2πfiτ1,…,2πfi(M-1)τp]T, where angle(·) is  to get phase part of a complex number. The estimated frequency vector g^(fi) (the ith column of the estimated delay matrix Φ) is processed through normalization, which also resolves the scale ambiguity, and then normalized sequence is processed to attain h^ according to above processing. Finally, we use leaset squares principle to estimatefi. Least squares fitting is PC=h^, where
(18)P=[11⋮102πτ1⋮2π(M-1)τP],C=[c0fi].
The least squares solution for C is
(19)C^=(PTP)-1PTh^.
Since Ax,Ay, and the delay matrix Φ are with Vandermonde characteristic, the direction matrix A has a quasi-Vandermonde structure, so we can use the same method with frequency estimation method to estimate DOAs. The estimated receive array steer vector a^(θk,ϕk,fk) (the kth column of the estimated matrix A^) is processed through normalization, which also resolves the scale ambiguity, then direction estimation matrix is divided into two submatrices, the first M rows form submatrices A^1.The last M-1 rows form sub-matrix A^2, the kth column of the sub-matrix A^1 and A^2 is a^1(θk,ϕk,fk) and a^2(θk,ϕk,fk), respectively. Then normalized sequence is processed to attain g^1,g^2(20)g^1=-angle(a^1(θk,ϕk,fk))=[0,2πdcosϕksinθkfkc,…,2πd(M-1)cosϕksinθkfkc]T=[0,mfkc,…,(M-1)mfkc]T,g^2=-angle(a^2(θk,ϕk,fk))=[2πdsinϕksinθkfkc,…,2πd(M-1)sinϕksinθkfkc]T=[nfkc,…,(M-1)nfkc]T,
where m=2πdsinθkcosϕk, n=2πdsinθksinϕk. Least squares fitting is P1C1=g^1, where
(21)P1=[101f^kc⋮⋮1(M-1)f^kc],C1=[m0m^],
where f^k is the estimated frequency through (19) and m^ is the estimated value of 2πdsinθkcosϕk. The LS solution to m is
(22)[m0m^]=(P1TP1)-1P1Tg^1.
Similarly, P2C2=g^2, where
(23)P2=[1f^kc12f^kc⋮⋮1(M-1)f^kc],C2=[n0n^],
in which n^ is the estimated value of 2πdsinϕksinθk. The LS solution to n is
(24)[n0n^]=(P2TP2)-1P2Tg^2.
The 2D-DOAs are estimated via
(25)θ^i=sin-1(m^2+n^22πd),ϕ^i=tan-1(n^m^).


## 3.3. Complexity Analysis and CRB

In contrast to ESPRIT algorithms in [11] and PM algorithm in [13], our algorithm has a heavier computational load, in our method, computing {Dp(Φ)}p=1P+1 in (14) takes a complexity of O((P+1)[2MK2+NK2+K3+2MN]), the complexity for computing S^ in (15) is O((P+1)[(2M-1)NK+NK2+2K2(2M-1)]+(P+2)K3+NK2]); the complexity for computing A^ in (17) is O((P+1)[(2M-1)NK+2NK2+(2M-1)K2+K3]+K3+(2M-1)K2). Therefore, the complexity per iteration in the ILS algorithm is O((P+1)[2(2M-1)NK+4NK2+(8M-3)K2+3K3+2MN]+2K3+(2M+N-1)K2); when the ILS algorithm converge within 20 steps, the total computational complexity of the proposed algorithm is about O(20((P+1)[2(2M-1)NK+4NK2+(8M-3)K2+3K3+2MN]+2K3+(2M+N-1)K2)), and ESPRIT algorithm in [11] needs O[(2M-1)2(P+1)2N+(2M-1)3(P+1)3+(2M-1)PK2+2(M-1)K2+6K3], and PM algorithm in [13] needs O(NK[(2M-1)(P+1)-K]+(4M-K-3)K2+4K3).According to [17], we derive the CRB for 2D-DOA and frequency estimation with L-shaped array
(26)CRB=σ22N{Re[(DHΠA⊥D)⊕PT]}-1,
where ⊕ stands for Hadamard product. ΠA⊥=I-A(AHA)-1AH, P=(1/N)∑n=1Ns(tn)sH(tn), D=[d1,d2,…,dK,f1,f2,…,fK,g1,g2,…,gK], dk=∂ak/∂ϕk, fk=∂ak/∂θk, gk=∂ak/∂fk.The advantages of the proposed algorithm can be summarized as follows. (1) The 2D-DOA and frequency can be paired automatically. (2) The proposed algorithm has better angle and frequency estimation performance than ESPRIT algorithm and PM algorithm.


## 4. Simulation Results

We present Monte Carlo simulations that are to assess joint 2D-DOA and frequency estimation performance of the proposed algorithm. The number of Monte Carlo trials is 500. There are three signals impinging on L-shaped array at (15°,10°, 1.2 MHz), (25°,20°, 1.6 MHz), (35°,30°, 1.8 MHz), respectively. We consider an L-shaped array with 15 sensors, which is shown in Figure 1. The spacing d between the adjacent elements in each uniform linear array is smaller than the half smallest wavelength of the incoming signals. Define root mean squared error (RMSE) of angle:
(27)RMSE=1K∑k=1K1500∑m=1500|αmk-αok|2+|βmk-βok|2,
where αm is the estimated elevation angle, αo is the perfect elevation angle, βm is the estimated azimuth angle, and βo is the perfect azimuth angle.Define the root mean squared error: (RMSE) of frequency(28)RMSE=1K∑k=1K1500∑m=1500|γmk-γok|2,
where rm is the estimated frequency and ro is the perfect frequency. Note that N is the number of snapshots; K is the number of sources; P is the number of delay outputs for received signal of array antennas; L is the number of antennas.Simulation 1.
We first investigate the convergence performance of the proposed algorithm. Defineδ=∥A^i-A^i-1∥F2, where A^i is the estimated matrix A of ith iteration. Figure 2 presents the algorithm convergence performance of the proposed algorithm with P=4,K=3,L=15, and SNR = 20 dB. Figure 2 shows that the proposed algorithm needs 20 iterations or so to achieve convergence.Simulation 2.
The performance of our proposed algorithm is investigated.P=4,K=3,L=15, and N=500 is in this simulation. Figure 3 shows 2D-DOA estimation of our proposed algorithm at SNR = 20 dB, and Figure 4 presents elevation angle and frequency scatter of our proposed algorithm at SNR = 20 dB. From Figures 3 and 4 we find that our proposed algorithm works well.Figure 3
Elevation angle and azimuth angle scatter.Figure 4
Elevation angle and frequency scatter.Simulation 3.
We compare our proposed algorithm with ESPRIT algorithm, MUSIC algorithm, propagator method and CRB. The simulation parameters are retained as Simulation1. From Figures 5 and 6 we find that our algorithm has much better angle and frequency estimation performance than ESPRIT algorithm and PM algorithm, and it has a very close angle and frequency estimation performance to MUSIC algorithm, but MUSIC algorithm needs three-dimensional spectral peak searching and the complexity is much larger than our proposed algorithm.Figure 5
Angle estimation performance comparison.Figure 6
Frequency estimation performance comparison.Simulation 4.
Our proposed algorithm performance withP=4,K=3,N=500 and different values of L is investigated. L is set to 13, 15, and 17 in this simulation. It is indicated from Figures 7 and 8 that 2D-DOA estimation performance of our algorithm is improved with the number of antennas increasing. When the number of antennas increases, our algorithm has higher receive diversity.Figure 7
Angle estimation with different antennas.Figure 8
Frequency estimation with different antennas.Simulation 5.
The performance of our proposed algorithm withP=4,L=15,N=500 and different values of K is investigated. K is set to 2, 3, 4 in this simulation. It is indicated from Figures 9 and 10 that 2D-DOA and frequency estimation performance of our algorithm degrade with the increasing of the source K.Figure 9
Frequency estimation with different sources.Figure 10
Frequency estimation with different sources.


## 5. Conclusions

In this paper, we develop a novel method for joint 2D-DOA and frequency estimation based on L-shaped array using iterative least squares technique. Without spectral peak searching and pairing, this algorithm works well. Furthermore, our algorithm has much better 2D-DOA and frequency estimation performance than conventional ESPRIT algorithm and PM algorithm, and it has a very close 2D-DOA and frequency estimation performance to MUSIC algorithm. The useful behavior of the proposed algorithm is verified by simulations.



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*Source: 983092-2012-09-12.xml*