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International Journal of Antennas and Propagation
Volume 2012 (2012), Article ID 983092, 8 pages
http://dx.doi.org/10.1155/2012/983092
Research Article

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

1College of Electronic Information Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China
2Nanjing Telecommunication Technology Institute, Nanjing 210007, China

Received 19 May 2012; Revised 20 July 2012; Accepted 24 July 2012

Academic Editor: Jung-Chieh Chen

Copyright © 2012 Ling-yun Xu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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.

1. Introduction

Antenna arrays have been used in many fields such as radar, sonar, and mobile communications, and so forth, [16]. 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. Section 2 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 at 2𝑀1 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.

983092.fig.001
Figure 1: The structure of L-shaped array.

The received signal of 𝑀 elements in 𝑋-axis is shown 𝐱1(𝑡)=𝐀𝑥𝐬(𝑡)+𝐰1(𝑡),(1) 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)]𝑀×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 of 𝑀 elements in 𝑌-axis is denoted as 𝐱2(𝑡)=𝐀𝑦𝐬(𝑡)+𝐰2(𝑡),(2) where 𝐀𝑦=[𝐚𝑦(𝜃1,𝜙1,𝑓1),𝐚𝑦(𝜃2,𝜙2,𝑓2),,𝐚𝑦(𝜃𝐾,𝜙𝐾,𝑓𝐾)](𝑀1)×𝐾 is the 𝑌-axis direction matrix and 𝐚𝑦(𝜃𝑘,𝜙𝑘,𝑓𝑘)=[𝑣(𝜃𝑘,𝜙𝑘,𝑓𝑘),𝑣2(𝜃𝑘,𝜙𝑘,𝑓𝑘),, 𝑣𝑀1(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 𝐱𝐱(𝑡)=1𝐱2=𝐀𝑥𝐀𝑦𝐬(𝑡)+𝐰(𝑡)=𝐀𝐬(𝑡)+𝐰(𝑡),(3) where the direction matrix 𝐀 and noise matrix 𝐰(𝑡) are shown as follows: 𝐀𝐀=𝑥𝐀𝑦𝐰,𝐰(𝑡)=1𝐰(𝑡)2(𝑡).(4)𝐀𝑥 and 𝐀𝑦 are Vandermonde matrices. The delayed signal for (3) with 𝜏𝑝 can be denoted as 𝐱𝑡𝜏𝑝=𝐀𝐬𝑡𝜏𝑝+𝐰𝑡𝜏𝑝=𝐀Φ𝑝𝐬(𝑡)+𝐰𝑡𝜏𝑝,(5) where Φ𝑝=diag[𝑒𝑗2𝜋𝑓1𝜏𝑝,𝑒𝑗2𝜋𝑓2𝜏𝑝,,𝑒𝑗2𝜋𝑓𝐾𝜏𝑝].

We assume that channel state information is constant for 𝑁 symbols. The received signal of array antennas can be denoted as 𝐗𝑝=𝐀Φ𝑝𝐒+𝐖𝑝,(6) where 𝐖𝑝=[𝐰(𝑡1𝜏𝑝),𝐰(𝑡2𝜏𝑝),,𝐰(𝑡𝑁𝜏𝑝)],𝐒=[𝐬(𝑡1),𝐬(𝑡2),,𝐬(𝑡𝑁)].

Define the delay matrix as 𝑒Φ=111𝑗2𝜋𝑓1𝜏1𝑒𝑗2𝜋𝑓2𝜏1𝑒𝑗2𝜋𝑓𝐾𝜏1𝑒𝑗2𝜋𝑓1𝜏𝑃𝑒𝑗2𝜋𝑓2𝜏𝑃𝑒𝑗2𝜋𝑓𝐾𝜏𝑃(𝑃+1)×𝐾.(7) Equation (6) is also denoted as 𝐗𝑝=𝐀𝐷𝑝(Φ)𝐒+𝐖𝑝,𝑝=1,2,,𝑃+1,(8) 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 𝑓𝐀,𝐷1(Φ),,𝐷𝑃+1=(Φ),𝐒𝑃+1𝑝=1𝐗𝑝𝐀𝐷𝑝(Φ)𝐒2𝐹,(9) where 2𝐹 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, fix 𝐀 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)=𝑒𝑗2𝜋𝑓𝑘𝜏𝑝1,𝑘=1,2,,𝐾. Then (9) can be rewritten as 𝑓=𝑃+1𝑝=1𝐗𝑝𝐾𝑘=1𝐚𝑘𝛽𝑘(𝑝)𝐬𝑘2𝐹=𝑃+1𝑝=1𝐗tr𝑝𝐾𝑘=1𝐚𝑘𝛽𝑘(𝑝)𝐬𝑘𝐗𝐻𝑝𝐾𝑘=1𝐬𝐻𝑘𝛽𝑘(𝑝)𝐚𝐻𝑘.(10) Let the gradient of 𝑓 with respect to 𝛽𝑘(𝑝), be equal to zero, one obtains 𝐚𝐻𝑘𝐾𝑘=1𝐚𝑘𝛽𝑘(𝑝)𝐬𝑘𝐬𝐻𝑘𝐚𝐻𝑘𝐗𝑝𝐬𝐻𝑘=0,(11) for 𝑝=1,2,,𝑃+1 and 𝑘=1,2,,𝐾. The above equation yields 𝐚𝐻𝑘𝐀𝐷𝑝(Φ)𝐒𝐬𝐻𝑘=𝐚𝐻𝑘𝐗𝑝𝐬𝐻𝑘,𝑝=1,2,,𝑃+1,𝑘=1,2,,𝐾.(12) Let 𝜷(𝑝)=[𝛽1(𝑝),𝛽2(𝑝),𝛽𝐾(𝑝)]𝑇,𝐄=(𝐀𝐻𝐀)(𝐒𝐒𝐻), and 𝐝(𝑝)=[𝐚𝐻1𝐗𝑝𝐬𝐻1,,𝐚𝐻𝐾𝐗𝑝𝐬𝐻𝐾]𝑇, where denotes the Hadamard product. Then (12) can be rewritten in matrix form 𝐄𝜷(𝑝)=𝐝(𝑝),𝑝=1,2,,𝑃+1.(13) Hence, the estimate of 𝐷𝑝(Φ) can be easily computed as 𝐷𝑝𝐄(Φ)=diag{𝜷(𝑝)}=diag1𝐝,(𝑝)𝑝=1,2,,𝑃+1.(14) Secondly, let 𝐀 and {𝐷𝑝(Φ)}𝑃+1𝑝=1 be fixed. Differentiation of (9) with respect to 𝐒 yields the estimate of ̂𝐒̂𝐒𝐻=𝑃+1𝑝=1𝐗𝐻𝑝𝐀𝐷𝐻𝑝(Φ)𝑃+1𝑝=1𝐷𝑝(Φ)𝐀𝐻𝐀𝐷𝐻𝑝(Φ)1.(15) Thirdly fix {𝐷𝑝(Φ)}𝑃+1𝑝=1, and 𝐒. Let the gradient of (9) with respect to 𝐀 be equal to zero, we have 𝑃+1𝑝=1𝐀𝐷𝑝(Φ)𝐒𝐗𝑝𝐒𝐻𝐷𝐻𝑝(Φ)=0,(16) which gives the estimate of 𝐀𝐀=𝑃+1𝑝=1𝐗𝑝𝐒𝐻𝐷𝐻𝑝(Φ)𝑃+1𝑝=1𝐷𝑝(Φ)𝐒𝐒𝐻𝐷𝐻𝑝(Φ)1.(17) Based on the above analysis, the ILS algorithm is summarized as follows.

Let the initial values 𝐀0=𝐀(2𝑀1)×𝐾 and ̂𝐒0=𝐒𝐾×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 𝐀𝑖1 and ̂𝐒𝑖1, for 𝑝=1,2,,𝑃+1.

Step 2. Calculate ̂𝐒𝑖 in (15) using {𝐷𝑝(Φ)𝑖}𝑃+1𝑝=1 and 𝐀𝑖1.

Step 3. Calculate 𝐀𝑖 in (17) using ̂𝐒𝑖 and {𝐷𝑝(Φ)𝑖}𝑃+1𝑝=1.

𝐀𝛿=𝑖𝐀𝑖12𝐹 with a threshold 𝜉(0<𝜉1); if 𝛿𝑖<𝜉, then stop; otherwise, go to Step 1.

The basic idea of the iterative least square algorithm is to minimize the cost function 𝑓(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=1𝐗𝑝𝐀𝐷𝑝(Φ)𝐒2𝐹0. Hence, the proposed algorithm is convergent, only 20 iterations are required to achieve convergence for this iterative algorithm (also shown in Figure 2).

983092.fig.002
Figure 2: Performance of algorithmic convergence.

We obtain the estimated matrices 𝐀=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 matrix 𝐀 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(𝑀1)𝑓𝑖𝜏𝑃]𝑇. 𝐠(𝑓𝑖) is the 𝑖th column of the delay matrix. We get 𝐡=angle(𝐠(𝑓𝑖))=[0,2𝜋𝑓𝑖𝜏1,,2𝜋𝑓𝑖(𝑀1)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 ̂𝐡𝐏𝐂=, where 1110𝐏=2𝜋𝜏12𝜋(𝑀1)𝜏𝑃𝑐,𝐂=0𝑓𝑖.(18) The least squares solution for 𝐂 is 𝐏𝐂=𝑇𝐏1𝐏𝑇̂𝐡.(19) 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 𝐀1.The last 𝑀1 rows form sub-matrix 𝐀2, the kth column of the sub-matrix 𝐀1 and 𝐀2 is ̂𝐚1(𝜃𝑘,𝜙𝑘,𝑓𝑘) and ̂𝐚2(𝜃𝑘,𝜙𝑘,𝑓𝑘), respectively. Then normalized sequence is processed to attain ̂𝐠1,̂𝐠2̂𝐠1̂𝐚=angle1𝜃𝑘,𝜙𝑘,𝑓𝑘=0,2𝜋𝑑cos𝜙𝑘sin𝜃𝑘𝑓𝑘𝑐,,2𝜋𝑑(𝑀1)cos𝜙𝑘sin𝜃𝑘𝑓𝑘𝑐𝑇=0,𝑚𝑓𝑘𝑐,,(𝑀1)𝑚𝑓𝑘𝑐𝑇,̂𝐠2̂𝐚=angle2𝜃𝑘,𝜙𝑘,𝑓𝑘=2𝜋𝑑sin𝜙𝑘sin𝜃𝑘𝑓𝑘𝑐,,2𝜋𝑑(𝑀1)sin𝜙𝑘sin𝜃𝑘𝑓𝑘𝑐𝑇=𝑛𝑓𝑘𝑐(,,𝑀1)𝑛𝑓𝑘𝑐𝑇,(20) where 𝑚=2𝜋𝑑sin𝜃𝑘cos𝜙𝑘, 𝑛=2𝜋𝑑sin𝜃𝑘sin𝜙𝑘. Least squares fitting is 𝐏1𝐂1=̂𝐠1, where 𝐏1=1𝑓10𝑘𝑐1𝑓(𝑀1)𝑘𝑐,𝐂1=𝑚0𝑚,(21) where 𝑓𝑘 is the estimated frequency through (19) and 𝑚 is the estimated value of 2𝜋𝑑sin𝜃𝑘cos𝜙𝑘. The LS solution to 𝑚 is 𝑚0=𝐏𝑚𝑇1𝐏11𝐏𝑇1̂𝐠1.(22) Similarly, 𝐏2𝐂2=̂𝐠2, where 𝐏2=1𝑓𝑘𝑐12𝑓𝑘𝑐1𝑓(𝑀1)𝑘𝑐,𝐂2=𝑛0̂𝑛,(23) in which ̂𝑛 is the estimated value of 2𝜋𝑑sin𝜙𝑘sin𝜃𝑘. The LS solution to 𝑛 is 𝑛0=𝐏̂𝑛𝑇2𝐏21𝐏𝑇2̂𝐠2.(24) The 2D-DOAs are estimated via ̂𝜃𝑖=sin1𝑚2+̂𝑛2,𝜙2𝜋𝑑𝑖=tan1̂𝑛.𝑚(25)

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)[(2𝑀1)𝑁𝐾+𝑁𝐾2+2𝐾2(2𝑀1)]+(𝑃+2)𝐾3+𝑁𝐾2]); the complexity for computing 𝐀 in (17) is 𝑂((𝑃+1)[(2𝑀1)𝑁𝐾+2𝑁𝐾2+(2𝑀1)𝐾2+𝐾3]+𝐾3+(2𝑀1)𝐾2). Therefore, the complexity per iteration in the ILS algorithm is 𝑂((𝑃+1)[2(2𝑀1)𝑁𝐾+4𝑁𝐾2+(8𝑀3)𝐾2+3𝐾3+2𝑀𝑁]+2𝐾3+(2𝑀+𝑁1)𝐾2); when the ILS algorithm converge within 20 steps, the total computational complexity of the proposed algorithm is about 𝑂(20((𝑃+1)[2(2𝑀1)𝑁𝐾+4𝑁𝐾2+(8𝑀3)𝐾2+3𝐾3+2𝑀𝑁]+2𝐾3+(2𝑀+𝑁1)𝐾2)), and ESPRIT algorithm in [11] needs 𝑂[(2𝑀1)2(𝑃+1)2𝑁+(2𝑀1)3(𝑃+1)3+(2𝑀1)𝑃𝐾2+2(𝑀1)𝐾2+6𝐾3], and PM algorithm in [13] needs 𝑂(𝑁𝐾[(2𝑀1)(𝑃+1)𝐾]+(4𝑀𝐾3)𝐾2+4𝐾3).

According to [17], we derive the CRB for 2D-DOA and frequency estimation with L-shaped array 𝜎CRB=22𝑁Re(𝐃𝐻𝚷𝐴𝐃)𝐏𝑇1,(26) where stands for Hadamard product. 𝚷𝐴=𝐈𝐀(𝐀𝐻𝐀)1𝐀𝐻, 𝐏=(1/𝑁)𝑁𝑛=1𝐬(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: 1RMSE=𝐾𝐾𝑘=11500500𝑚=1||𝛼𝑚𝑘𝛼𝑜𝑘||2+||𝛽𝑚𝑘𝛽𝑜𝑘||2,(27) 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 1RMSE=𝐾𝐾𝑘=11500500𝑚=1||𝛾𝑚𝑘𝛾𝑜𝑘||2,(28) 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. Define 𝐀𝛿=𝑖𝐀𝑖12𝐹, where 𝐀𝑖 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.

983092.fig.003
Figure 3: Elevation angle and azimuth angle scatter.
983092.fig.004
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 Simulation 1. 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.

983092.fig.005
Figure 5: Angle estimation performance comparison.
983092.fig.006
Figure 6: Frequency estimation performance comparison.

Simulation 4. Our proposed algorithm performance with 𝑃=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.

983092.fig.007
Figure 7: Angle estimation with different antennas.
983092.fig.008
Figure 8: Frequency estimation with different antennas.

Simulation 5. The performance of our proposed algorithm with 𝑃=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.

983092.fig.009
Figure 9: Frequency estimation with different sources.
983092.fig.0010
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.

Acknowledgments

This work is supported by National Nature Science Foundation of China (nos. 61179006, 60801052), Jiangsu Planned Projects for Postdoctoral Research Funds (no. 1201039C), Open project of key laboratory of underwater acoustic communication and marine information technology (Xiamen University) and Nanjing University of Aeronautics and Astronautics Research Funding (nos. NP2011036, NZ2012010, kfjj120115, kfjj20110215). The authors wish to thank the anonymous reviewers for their valuable suggestions on improving this paper.

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