Vol.23 No.1

JOURNAL OF ELECTRONICS (CHINA)

January 2006

DOA ESTIMATION FOR WIDEBAND SOURCES BASED ON UCA Yang Liming

Zhang Hou

1

Yang Xiaorong

(The Missile Institute, Air Force Engineering University, Sanyuan 713800, China) Abstract A new Direction Of Arrival (DOA) estimation algorithm for wideband sources based on Uniform Circular Array (UCA) is presented via analyzing wideband performance of the general ESPRIT. The algorithm effectively improves the wideband performance of ESPRIT based on the interpolation principium and UCA-ESPRIT. The simulated results by computer demonstrate its efficiency. Key words Wideband; Uniform Circular Array (UCA); Direction Of Arrival (DOA)

I. Introduction Dirctions Of Arrival (DOA) estimation for wideband sources develops into oneof the most important techniques in modern Electronic War (EW). The DOA estimation technique is paid more and more attention and lots of reserch results are presented[1−7].There are mainly two types of algorithms for wideband DOA estimation: one is based on the method of dividing the whole bandwidth into several narrow bands which are not overlapped. The combination of the results of DOA estimation for different bands is the DOA estimation of wideband sources. The weakpoint of the technique is that it is the so-called noncoherent processing techniques and cannot be applied to DOA estimation of the coherrent sources. The other one is the coherent processing techniques which is based on the use of focusing matrices. It combines the covariances of the narrowband components into a wideband matrix, to which the subspace processing techniques is applied by the eigenvalues decomposition for the DOA estimation. The main embarrassment of this technique is to derive the focusing matrix which is developped quickly. And the focusing matrix must be a unitary matrix in order to assure the noise subspace structure after the transformation. Following that, the interpolation array approach is applied to the wideband sources DOA estimation which obtains the wideband sources covariance matries by the interpolation of the real array. The approch is also used here. It is well known that the performance of Uniform Circular Array(UCA) for DOA estimation of wideband sources is superior to that of Uniform Linear Array(ULA) because of the unique steering vectors of UCA for different frequences. In the letter, 1

Manuscript received date: December 27, 2004; revised date: March 22, 2005. Communication author: Yang Liming, born in 1976, male, Ph. D. student. The Missile Institute, Air Force Engineering University, Sanyuan 713800, China. [email protected].

the improved UCA-ESPRIT[7] algorithm is introduced for the wideband sources, and the application of ESPRIT to UCA can be figured out.

II. Wideband Signal Model of UCA Consider a UCA composed of N isotropic elements, shown as Fig.1. Let K wideband sources whose incident angles are (θ1 , ϕ1 ),(θ2 , ϕ2 )"(θK , ϕK ) respectively impinge on this array, the output signal of array at time t can be discribed by X (t ) = AS (t ) + N (t ) (1) where matrix A =[a (θ1 , ϕ1 ), ", a (θK , ϕK )] is the array mainfold matrix; a (θi , ϕi ) =[e j 2π fR sin θi cos(ϕi −γ1 ) / c , e j 2 π fR sin θi cos(ϕi −γ 2 ) / c ,", e j 2π fR sin θi cos(ϕi −γ N ) / c ]T , i = 1, 2,", N ; γ n = 2πn / N , n = 1, 2,", N , represents the location of elements; f L < f < f H is the frequency of the wideband signal; S (t )= [ s1(t ), s2 (t ),", sK (t )]T and N (t ) = [n1 (t ), n2 (t ),", nN (t )]T are the signal vector and the noise vector respectively, and the noise is zero-mean spatially white noise whose variance is σ2.

Fig.1 Geometry of uniform circular array

III. UCA-ESPRIT Algorithm[7] As we all know that the general ESPRIT algorithm is used only for the ULA[5], and it needs amendments for the applicability to the UCA.The beamformer FrH is employed to make the transformation from element space to beam space, and the array-out signal (narrowband) can be written as y (t ) = FrH AS (t ) + FrH N (t ) = BS (t ) + N (t ) (2) where B = FrH A is the array mainfold matrix after the transformation; and

YANG et al. DOA Estimation for Wideband Sources Based on UCA

FrH = W H C vV H

(3)

where: C v = diag( j −M ,", j −1 , j 0 , j1 ,", j M ). while the matrix V = N [ w−M ,", w0 ,", wM ] composed of vectors wm = (1/ N )[1, e jmγ1 , e jmγ2 ,", e jmγ N ]T , m = −M ,", M a sets columns, M ≈ k0 r = 2πr / λ is the highest excited mode of the UCA. Another factor of Eq.(3) is W = 1/ M [v (α−M ), ", v (α0 ),", v (αM )], αi = 2πi / M ', i = −M ,", M , M ' = 2M + 1, and the columns of W is v (ϕ ) = [e− jM ϕ ,", e− jϕ , e− j 0 , e jϕ , e jM ϕ ]T . The beamspace is real after the transformation and the real-valued eigvalue Decomposition(EVD) can be performed which is required by UCA-ESPRIT. The output covariance matrix now can be written as Ry = E[ y (t ) y H (t )] = BPB T + σ 2 I with σ 2 I as the white noise covariance matrix. The real part of the matrix is R = Re{Ry } = B Re( P ) B T + σ 2 I . To perform the real-valued EVD of the matrix R, we can get the matrices S and G which span the signal and noise space respectively by the number of sources. The matrix S can be written as S = BT , where T is a real nonsingular matrix. Then, another beamformer F0H = C 0WFrH is employed to transfer the beamspace mainfold to the form required by UCA- ESPRIT, where C 0 = diag{(−1) M ,",(−1),1, 1,",1} and W is defined as Eq.(3). Three vectors are extracted from the matrix S0 = C0WS as the following: S (−1) , S (0) and S (1) , which are the first, middle and last M e = M ' − 2 vectors of S0 . Now, we have the relationship as follows:  (1)∗Ψ ∗ ΓS (0) = S (−1)Ψ + DIS (4)

Γ = (λ / π R)diag{−( M −1),", −1,0,1,",( M −1)}, D = diag{(−1) M −2 ,",(−1),(−1)0 ,(−1),(−1) M }, I is the reverse permutation matrix Ψ = T −1ΦT , Φ = diag {μ1 ,"μK }. Eq.(4) can also be written as block matrix form as EΨ ' = ΓS 0 (5) (−1) (−1)∗ ' T H  DIS ],Ψ = [Ψ Ψ ]. This where E = [ S equation has unique solution while M e > 2 K , and the eigenvalues of the solution has the form μk = e jϕk sin θk , k = 1,", K . The estimation of the elevation and azimuth angles of k -th sources can be obtained by θk = sin −1 ( μk ), and φk = arg(μk ).

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sources. The interpolation array[3−5] technique is employed in the letter to modify the UCA-ESPRIT. The frequency domain model of the array output signal can be obtained by the DFT transformation of Eq.(1): X ( fl ) = AS ( fl ) + N ( fl )

(6)

where S ( fl ) = [ S1 ( fl ), S2 ( f l ),", S K ( fl )] , N ( f l ) = [n0 ( fl ), n1 ( f l ),", nN ( fl )]T . The reason why the algorithm mentioned in Section II fail in the wideband case is that the array mainfold varies via the signal frequency. So the goal of the wideband array signal processing is to get the fixed virtual mainfold from the real mainfold, as discussed in the interpolation technique. As for the UCA mainfold, the array mainfold hold on if the product of frequency and the radius of the circular is constant i.e., f r = const. The virtual array output can be obtained via the interpolation of the real array output in the discrete frequency which is obtained by dividing the whole frequency band into L points as fl , l = 1,", L. The interpolation array is needed to transfer the real array mainfold to the contant virtual array mainfold, as  = Bl Al , l = 1,", L A (7) T

where Al is the real array mainfold at the l -th fre is the virtual array mainfold which quency, while A is the same for all the frequency within the bandwidth of the sources. Bl is the interpolation matrix . which maps Al to A The covariance of the virtual array can be calculated by the interpolation matrix. If we denote the covariance of the real array and the virtual array by l respectively, then we have the relationRl and R ship as follows: l = Bl Rl BlH R (8) The covariances of the virtual array at different frequency should be summed up[4] to obtain the covariance of the wideband signal composed of fl , l = 1,", L : L

=∑R l R l =1 L

= ∑ Bl Rl BlH l =1

L

L

l =1

l =1

IV. Wideband ESPRIT Algorithm Based on UCA

{ R s } A =A ∑ l  H + ∑ Bl Rln BlH

The UCA-ESPRIT will fail if the sources are wideband signals because the algorithm above is based on the narrow band. It needs some amendement so that it can be used for the wideband

where Rls , Rln are the signal covariance and the noise covariance respectively. Now we can use the beamformer FrH to obtain covariance of the beamspace of the wideband sources as follows:

(9)

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JOURNAL OF ELECTRONICS (CHINA), Vol.23 No.1, January 2006

 r Ry = FrH RF

(10)

The matrices S and N which span the beamspace signal and noise subspace are obtained via the real-valued eigenvalues decomposition of Re( Ry ). The same processing as descriped in Section II is applied to the matrix S to obtain the matrix S0 = C0WS . Then the automatically paired wideband source azimuth and elevation angles estimation can be obtained by the processing discribed in Section II. As for the maximum phase excited mode, it is the same for all the frequency components and the phase excited mode of the highest frequency component is used in the following processing. The highest frequency component of the sources is denoted by f 0 and the corresponding phase excited mode is M 0 = 2π f 0 R / c. The algorithm discribed above can be summed up as follows: Step 1 Obtaining the sample covariance matrix at different frequency by the DFT of the T sources snapshots as Rl = (1/ T )∑ Tl =1 X (ϖl , t ) X H (ϖl , t ); Step 2 Calculating the corresponding interpolation matrix Bl ; Step 3 Calculating the beamspace covariance from Eqs.(8), (9) and (10), performing real-valued eigenvalue decomposition of the matrix Re( Ry ), and obtaining the matrix S which span the signal subspace. Step 4 Calculating the matrix S0 = C 0WS and obtaining the matrix S (−1) , S 0 and S 1 ; Step 5 Estimating the DOA of sources by solving Eq.(5).

and the relative bandwidth varies from 0 to 2. The number of frequency bins and snapshots is the same as that of Simulation 1. Fig.3 shows the standard deviation of azimuth angle in different relative bandwidth. The elevation case is omitted due to the similarity of the azimuth case.

Fig.2 Standard deviation of DOA

V. Simulation Results In this Section, we perform two Monte-Carlo simulations to test the performance of the algorithm discussed above. Simulation 1 The radius of a UCA R = λmin is employed for the simulations. The number of elements is 13, and the maximum phase excited mode is M = 6. The two wideband sources are assumed as zero-mean Gauss stochastic process which has the same flat spectra. The relative bandwidth is 0.5. The arrival angles are (25°,20°), (63°,120°) respectively. The whole bandwidth is divided into L = 10 frequency bins with 200 snapshots in each bin. The SNR is 0~20dB. Fig.2 depicts the DOA standard deviation of the estimation errors as a fuction of SNR. Fig.2(a) is the azimuth case and Fig.2(b) is the elevation case. Simulation 2 The UCA is the same one used in simulation 1. Only one source with arrival angle(25°,20°) is simulated. The SNR is fixed in 10dB,

Fig.3 The performance of the proposed algorithm versus wideband sources

The curves shown in Fig.2(a) and Fig.2(b) depicts the performance of the algorithm for the wideband sources. The DOA standard deviations are both inversly proportional to the SNR, and have good accuracy. The results of Simulation II shows that the DOA standard deviation fluctuates in a interval small enough compared with the relative bandwidth. Compared with the general UCA-ESPRIT algorithm, the performance is improved. And we can say that

YANG et al. DOA Estimation for Wideband Sources Based on UCA

the proposed algorithm is not sensitive to the wideband signal over a substantial large bandwidth. The efficiency of the proposed alorithm can be verified by the results of simulations.

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[3]

VI. Conclusions In this letter, the DOA estimation algorithm for wideband sources based on the UCA is presented. The algorithm is derived from the principium of interpolation and the UCA-ESPRIT algorithm. The algorithm is insensitive to the bandwidth in a substantial large bandwidth and the performance of this algorithm is verified by the simulation results.

[4]

[5]

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