ASYMPTOTIC PERFORMANCE ANALYSIS OF DOA ESTIMATION METHOD FOR AN INCOHERENTLY DISTRIBUTED SOURCE Jooshik Lee and Doo Whan Sang LG Electronics, Inc. Seoul, Korea Jingon Joung School of EECS, KAIST Daejeon, Korea ABSTRACT Estimation method, based on the beamforming approach, for the nominal direction of arrival (DOA) of an incoherently distributed source is proposed. The proposed method is computationally more attractive than the conventional redundancy averaged covariance matching (RACM) method. The asymptotic performance of the proposed method and the RACM is compared analytically as well as numerically. By using the computer simulation, it is verified that the asymptotic performance of the proposed method is better than that of the RACM. INTRODUCTION In wireless system, the sources can be viewed either as coherently distributed or incoherently distributed according to the relationship between the channel coherency time and the observation period. If the channel coherency time is much longer than the observation period then the coherently distributed or partially coherent model is relevant. In the opposite case, the incoherently distributed model can be used. In [] and [], the estimation of two-dimensional (azimuth and elevation) direction-ofarrival (DOA) is considered by using a pair of uniform circular arrays (UCA) under a coherently distributed source model. In this paper, we consider a generally, incoherently distributed source model and propose a spectral-based DOA estimation method on the basis of conventional beamformers. The proposed method is based on the fact that the spatial covariance matrix can be decomposed by Cholesky factorization [3]. We derive the asymptotic performance of the proposed method and compare the performance with that of the redundancy averaging covariance matching (RACM) algorithm [4] and the Cramer-Rao bound (). A DISTRIBUTED SOURCE MODEL An observation date vector x(t) for the incoherently distributed sources can be modelled as follows: x(t) = a(θ)ς(θ, t)dθ + n(t), () where the additive noise n(t) in () is temporally and spatially independent and identically distributed, zeromean, complex Gaussian with covariance σn; the steering vector a(θ) =[e j sinθ e j (L ) sin θ ] T, for a direction θ and a uniform linear array (ULA), here = πd λ, d is the distance between two adjacent elements, and λ is the wavelength of a propagation wave; and the complex-valued angular-temporal signal intensity ς(θ, t) =s(t)ϱ(θ, t), heres(t) is the transmitted signal from the source and ϱ(θ, t) is the spatially continuous distribution of the sources having correlation E[ϱ(θ, t)ϱ (θ,τ)] = γ(θ; µ)δ(θ θ )δ t,τ, () where γ(θ; µ) can be interpreted as the spatial power density of the source, δ( ) is the Dirac delta function, and δ t,τ is the Kronecker delta function. The parameter vector µ contains the nominal DOA θ and angular extension σ θ of an incoherently distributed source. Equation () implies that the signal components of the source at different angles are uncorrelated. In terms of the array response vector, the incoherently distributed source model () can be rewritten by x(t) =s(t)b(t, θ,σ θ )+n(t), where b(t, θ,σ θ ) is the array response vector defined as b(t, θ,σ θ )= a(θ)ϱ(θ, t)dθ. The covariance matrix of observation data vector is expressed as R = E[x(t)x H (t)] = pe[b(t, θ,σ θ )b H (t, θ,σ θ )] + E[n(t)n H (t)] = pr s + σni, (3) of5 Authorized licensed use limited to: Univ of Calif Los Angeles. Downloaded on March 9, 9 at 4:7 from IEEE Xplore. Restrictions apply.
where signal power p = E[ s(t) ], the signal covariance matrix R s = γ(θ; µ)a(θ)a H (θ)dθ, and I is the L- dimensional identity matrix. Thus it is important to know the spatial power density function γ(θ; µ) in practical environment. Many researchers have assumed that the spatial power density function is in the form of a Gaussian density function because knowing exactly the spatial power density function is difficult. In this paper, we assume that the spatial power density function is a Cauchy density function, since the choice of spatial power density function is not critical for estimation performance for small spreading [5], as follows: γ(θ; µ) = { π σ θ (θ θ ) +σ θ θ θ <ɛ θ, θ θ >ɛ θ, where ɛ θ is a small number possibly depending on σ θ. Therefore, for R s in (3), we have R s (k, l) = γ(θ; µ)e j(k l) sin θ dθ e σθ (k l)cosθ j(k l) sin θ e = ρ k l e j(k l)ω, (4) where R s (k, l) indicates the (k, l)th element of the matrix R s, ρ = e σθ cosθ,andω = sinθ.the result of (4) is the same with that of [6] and [7]. Thus, we have shown how the signal covariance matrix like (4) is obtained clearly. The covariance matrix (3) can now be rewritten as R a(ω )a H (ω ) B + σ ni, where B is a real-valued symmetric Toeplitz matrix with B ρ (k, l) = ρ k l and indicates the element wise matrix product. As ρ, the matrix R tends to the observation covariance matrix under the point source model. PARAMETER EXTIMATION In general, an optimum estimation for the distributed sources would provide the best performance at the cost of intensive computation. Subspace-based methods may not be suitable for the full-rank data model such as the incoherently distributed source. As a computationally attractive alternative, we propose a spectral-based method based on conventional beamforming approach. In the conventional beamforming approach to DOA estimation, the beam is scanned over the angular region of interest in discrete steps by forming weights w (= a(θ) at different θ), and the output power is measured. The output power of the conventional beamformer, as a function of the direction of arrival, is given by P (θ) =w H ˆRw = a H (θ) ˆRa(θ), (5) where ˆR is a sample covariance matrix obtained from ˆR = N N x(t)x H (t). t= Thus, if we have an estimate of the covariance matrix and the steering vectors a(θ) are known at all θ s of interest, it is possible to estimate the output power, termed as the spatial spectrum, as a function of the direction of arrival θ. Clearly, the direction of arrival can be estimated by locating peaks in the spatial spectrum defined in (5). For the distributed sources, the signal covariance matrix R s can be decomposed through Cholesky factorization as follows: R s = MM H, (6) where M is an L L lower triangular matrix as shown in (7). Note that M is a full-rank matrix when ρ. We can expect that the rank of M decreases as ρ in (7). When ρ =, that is, when the source is a point source, R s = a(ω )a H (ω ). From (3) and (6), R = pmm H + σni L = p m k m H k + σ ni, (8) k= where m k is the kth column vector of the matrix M = [m m m L ]. Noting that (8) is analogous to the expression of the equipower point sources and that m k s are linearly independent if ρ,thenm k s can be regarded as the steering vectors of L uncorrelated point sources with equipower. Therefore, following the conventional beamformer approach, we can define the parameter spectrum as follows: L = m H k (ω, ρ) ˆRm k (ω, ρ). (9) k= The nominal DOA and angular spread can be estimated by locating peaks in the parameter spectrum defined in (9) as follows: (ˆω, ˆρ) = arg max. ω,ρ The proposed method requires only the covariance matrix and does not possess the ambiguity of the estimation of5 Authorized licensed use limited to: Univ of Calif Los Angeles. Downloaded on March 9, 9 at 4:7 from IEEE Xplore. Restrictions apply.
M = ρe jω ρ e jω ρ e jω ρ ρ e jω...... ρ L e j(l )ω ρ L ρ e j(l )ω ρ e j(l )ω (7) for the number of signal source associated with prior information of subspace based methods. We adopt the covariance matching method based on the redundancy averaging of the covariance matrix estimate [4] and compare it with the proposed method..4. STATISTICAL PROPERTIES We derive the asymptotic covariance of the nominal DOA estimates for the proposed methods. In general, the performance analysis is based on the asymptotic distribution of the estimate error for each estimator. To derive the asymptotic distribution of the estimates ˆω and ˆρ under the proposed method, once it is assumed that ˆω and ˆρ converges to true values ω and ρ, respectively. When the gradient and Hessian of are defined as ) V (ω, ρ) = ( ω ρ and ( ) ω H(ω, ρ) = ωρ ρω, ρ respectively, we have = V (ˆω, ˆρ) = V (ω,ρ )+H η ( ˆω ω ˆρ ρ ), through a first-order Taylor series expansion of V around (ω,ρ ),wherev (ω,ρ )=V (ω, ρ) ω=ω,ρ=ρ and H η = H(ω, ρ) ω=ωη,ρ=ρ η. Here, ω η is a point on the line segment joining ω and ˆω and ρ η a point on the line segment joining ρ and ˆρ. Wehaveby[8] ( ) ˆω ω { H ˆρ ρ }V (ω,ρ ) for N is large. The asymptotic Hessian matrix is defined as H = lim H(ω,ρ ). N. 3 4 5 6 Angular extension (degree) Fig.. Theoretical RMSE of nominal DOA for the covariance matching method (*), the proposed method ( ), and (no marker) versus the angular extension. θ =, N =, and SNR=dB. Using the asymptotic normality of the covariance matrix of received signal vector, NV (ω,ρ ) is also asymptotically normal with zero mean and covariance matrix Q given by Q = lim N NE[V (ω,ρ )V H (ω,ρ )]. It follows that the normalized estimation error vector N[ˆω ω, ˆρ ρ ] T is asymptotically Gaussian distributed with zero mean and covariance matrix Ξ given by Ξ = ( H ) Q( H ). SIMULATION RESULTS In this section, we illustrate the performances of the proposed and covariance matching methods [4] and demonstrate the theoretical analysis. We consider a ULA having eight sensors (L =8) with d = λ/. A single incoherently distributed source scenario is considered with the Cauchy spatial power density function. The 3of5 Authorized licensed use limited to: Univ of Calif Los Angeles. Downloaded on March 9, 9 at 4:7 from IEEE Xplore. Restrictions apply.
.9.75.7.7 5.5.55.5 5.3..35. 5 5 5 3 35 4 45 5 Number of snapshots.3 3 4 5 6 7 Nominal DOA (degree) Fig.. Theoretical RMSE of nominal DOA for the covariance matching method (*), the proposed method ( ), and (no marker) versus the number of snapshots. θ =, σ θ =3,and SNR=dB. Fig. 4. Theoretical RMSE of nominal DOA for the covariance matching method (*), the proposed method ( ), and (no marker) versus true nominal DOA. σ θ = 3, N =, and SNR=dB..8.6.4..4... 5 5 5 5 SNR (db) 3 4 5 6 7 8 9 Number of sensors Fig. 3. Theoretical RMSE of nominal DOA for the covariance matching method (*), the proposed method ( ), and (no marker) versus SNR. θ =, σ θ =3,andN =. Fig. 5. Theoretical RMSE of nominal DOA for the covariance matching method (*), the proposed method ( ), and (no marker) versus the number of sensors. θ =, σ θ =3, N =, and SNR=dB. emitter signal is assumed as the unit-power phase modulated signal. The signal-to-noise ratio (SNR) is defined as log σ. We concentrate on the nominal DOA estimation problem since it is the most important issue in n practical communication environment. The plots in Figs. 5 show the theoretical rootmean square error (RMSE) of the nominal DOA for the proposed and covariance matching methods, when the different parameters are varied one at a time. Also, is illustrated. The point to be noted is that although the number of sensors increases the proposed method has a bias from the. Since the resolution of spectral based beamforming method increases by adding more sensor elements in general, as the number of sensors increases, more than two peaks appear on spatial spectrum of spectral based beamforming methods. Although these peaks do not differ significantly from the nominal DOA, the biased peaks yield the performance degradation in nominal DOA estimation for an incoherently distributed source. 4of5 Authorized licensed use limited to: Univ of Calif Los Angeles. Downloaded on March 9, 9 at 4:7 from IEEE Xplore. Restrictions apply.
In summary, the proposed method can lighten the computational load with a cost of the negligible performance degradation compared to the optimal method. Furthermore, the proposed method provides better performance than the conventional covariance matching method. CONCLUDING REMARK In this paper, for the case of incoherently distributed source, we have proposed the nominal DOA estimation method based on conventional beamforming approach and shown the performance of spectral based methods in full-rank data model, such as the incoherently distributed source. The proposed method provides better performance than the covariance matching method based on least squares and is comparable to the optimal maximum likelihood. REFERENCES [] J. Lee, I. Song, and J. Joung, Uniform circular array in the parameter estimation of coherently distributed sources, in Proc. st IEEE Mil. Comm. Confer. (MILCOM), Anaheim, CA, Oct., pp. 58 6. [] J. Lee, I. Song, H. Kwon, and S.R. Lee, Low-complexity estimation of -D DOA for coherently distributed sources, Signal Processing, vol. 83, pp. 789 8, 3. [3] G. H. GolubandC. F. VanLoan, Matrix Computations, nd ed., Baltimore, MD: Johns Hopkins University Press, 989. [4] M. Ghogho, O. Besson, and A. Swami, Estimation of directions of arrival of multiple scattered sources, IEEE Trans. Signal Processing, vol. 49, pp. 467 48, Nov.. [5] M. Bengtsson and B. Ottersten, Low-complexity estimators for distributed sources, IEEE Trans. Signal Processing, vol. 48, pp. 85 94, Aug.. [6] Y. U. Lee, J. Choi, I. Song, and S. R. Lee, Distributed source modeling and direction-of-arrival estimation techniques, IEEE Trans. Signal Processing, vol. 45, pp. 96 969, Apr. 997. [7] O. Besson, F. Vincent, and P. Stoica, Approximate maximum likelihood estimators for array processing in multiplicative noise environments, IEEE Trans. Signal Processing, vol. 48, pp. 56 58, Sept.. [8] M. Viberg and B. Ottersten, Sensor Array Processing Based on Subspace Fitting, IEEE Trans. Signal Processing, vol. 39, pp., May 99. 5of5 Authorized licensed use limited to: Univ of Calif Los Angeles. Downloaded on March 9, 9 at 4:7 from IEEE Xplore. Restrictions apply.