Joint Azimuth, Elevation and Time of Arrival Estimation of 3-D Point Sources

Transcription

1 ISCCSP 8, Malta, -4 March 8 93 Joint Azimuth, Elevation and Time of Arrival Estimation of 3-D Point Sources Insaf Jaafar Route de Raoued Km 35, 83 El Ghazela, Ariana, Tunisia insafjaafar@infcomrnutn Hatem Boujema Route de Raoued Km 35, 83 El Ghazela, Ariana, Tunisia boujemaahatem@supcomrnutn Mohamed Siala Route de Raoued Km 35, 83 El Ghazela, Ariana, Tunisia mohamedsiala@supcomrnutn Abstract In this paper, we estimate the azimuth, the elevation and the Time Of Arrival (TOA) of point sources by using the MUltiple SIgnal Classification (MUSIC) algorithm The estimated channel matrix is used to jointly estimate the azimuth, elevation and TOA The performance of the proposed algorithm is then evaluated in terms of Mean Square Error of the azimuth, elevation and the TOA estimates The obtained results are also compared to the Cramer-Rao Bound I INTRODUCTION Source localization is an issue of interest in wireless communications Methods for mobile localization are based on Direction Of Arrival (DOA) and/or Time Of Arrival (TOA) estimation Parametric joint angle/delay estimation has received increased research interest lately [], [] Many of the proposed algorithms are based on Maximum-Lielihood or multidimensional MUSIC Previous wors focused on D localisation; DOA (azimuth) and TOA estimation which is usually obtained by using subspace algorithms such as JADE-MUSIC [] and JADE-ESPRIT [5], [6] Unlie the conventional MUSIC and ESPRIT [3], [4] algorithm, a major contribution of JADE algorithm is its ability to wor with a number of paths exceeding the number of antennas In this paper, we treat the problem of tridimensional localization in the presence of point sources We use the JADE-MUSIC algorithm to jointly estimate the azimuth, the elevation and the time of arrival in a Rayleigh channel transmission context The performance of the obtained algorithm is evaluated in terms of the Mean Square Error (MSE) of azimuth, elevation and time of arrival estimates The paper is organized as follows In section, the system model is described Section 3 gives the definition of the space-time array manifold Section 4 describes the adaptation of JADE to the proposed algorithm In section 5, the Cramer Rao Bound () of the joint azimuth, elevation and TOA estimates is derived Simulation results are given in section 6 Finally, Section 7 draws some conclusions II SYSTEM MODEL The received baseband signal at an M-elements antenna array at time t, x(t), can be written as the convolution of the transmitted digital sequence {s l with a multipath channel h(t) x(t) = l s l h(t lt)+n(t), () where T is the symbol period, n(t) is a zero mean white Gaussian noise of covariance matrix σ I M The channel impulse response can thus be modeled as h(t) = Q a(θ i, i )β i g(t τ i ) () i= where Q is the number of paths, β i is the i-th path attenuation, θ i, i and τ i are respectively the nominal azimuth, elevation and time of arrival of the i-th path, g(t) is the pulse shaping function and a(θ, ) is the array manifold defined by ( [ a(θ, ) = exp πj ] λ cos( ) (p x cos(θ)+p y sin(θ)), [, exp πj ]) T λ cos( ) (p x M cos(θ)+p ym sin(θ)) where λ is the wavelength, p xm and p ym are the coordinate of the m-th sensor in the antenna plane Let LT = L g T + δτ be the length of the channel impulse response where L g T is the length of the pulse shaping function and δτ is the maximum path delay The sampled version of x(t) at rate P/T, where P is the oversampling factor, over N symbol periods can be written as [5] X = HS + N, (3) where x() x((n )T ) X = x(( P )T ) x((n P )T ) h() h((l )T ) H = h(( P )T ) h((l P )T ) MP N MP L,, /8/$5 c 8 IEEE

2 93 ISCCSP 8, Malta, -4 March 8 S = s s s N s s s s L+ s L+ s N L L N and N is the noise matrix defined similarly to X It will be convenient to rearrange the matrix H into H =[h() h(t/p) h((l /P )T )] M LP (4) According to (), H satisfies the factorization H = [a(θ, ),, a(θ Q, Q )] {{ M Q β {{ β Q Q Q g T (τ ) g T (τ Q ) {{ Q LP = A(θ, )DG T (τ ), (5) where g(τ i ) = [g(t τ i )] =,/P,,L /P is an LP - dimensional column vector containing the samples of g(t τ i ), θ = [θ,,θ Q ], = [,, Q ], τ = [τ,,τ Q ], D = diag[β], andβ T =[β,,β Q ] III BASIC EQUATIONS A Space-time manifold Let h = vec( H) be a column vector of length MPL obtained by taing the transpose of each row of the matrix H and stacing it below the transposes of the previous rows Applying the vec() operation to (5) and using the general relation vec(a diag[b]c) =(A C T )b yields h =(A(θ, ) G(τ ))β = U(θ,,τ)β (6) where represents the Khatri-Rao (column-wise Kronecer) product, U(θ,,τ) = A(θ, ) G(τ ) (7) = [u(θ,,τ ),, u(θ Q, Q,τ Q )], and u(θ,,τ) is an MPL vector called the space-time array manifold given by u(θ,,τ)=a(θ, ) g(τ), (8) where denotes the Kronecer product B Method outline Since the azimuth, elevation and delay parameters vary very slowly, we assume that the space-time matrix U(θ,, τ ) is time-invariant over S time slots Thus, we have at slot n, vec( H (n) )=U(θ,, τ )β (n), n =,,,S (9) The channel matrix, H, can be estimated slot by slot using N training symbols as follows H (n) est = X (n) S (n) () = H (n) + V (n), n =,,,S, () where X (n) = H (n) S (n) + N (n), () V (n) = N (n) S (n) is the channel estimation noise matrix at time slot n and is the matrix pseudo-inverse Equivalently, using (4) we can write H (n) est = H (n) + V (n), n =,,,S (3) We denote by v (n) the -th column vector of V (n) Wehave E{v (n) v(n)h l = σ E((S (n) S (n)h ) ) l I MP Assuming that the training data is perfect ie SS H = NI L,wehave E{v (n) v(n)h l = σ N δ li MP (4) Applying the vec operation to (3), we obtain y (n) = U(θ,, τ )β (n) + v (n), n =,,,S, (5) where y (n) = vec( H (n) est ) and v (n) = vec( V (n) ) under a matricial representation, the above equation becomes Y =[y (), y (S) ]=U(θ,, τ )B + V, (6) where B = [β () β (S) ], and similarly for V The joint azimuth, elevation and time of arrival problem amounts for given channel estimates { y (),, y (S), find the azimuth θ, elevation and delays τ using the model (6) The next and last step of the method thus consists in jointly estimating the parameters η =[θ τ ] T that satisfy the model in (6) When we have more than one user, we can independently estimate the corresponding channel matrices H using each user s unique training signal We can then proceed as above for each user IV ADEQUATION OF JADE-MUSIC ALGORITHM TO 3-D POINT SOURCES Similarly to the conventional MUSIC algorithm [3], the proposed algorithm is based on the decomposition of the theoretical correlation matrix, R Y, into a signal subspace E s and a noise subspace E n R Y = E s Λ s E H s + E ne H n σ e, (7) where σe = σ /N is the variance of each entry in the channel estimation noise matrix In practice, R Y must be estimated using snapshots as follows ˆR Y = YY H /S, The eigenvalues have been ordered so that ˆΛ s is a diagonal matrix containing the Q largest eigenvalues of ˆR Y in decreasing order, and the columns of Ê s are the corresponding eigenvectors The columns of Ê n are the remaining MPL Q eigenvectors We now that the true space-time channel vectors u(η i ), i = Q, are orthogonal to the noise subspace Ê n Sothe proposed algorithm consists in searching the Q maxima of the following criterion u(η) H u(η) P (η) =, (8) u H (η)ê n Ê H n u(η) For a full ran matrix A =(A H A) A H

3 ISCCSP 8, Malta, -4 March where η = [θ τ] T This algorithm has a 4-D spectrum We present in figure the algorithm spectrum for θ =[,, 5], =[, 5, ] and τ =[,, 5]µs For illustration purpose, we plot in figure (a) (resp (b)) the evolution of P (η) with respect to elevation and time of arrival (resp azimuth and elevation) for θ = (respectively τ =µs In figure (a) we notice the presence of a pea corresponding to = and τ =µs for θ = and similarly in figure (b) Adapting the results of [7], we obtain { CBR(η) = σ S e Re { B(n) H D H UP UD U B(n) n= (9) where B(n) =I 3 diag[β (n) ], P U = I UU,andD U = [G A θ, G A, G A] TOA[µs] 5 3 Elevation[degrees] 4 MSE of azimuth estimates[degrees] (a) Fig MSE of azimuth estimates with respect to SNR Elevation[degrees] 4 Azimuth[degrees] 4 MSE of elevation estimates[degrees] Fig (b) Spectrum of the proposed algorithm for SNR=dB V DERIVATION OF THE CRAMER-RAO BOUND The Cramer-Rao Bound (CBR) provides a lower bound on the covariance matrix of any unbiased estimator In [7], the was derived for only azimuth and TOA estimation only Fig 3 MSE of elevation estimates with respect to SNR Here, the prime denotes differentiation Each column is differentiated with respect to the corresponding parameter and all matrices are evaluated at the true parameter values More precisely, we have [ A θ = A da(θ ) (θ) =,, da(θ ] Q), () dθ dθ Q

4 934 ISCCSP 8, Malta, -4 March 8 3 x 6 5 MSE of TOA estimates[µs] MSE of elevation estimates[degrees] Fig 4 MSE of TOA estimates with respect to SNR Fig 6 MSE of elevation estimates with respect to the number of slots MSE of azimuth estimates[degrees] MSE of TOA estimates[µs] x Fig 5 MSE of azimuth estimates with respect to the number of slots Fig 7 MSE of TOA estimates with respect to the number of slots and similarly for A = A ( ) and G = G (τ ) The proof of this claim is similar to the one in [] VI SIMULATION RESULTS In this section, we evaluate the performance of the proposed algorithm in terms of MSEs of azimuth, elevation and TOA estimates We assume a single user, a 3-path channel and an antenna array with 5 elements The azimuths of the paths are [,, 5], the elevations are [, 5, ] relative to the array broadside and the corresponding path delays are [,, 5]µs and T =37µs The path gains are generated from a complex Gaussian distribution with zero mean and variances [, 6, 4] respectively for the three rays The modulation waveform is a raised cosine pulse with excess bandwidth 35 The over sampling factor is set to P = Data is collected over S = 4 time slots, and at each time slot, the channel is estimated using N = 6 training symbols The experimental MSE of the azimuth, elevation and delay estimates are computed from Monte Carlo simulations The Experiments are run with the proposed algorithm for various Signal to Noise Ratios (SNR) The SNR is defined as the ratio of the power of the strongest path to the variance of the noise: β σ In figures -4, we present respectively the evolution of the MSE of azimuth, elevation and TOA estimates of the first path with respect to the SNR We note that for large SNRs, the algorithm reaches nearly the In figures 5-7 we present the performance of the proposed algorithm in terms of MSE of azimuth, elevation and TOA estimates with respect to the number of slots for SNR = db We notice that the proposed algorithm reaches the for a large number of slots

5 ISCCSP 8, Malta, -4 March VII CONCLUSION In this paper, we proposed a new algorithm for joint azimuth, elevation and time of arrival estimation in the presence of point sources The concept of space-time array manifold was introduced and the Cramer-Rao bound for the proposed estimator was provided, along with simulation results showing the performance of the algorithm in terms of MSE This wor is expected to be extended to joint azimuth, elevation and delay estimation of diffuse sources REFERENCES [] Y Ogawa, N Hamaguchi, K Ohshima and K Itoh, Highresolution analysis of indoor multipath propagation structure, IEICE Transactions on Communications, vol E78-B, pp , Nov 995 [] MC Vanderveen, CB Papadias and A Paulraj, Joint angle and delay estimation (JADE) for multipath signals arriving at an antenna array, IEEE communication Letters,pp:-4, January 997 [3] R O Schmidt A Signal Subspace Approach to Multiple Emitter Location and Spectral Estimation PhD thesis, Stanford University, Stanford, CA, November 98 [4] R Roy and T Kailath, ESPRIT-Estimation of Signal Parameters via Rotational Invariance Thechniques, IEEE Trans Acoust, Speech, Signal Processing, vol 37, pp , July 989 [5] A J Van der Veen, M C Vanderveen and A Paulraj, Joint angle and delay estimation (JADE) using shift invariance properties, IEEE Signal Processing Letters, pp 4-45, May 997 [6] M C Vanderveen, A J Van der Veen, A Paulraj, Estimation of multipath parameters in wireless communications, IEEE Trans Signal Processing, Vol 46, pp 68-69, Mar 998 [7] MC Vanderveen, AJ van der Veen, and A Paulraj Estimation of multipath parameters in wireless communications, IEEE Trans Signal Processing, VOL 46, NO 3, pp68-69, March 998