ARRAY SHAPE CALIBRATION USING EIGENSTRUCTURE METHODS

Transcription

1 ARRAY SHAPE CALIBRATION USING EIGENSTRUCTURE METHODS Anthony J. Weiss Department of Electronic Systems Faculty of Engineering Tel Aviv University Ramat Aviv, 69978, Israel Benjamin Friedlander Signal Processing Technology, Ltd 703 Coastland Drive Palo Alto, CA USA Abstract Sensor location uncertainty can severely degrade the accuracy of direction finding systems. An eigenstructure based method for simultaneously estimating directions of arrival (DOA) and sensor locations is developed to alleviate this problem. The proposed technique does not require calibration sources at known positions, and can handle non-disjoint sources, (i.e. sources occupying the same frequency band and the same time interval). The procedure is guaranteed to converge and offers an alternative to the procedure presented in [5]. Numerical examples and Monte-Carlo simulations are used to study the performance of the proposed technique. 1. Introduction Source localization techniques based on eigenstructure methods have been discussed extensively in the literature since the beginning of the decade. Computer simulations and a relatively small number of experimental systems have demonstrated that in certain cases these techniques have superior performance compared to conventional direction finding techniques. In this paper we are interested in source localization in the presence of sensor location uncertainty. Such uncertainty occurs, for example, in towed arrays or in tactical systems which require that the array be dismantled and reassembled in the field. Rockah and Schultbeiss [2]-[3] have studied this problem in detail, using CramBr-Rao bounds for the direction of arrival (DOA) estimates in the presence of sensor location uncertainty. They have shown that finding the direction to the sources is possible if the location of one sensor and the direction to another sensor are known, provided the array is non-linear and the DOAs are distinct. Moreover, they presented an algorithm for self- Calibration which is based on observing disjoint sources (i.e.sources which can be separated either in the time domain or in the frequency domain). In this work we emphasize the case of non-disjoint sources. This work was supported by the Army Research Office U- der contract No. DAAL03-8PC-OW7. Lo and Marple [4] discussed a calibration technique which requires at least two calibration sources whose DOAs are known. Therefore their technique is not a true self calibration procedure. In a previous publication [5] we proposed an exact Maximum-Likelihood method for solving this problem. Although the solution is optimal in the maximumlikelihood sense, the algorithm typically requires more computations than the method proposed here. The outline of the paper is as follows. In Section 2, we formulate the problem for the general wideband case. Section 3 describes an algorithm for simultaneously estimating the DOAs and the unknown sensor locations. Section 4 presents numerical examples and Monte-Carlo simulations. Section 5 includes some conclusions. 2. Problem Formulation Consider N radiating sources observed by an array of M sensors. The signal at the output of the m-th sensor can be described by N Zm(t) = Csn(t - ~ mn) + ~ m(t) (1) n=1 where m = 1,2;..,M and -TI2 5 t 5 Tf2 and where {~~(t)}f=~ are the radiated signals, {u,(t)},=, M are additive noise processes, and T is the observation interval. The delays T," are parameters related to the relative location of the n-th source and the m-th sensor. Taking the Fourier transform of (1) we obtain N x,,,(~~) = ~ e - j ~ sn(wf)+vm(wl); ~ r - e= i,...,~; n=1 (2) where Sn(w,) and Vm(wL) are the Fourier coefficients of s,(t) and v,(t) respectively. To further simplify the exposition we assume that the sensors and sources are cwplanar and the sources 23ACSSC-12/89/0925 $ MAPLE PRESS 925

2 are far enough from the observing array so that the signal wave-fronts are effectively planar over the array. Under these restrictions the delays T,, are given by ~ m= n - dmn = xm sin + ym cos, (3) where c is the propagation velocity, d,, is the distance from the m-th sensor to the first sensor (which we selected as a reference sensor) in the direction of the n-th source, (x,,,,ym) are the Cartesian coordinates of the m-th sensor, yn is the DOA of the n-th source relative to the y axis, and the origin of the Cartesian coordinate system coincides with the location of the reference sensor. Equation (2) may be expressed using vector notation as follows: The problem addressed here can be summarised as follows. Given the data X(wc), = 1,2,..., L; estimate the unknown directions of arrival y,, n = 1,2,..., N, as well as the unknown sensors coordinates (xm,ym), m= 1,2,...,M. 3. The Estimation Procedure The proposed methad is based on the eigen decomposition of the sample covariance matrix of the vector of received signals. We make the standard assumptions underlying the MUSIC algorithm [l] in addition to the assumptions that are required for array shape calibration as given in [2]: (a) The signals and noises are stationary over the observation interval. (b) The number of sources is known or can b-e estimated. (c) The columns of A(w0 are linearly independent. (d) The signals are not perfectly correlated. (e) The noise covariance matrix is known except for a multiplicative constant U:. (f) The sensors are not placed on a straight line. (9) There are more sensors than sources (N < M). (h) The location of one of the sensors and the direction to another sensor are known. The covariance matrices of the signal, noise and observation vectors are given by &(ut) %{s(wc)s(wc)h}, a:% =E{v(wc)v(wc)H}, &(Ut) 5E{X(wc)X(wdH} = A(wc)Rs(wc)A(wc)H + &, (5) where (.)H represents the Hermitian transpose operation. The following theorem forms the basis for the eigenstructure approach. Note that for notational simplicity we dropped the frequency index. The theorem holds for each frequency cell. Theorem: Let Xi and U;, i = 1,2,..., M be the eigenvalues and corresponding eigenvectors of the matrix pencil (&,E), (i.e., the solutions of R,u = Ah), where the Xis are listed in descending order. Then,.,. (1) XN+1 =.X,V+z = '., = = U 2. (2) Each of the columns of A is orthogonal to the ' matrix U = [u,v+~,u,v+~,...,um]. Proof: The proof is presented in [l] This theorem suggests that one should first estimate &(wc) and use the estimates of Xi(wf) to determine the number of signals. Once N is known, we can construct an estimate U (q) of U(wc) and generate reasonable estimates of the DOAs and the sensors coordinates by finding the values that will best satisfy 'the second part of the theorem. Therefore, we propose the minimization of the following cost function: Three questions arise immediately: (i) What are the conditions for a unique solution? (ii) If a unique solution exists, how it can be found without being trapped in one of the local minima of Q?. (iii) What are the advantages/disadvantages of the proposed approach over other approaches? Equation (6) represents the least squares solution of L(M - N)N complex equations. The number of unknowns is 2M-3 sensor coordinates (see assumption h) plus N DOAs. Hence, a necessary condition for uniqueness is given by 2M -3+ N 5 2L(M - N )N. (7) 926

3 In general, the cost function Q is not unimodal, i.e. it exhibits more than a single minimum. There are no computationaly efficient methods that can avoid the local minima while searching for the global minimum. However, in the problem at hand we assumed only relatively small sensor location displacements. Therefore, if one uses the nominal sensor coordinates to generate initial DOA estimates, the search for the DOAs and sensor coordinates that minimize Q starts close to the true values. This suggests that the minimumof Q that is closest to the initial search point is the desired estimate. This claim is verified experimentally in the next section. The search for the minimum of Q can he performed by many optimization algorithms. The technique used here is an extension of the well known MU- SIC algorithm [I]. The algorithm iterates between two steps. The first step assumes known sensor coordinates and estimates the DOAs, while the second step assumes known DOAs and estimates the sensor coordinates. The first step is identical to the standard MU- SIC procedure, while the second step is new. The prcess is guaranteed to converge, as will be shown later. The First Step Given the nominal sensors coordinates (or the last estimated values), we first evaluate the function defined by by (x,o,y,o), m = 1,2,..., M, while the sensors CD ordinates that minimize the cost function are given by (Xm,Ym)=(~mo,~mo)+(A~m,A~m)~ (9) where m = 1,2,..., M. If (Ax,, Ay,) are small enough, the matrix A(wf) can be expanded as follows A(wc) = + A,Al(wl) + A~Az(wL), (10) where Ao(wc) is the matrix A(w0 computed with the nominal sensor coordinates (~,~,y,~), and A. As = diag{axl, Axz,..., AXM], Ay ~diaglayl,ay~,...,ay~), Of. Al(wc) ~jaa(wc)-diag{siny,,...,sin7~), A. Az(wc) =jao(wc)-diag{cosyl,. C Now Q can he rewritten as L Q =E..,cOsy,v] Ilfi(wc)HIAo(wc) + A=Ai(w<) + AyAz(w~))] f=1 L N =CCI1O(wc)~[ao(e,y,,)+~~ai(e,7"~n) kl n=1 + Ayaz(&Yn)1 11' /Iz We select as the temporary DOA estimates the values of 7 associated with the N largest peaks of this function. In practice, one selects a fairly dense set of 7is, say every two degrees, over the field of interest, and computes the (8) these values. The exact peak location is obtained by interpolation. Note that this method of selecting the 00.4 estimates minimizes the cost function for the assumed sensors coordinates. The Second Step In this step we use the DOA estimates provided by the first step to find the sensor coordinates which minimize the cost function. The minimization of Q with respect to (xm,y,,,),m = 1,2,...,M is a nonlinear problem which can be solved in several ways. See [7] and [SI for a wide selection of methods. We chose a closed form solution that is related to the Gauss- Newton technique. Suppose that the last estimates of the sensor coordinates (or the nominal sensor coordinates) are given + diag{a~(e,m)}v~l 112, (11) where ao(e,7"),al(e,7,,) and az(!,yn) are the n-th column vectors of Ao(wl),Al(wc) and Az(w(), respectively, while v. and vy are column vectors whose m-th element is Ax,,, and Ay,,,, respectively. We construct the following vectors and matrices A T TT VCY =[v, 3 vy 1 3 A B(t, n) = - [fi(wl)h diag{al(e,y,)), U(wc)H diaglaz(e,m)}l, z(e, n) fiq~)~ao(!,~,,), (12) which simplify the expression for the cost function: The ~ ~lv,, that minimizes the expression in (12) is given by the classic result B,, = [Re{BHB]]-'Re{BHZ),, (14) 927

4 where B A[B(l, l)t, B(1, 2)T,..., B(L,AV)~]~, z &[z(1, l)t,z(1,2)t,..., ~(L,N)T]T. (15) The proposed procedure may he summarized as follows: Initialization (a) Collect the data vectors {Xj(wi)}f=, over J snbintervds and estimate the data covariance matrices &(ut) for e = I, 2,...L, (b) Estimate the number of signal N by inspection of the eigepvalues of the matrix pencil (&,CO), and construct U using the appropriate eigenvectors. (c) Using the nominal sensor coordinates generate the array manifold at(7) for a fairly dense set of yis over the field of interest. Using this manifold compute the function given by equation (8). Select the DOA estimates as the values of 7 associated with the N largest peaks of P(7). - Loop: (d) Compute the cost function value, using equation (6). (e) Use the estimated DOAs and the last estimates of sensor coordinates together with U(wt) to construct B and 2 as defined above and estimate the sensor coordinates according to equation (14). (f) Verify the decrease in Q. If Q did not decrease reduce the change in the sensor coordinates until1 a reduction is observed. (g). Using the new sensor coordinates generate the array manifold in the vicinity of the estimated DOAs. Generate new DOA- estimates using P(y).. (h) Compute the cost function Q. If Q converged stop. If not, go back to step d). This procedure generates a monotonic decreasing sequence of the cost function Q. Since Q can not he negative, the convergence of the algorithm is guaranteed. Note that step f) above is used only to insure the convergence, usually in practice this step can be skipped. 4. Monte Carlo Experiments In this section we present some numerical examples for the narrowband case (L = 1). The computer simulations presented here are meant to check the statistical efficiency of the proposed approach, and its spatial resolution. We consider a uniform circular array with six sensors,. seuarated. by half a wavelength. The nominal sensor coordinates are given in half wavelength units by: (xi,yi) = (O,O), > > ( ~ 2 ~ = ~ (LO), 2 ( ~ 3 ~ ~ = 3 ) (3/2,4/2), (x4,~4 = (1, 31, (Xs1Y5) = (O)d% (X6,ys) = (-1/2, 3/2). The array is perturbed randomly and the coordinates of the perturbed array are: (xi,yi) = (O,O), (XZ,YZ) = (0.8434,0), ( ~ 3 ~ ~ = 3 ) (1.2408,1.0929),(xq,y4) = (0.7487,1.5180),( ~ 5, ~ = s ) ( ,1.5675), (xg,y6) = ( ,0.9171). Note that these perturbations are as Large as 26 percent of the nominal distance between the sensors. Three equal power narrow-band (single frequency cell) far-field sources are at directions 71 = - 35O, 72 = On, 73 = +35O. The sources generate zero mean Gaussian signals. The noise is also zero mean, Gaussian, nncorrelated from sensor to sensor and uncorrelated with the signals. The application of a heamforming technique to this array yielded discouraging results. The beamformer failed to resolve the signals even for high (e.g. 40 db) Signal-bNoise Ratios (SNR). However, the proposed technique performed well and resolved the 'sources (for SNR above 10 dh) in all the experiments. To demonstrate the statistical efficiency of the proposed procedure for different SNRs, we collected 120 data sets of random data. For each of the SNR values (10 db, 20 db and 30 db) we collected 40 data sets. Each data set contained J = 100 snapshots. The proposed algorithm was applied to each of the data sets and the estimated DOAs and sensor coordinates were collected. The algorithm termination criterion was to stop when the value of the cost function reduces by less than one percent in a single step. The standard deviation and bias of the estimated DOAs and sensor coordinates were computed. Their values are plotted in the following figures together with the 0.99 confidence intervals and the tbeoreticalstandard deviations computed from the Cram&Rao lower hound (CRLB). In the appendix [5] we developed simple formulas for exact numerical evaluation of this hound. Figure 1 depicts the standard deviations for the DOA estimates of the source at -35 degrees. It is clear that the performance of the algorithm is nearly optimal. Figure 2 depicts the standard deviations for the 928

5 10 ~ IO, i :... i... j... I... I I:::... Figure 1: 111 I5 20 ZI SNRIdBI Standard Deviation (in degrees) vs. SNRfor the DOA of the Source at -35 : (i) Monte Carlo Results With 99% Confidence Intervals (Point Estimates) and (ii) The CRLB (Solid Curve). IW-, t 1 1.~......, Figure 2: i...: 15. 1D?J 30 3s SNR IdBI Standard Deviation (in multiples of X/2) vs. SNR for the X Coordinate of Sensor #6: (i) Monte Carlo Results With 99% Confidence Intervals (Point Estimates) and (ii) The CRLB (Solid Curve). X coordinate of the 6-th sensor. The figures corresponding to the other DOAs, and to the coordinates of the other sensors are quite similar to figures 1-2 and are therefore omitted. 9. Conclnsions We presented an eigenstructure approach for estimating the DOAs of wavefronts impinging on an array with uncertain sensor locations. The algorithm belongs to the so-called family of super-resolntion techniques and offers an alternative to the technique proposed in [5]. Monte-Carlo simulations verified the statistical efficiency of the procedure in the selected examples. Further study is required in order to evaluate more completely the performance of the algorithm and compare it with the performance of alternative techniques. References R. 0. Schmidt, A Signal Subspace Approach to Multiple Emitter Location and Spectral Estimation, Ph.D. dissertation, Stanford University, Stanford, California, Y. Rockah, P.?GI. Schultheiss, Array Shape Calibration Part I: Far-Field Sources, IEEE Trans. Acoustics Speech and Signal Processing, Vol. ASSP-35, No. 3, pp , March Y. Rockah, P. M. Schultheiss, Array Shape Calibration Part 11: Near-Field Sources, IEEE Pans. Acoustics Speech and Signal Processing, Vol. ASSP-35, No. 6, pp , June J. T. Lo, S. L. Marple, Eigenstructure Methods for Array Sensor Localization, ICASSP S7, pp A. J. Weiss and B. Friedlander, Array Shape Calibration A Maximum Likelihood Approach, IEEE Trans. Acoustics Speech and Signal Processing, In press. A. J. Barabell et. al. Performance Comparison of Super resolution Array Processing Algorithms, Project Report TST-72, Lincoln Laboratory, MIT, Lexington, MA, May D. G. Luenherger Introduction to Linear and Nonlinear Programming, Addison-Wesley Publishing Company, G. Dahlquist, A. Bjorck, Numerical Methods, Prentice Hall, Inc., Englewood Cliffs, New Jersy, M. Wak, T. J. Shan, and T. Kailath, Spatio Tempmal Spectral Analysis By Eigenstructure Methods, IEEE fins. Acoustics Speech and Signal Pmcessing, Vol. ASSP-32, No. 4, pp , August