1 Introduction Modern superresolution direction finding techniques such as MUIC [4], Minimum Variance [1], Maximum Likelihood [12], and subspace fitti

Transcription

1 Array Calibration in the Presence of Multipath Amir Leshem 1 Λ and Mati Wax 2 y 1 Delft University of Technology Dept. Electrical Engineering Mekelweg 4, 2628CD Delft, The etherlands 2 RAFAEL P.O.B 2250 Haifa, Israel January 24, 2000 Abstract We present an algorithm for the calibration of sensor arrays in the presenceofmultipath. The algorithm is based on two sets of calibration data obtained from two angularly separated transmitting points. We show the similarity between the calibration problem and blind identification of IMO systems, and analyze the identifiability of the problem. imulation results demonstrating the performance of the algorithm are included. Keywords: Array calibration, blind channel identification, multipath, DOA estimation. P EDIC : Λ While preparing this work Amir Leshem was with RAFAEL and the Institute of Mathematics, Hebrew university of Jerusalem. leshem@cas.et.tudelft.nl y mati@uswcorp.com 1

2 1 Introduction Modern superresolution direction finding techniques such as MUIC [4], Minimum Variance [1], Maximum Likelihood [12], and subspace fitting methods [8], presume the knowledge of the array response. As the analysis of these techniques show, [11], [6], [7], any inaccuracy in the presumed array response results in severe degradation of performance. The measurement of the array response, referred to as array calibration, is therefore a crucial step in the implementation of these techniques. The existing calibration techniques, [5], [9], are based on modeling the array response by a free-space model perturbed by an unknown coupling matrix and sensor location uncertainty. These unknown parameters are estimated together with the unknown signal parametrs, assuming known or unknown source location. Yet, for general arrays with arbitrary sensor responses, these methods are no longer adequate, since these modelling assumptions are no longer valid. In this paper we address the problem of measuring the array response of arrays, with arbitrary sensor response in the presence of multipath. This problem is important since multipath is essentially unavoidable in practice and it sets the limit on the achievable calibration accuracy. The organization of the paper is as follows. In section 2 we formulate the problem. In section 3 we present the proposed solution. In section 6 we present simulation results demonstrating the performance of the algorithm. Finally, in section 7 we present some concluding remarks. In sppendix 5 we consider the similarity and the differences between the calibration problem as presented here and the problem of blind identification of multiple FIR channels. 2 Problem Formulation Let a( ) denote the p 1 vector of the array response to a source impinging from direction. The array calibration problem amounts to measuring a( ) for 2 [0; 2ß). It is usually performed by transmitting a signal from some location, rotating the array and measuring the array response at each angle. Unfortunately, in many cases the measured response is composed not only of the direct path from the transmitting point, but also of multiple reflections from the surroundings, see Figure 1. In the case of arbitrary array response we can no longer resolve the multipath from a single set of measurements, since the measured data can be considered as the "true" array manifold. This situation is similar to the problem of blind identification of IMO systems where in without any a-priory knowledge of the signal, a single channel is not identifiable while two channels are identifiable even using second order statistics only. To cope with the multipath problem we propose to carry out the calibration twice, i.e. rotate the array and measure the received array vector as a function of, yet each time use a different transmitting point. Let y l ( ) denote the p 1 vector recieved at the angle from the l'th transmitting point (l = 1; 2). Assuming that the reflections are considered as point sources, and all multipath effects are completely coherent with calibrating signals, i.e., each path differs by a complex reflection coefficient from the direct path, we get where y l ( ) =ρ 1;l a( + 1;l )+ Xr l i=2 i;l - the direction of the i'th reflection, in the l'th set. ρ i;l a( + i;l )+n l ( ) l =1; 2 (1) 2

3 ρ i;l - r l - the complex coefficient representing the phase shift and the amplitude of the i'th reflection, in the l'th set. the number of reflections in the l'th set. n l ( ) - the noise vector for the angle in the l'th set. ince the array manifold is measured relative to some arbitrary point, and the relative angle between the measurement points is known, we can assume without loss of generality that ρ 1;1 = 1 and 1;1 = 1;2 = 0 o. Also, since the reflecting objects remain fixed while the transmitting point change, the relative directions of the reflections are different i.e., i;1 6= i;2 (i 6= 1). Assuming that the calibration process consists of measurements taken uniformly on 2 [0; 2ß), it follows from (1) that the measured data is given by X y l ( 2ßk )= r l i=1 ρ i;l a( 2ßk + i;l)+n l (k) ; l =1; 2; k =0;:::; 1 (2) where we use n l (k), to emphasize that the noise is not angle dependent.ote that we have included the direct path with the multipath. In our solution we make the following assumption : A1: All reflections i;l are a multiple of the basic rotation 2ß. Assumption A1 serves as a very good approximation when the grid is fine. The array calibration problem can now be formulated as follows. Given the two measured data sets Φ y l ( 2ßk )Ψ 1 k=0 (l =1; 2), estimate the array manifold Φ a( 2ßk )Ψ 1 k=0. 3 The Maximum Likelihood Estimator The proposed solution is based on two steps : (i) Estimating the reflections' parameters fρ l ; l g (ii) Estimating the array manifold using the estimated reflections. where ρ l =[ρ 1;l ;:::;ρ rl ;l] is the vector of the reflection coefficients at the l'th set of measurements. l =[ 1;l ;:::; rl ;l] isthevector of the reflections' DOAs at the l'th set of measurements. For the first step we have two approaches. The first approach uses the L estimator which is identical to the MLE under the assumption of white gaussian noise. This estimator is derived in this section. The second approach uses a simplified L, which we derive in the next section. The second step is derived by a least squares solution, which is the MLE under the assumption of white gaussian noise. This step is performed identically in the two approaches. using the results of the first step. To carry out the derivation of the MLE, let w l denote the 1vector whose k'th element is ρ i;l if 2ßk = i;l, and zero otherwise. Mathematically this is expressed as w l (k) = Xr l i=1 ρ i;l ffi( 2ßk i;l) (3) 3

4 where ffi( ) is the delta function. Let a m be the 1 array manifold of the m'th sensor,» 2ß 2ß ( 1) T a m = a m (0) ;a m ;:::;a m (4) y m;l =[y m;l (1);:::;y m;l ()] T (5) n m;l =[n m;l (1);:::;n m;l ()] T (6) and y m;l (k) and n m;l (k) are the m'th element ofy l ( 2ßk )andn(k), respectively. With this notation we can rewrite equation (2) as y m;l = P r l i=1 ρ i;l Pki;l am + n l = P r l = P r l i=1 w l(k i;l )P ki;l Λ am + n l = h P 1 w i=1 l(k i;l )P ki;l ia m + n l w (7) k=0 l(k)p k am + n l where k i;l = i;l 2ß and P k isapermutation matrix which rotates the 0'th element ofa into the k'th position defined by ρ 1 (P k ) m;n = if m + k = n or m + k = n 0 otherwise (8) Thelastequality in (7) is due to the fact the w(k) =0if69i s.t. 2ßk A l = 1 X it follows that A l is an circulant matrix generated by w l : A l = Thus we can rewrite (2) as k=0 = i;l. Denoting w l (k)p k (9) w l (0) w l ( 2) w l ( 1) w l ( 1) w l (0) w l ( 2).. w l (1) w l ( 1) w l (0) (10) y m;l = A l a m + n m;l 1» m» p (11) ince A l is a circulant matrix it is diagonalized by the DFT matrix of order and its eigenvalues are given by the DFT of the generating vector w l [2]. Therefore, F H A l F = diagffw l g =diagf^w l g; (12) where F is the normalized DFT matrix of order (FF H = I) and ^w l = Fw l is the DFT of w l, given by ^w l (k) = p 1 Xr l 2ßk j ρ i;l e k i;l : (13) Hence i=1 A l = Fdiagf^w l gf H : (14) With this representation of the matrices A l we can derive a somewhat simplified expression of the MLE. 4

5 »» ym;1 A1 Let y m = and A =. y m;2 A 2 Assuming that the noise is white and gaussian from (11), the MLE is given by [~a 1 ;:::;~a p ; 1 ~ ; 2 ~ ; ~ρ 1 ; ~ρ 2 ] = arg min a 1 ;:::;a p; 1 ; 2 ;ρ 1 ;ρ 2 Minimizing first with respect to a m weobtain ow, from the definition of A and (14) we obtain px m=1 ky m Aa m k 2 : (15) ~a m =(A H A) 1 A H y m : (16) A H A = F(j ^W 1 j 2 + j ^W 2 j 2 )F H ; (17) where ^W l =diag(fw l ). ubstituting (17) and (14) into (16) yields where ~a m = FD( ^W H 1»y m;1 + ^W H 2»y m;2 ); (18) D =(j ^W 1 j 2 + j ^W 2 j 2 ) 1 ; (19) and»y m;l = F H y m;l. Finally, substituting (18) and (14) into (15) we obtain [~ 1 ; 2 ~ ; ~ρ 1 ; ~ρ 2 ]=arg min 1 ; 2 ;ρ 1 ;ρ 2 which can also be rewriten as [~ 1 ; 2 ~ ; ~ρ 1 ; ~ρ 2 ]=arg min 1 ; 2 ;ρ 1 ;ρ 2 px 2X m=1 l=1 px m=1 l=1 ky m;l F ^W l D ^W H l F H y m;l k 2 (20) 2X fl fl fl fl I ^W l D ^W H l»ym;lflfl 2 : (21) otice that this estimator involves all the reflections parameters, i.e. the DOA's and the reflection coefficients, in a highly non-linear fashion, and hence is computationally unattractive. 4 imple L estimator In this section we derive a simplified L estimator for the reflections DOA's and reflection coefficients. This estimator together with the estimator for a m given in equation (18) consists the simplified L estimator for the array manifold. ubstituting (14) into (11), we obtain F H a m =diag(^w l ) 1 (»y m;l»n m;l ) (22) where»n m;l = F H n m;l ince this holds for both sets of measurements we obtain which can be rewritten as ^W 1 1 (»y m;1»n m;1 )= ^W 1 2 (»y m;2»n m;2 ) (23) ^w 1 ffi»y m;2 ^w 2 ffi»y m;1 = ^w 1 ffi»n m;2 ^w 2 ffi»n m;1 (24) 5

6 where ffi denotes elementwise multiplication. ince the right hand side of (24) is noise", a possible L estimator for the reflections' parameters is given by [~ 1 ; 2 ~ ; ~ρ 1 ; ~ρ 2 ]= min k^w 1 ffi»y m;2 ^w 2 ffi»y m;1 k 2 : (25) 1 ; 2 ;ρ 1 ;ρ 2 ubstituting (13) into (25) yields X 1 [~ 1 ; 2 ~ ; ~ρ 1 ; ~ρ 2 ]= min 1 ; 2 ;ρ 1 ;ρ 2 k=0 k»y m;2 (k) r 1 X i=1 2ßk j ρ i;1 e ki;1»y m;1 (k) r 2 X i=1 2ßk j ρ i;2 e ki;2 k 2 : (26) where k i;l = i;l 2ßk. Denoting 1 (k; ) =»y m;1 (k)e jk (27) and B m = 0 2 (k; ) =»y m;2 (k)e jk (28) 2 (0; 2;1 ) ::: 2 (0; r1 ;1) 1 (0; 1;2 ) ::: 1 (0; r2 ;2).. 2 ( 1; 2;1 ) ::: 2 ( 1; r1 ;1) 1 ( 1; 1;2 ) ::: 1 ( 1; r2 ;2) ρ =[ρ 2;1 ;:::;ρ r1 ;1;ρ 1;2 ;:::;ρ r2 ;2] T (29) we can rewrite (26) as a linear problem in ρ (recall that we have assumed ρ 1;1 =1and 1;1 =0 0 ) [~ 1 ; ~ 2 ; ~ρ] = min 1 ; 2 ;ρ kb m( 1 ; 2 )ρ +»y m;2 k 2 (30) This estimator is based on the data of the m'th sensor only. Clearly, we can improve the performance by combining the information from all sensors. This yields [~ 1 ; 2 ~ ; ~ρ] = arg min 1 ; 2 ;ρ px m=1 To evaluate this estimator we first rewrite it in matrix form as where and 1 C A kb m ( 1 ; 2 )ρ +»y m;2 k 2 (31) [~ 1 ; ~ 2 ; ~ρ] =arg min 1 ; 2 ;ρ kb( 1; 2 )ρ +»y 2 k 2 (32) B( 1 ; 2 )= B T 1 ( 1 ; 2 );:::;B T m( 1 ; 2 ) Λ T (33)»y 2 =»y T 12;:::;»y T p2λ T (34) Minimizing first with respect to ρ, with 1 ; 2 being fixed, we obtain the well known least squares solution ~ρ = (B( 1 ; 2 ) H B( 1 ; 2 )) 1 B( 1 ; 2 ) H»y 2 (35) ubstituting (35) back into (32), the resulting estimator of the directions-of-arrival of the reflections is h i ~ 1 ; 2 ~ = arg min kp? 1 ; B( 1 ; (»y 2 ) m;2)k 2 (36) 2 6

7 where P? B( 1 ; is the projection on the orthogonal complement of the subspace spanned by the 2 ) columns of B( 1 ; 2 ), P? B( 1 ; 2 ) = I B( 1; 2 ) B H ( 1 ; 2 )B( 1 ; 2 ) 1 B H ( 1 ; 2 ): (37) The structure of this estimator is similar to that of the deterministic signal maximum likelihood DOA estimator. Hence, the optimization methods developed for this problem, including, the Alternating Projections [12], and the clustering methods [3], can be used. With the estimated parameters at hand we can use equation (16) to estimate the array manifold. First we obtain an estimate ~w of ^w by substituting the ρ's and the 's into equation (13). Thid yields ~w l (k) = 1 p Xr l By substituting ~W = diag(~w l )into equation (14) we obtain i=1 ~ρ i;l e jk~ i;l (38) ~A l = F ~W l F H : (39) which when substituted into (16), yields ~a m =(~A H ~A) 1 ~A H y m = F~D ~W H 1»y m;1 + ~W H 2»y m;2 (40) where ~A =» ~ A 1 ~A 2 : (41) 5 The relation to the IMO blind equalization and identifiability results In this section we cast the calibration problem as the identification of a ingle Input Multiple Output (IMO) system. This will enable us to derive identifiability conditions, as well as present an alternative derivation of the L estimator for reflections parameters. First note that we can rewrite (11) as where y m;l = h l Λ a m + n m;l l =1; 2 (42) h l (k) = ρ ρi;l if i;l = 2ß( k) 0 otherwise That is the measurements are just a spatially filtered version of the "signal" a m by FIR filters with coefficients ρ i;l at i;l and zeros otherwise. Our problem can now be stated as: Given the output of two linear systems driven by the same signal reconstruct the input signal. The problem is in the form of blind identification. However several differences between our problem and the conventional blind identification problem exist: (1) The signal is periodic with known period. (2) The measurements are taken along a single period. (43) 7

8 (3) We have several pairs of output signals, one for each element of the array. (4) The filters are sparse, i.e. most of the coefficients are zero. (5) The length of the filters may be the same as the number of samples. We next develop the L estimator as a natural variation on the method of [10] in the frequency domain. Using the convolution theorem (remembering that our signal is periodic) we obtain where ffi denotes element-wise multiplication. Hence ^y m;l = ^h l ffi ^a m (44) ^y m;1 ffi ^h 2 = ^y m;2 ffi ^h 1 (45) After some algebraic manipulations, using the relation between w and h, we obtain the noiseless version of equation (24). The fact that our L estimator can be derived using the approach of [10] enables us to give a sufficient condition for identifiability. This condition is obtained by translating the sufficient condition for identifiability of[10]. However since our channel is sparse we will be able to obtain stronger identifiability conditions. To that end note that the problem is identifiable for channels with signature (r 1 ;r 2 ) (i.e., has a unique solution with at most r 1 reflections in the first set of measurements, and at most r 2 reflections in the second set in the noiseless case) if (32) has a unique solution with the true ( 1 ; 2 ), while having no solution with any other substitution of 0 ; For this condition to hold, it is sufficient and necessary that for any pair ( 1 ; 2 ) where 1 2 C 2r 1 and 2 2 C 2r 2 the matrix B( 1 ; 2 ) has full column rank. We shall elaborate on this to obtain some further conditions, easier to verify. To simplify notation we shall work with a single matrix B k, instead with the full matrix B. The generalization is straightforward though notationally complex. Let Λ l = 2 6 4»y m;l (0)...»ym;l( 1) (46) and C( 1 ; 2 )= e j0 21 e j0 2r 1 ;1.. 0 e j( 1) 11 e j( 1) 2r 1 ;1 e j0 12 e j0 2r 2 ;2 0.. e j( 1) 12 e j( 1) 2r 2 ; (47) ote that B m ( 1 ; 2 )=[Λ 1 Λ 2 ] C( 1 ; 2 ) (48) The second matrix is always full column rank due to the Vandermonde structure of each block. Thus the identifiability condition boils down to having the first matrix preserve the column rank. imilarly to the condition in [10] we can now split this condition into two conditions. The first 8

9 demanding informative array manifold, and the second is a condition on identifiable channels. Factoring Λ l similarly to equation (7) we obtain Λ l = 2 6 4»a m (0)...»a m;l( 1) ^w m (0)... ^w m;l( 1) (49) Thus the following immediately follows: Theorem 5.1 Let L =2(r 1 + r 2 1). Assume that for every 0» k» L 1»a m (k) 6= 0and ^w 1 (k) and ^w 2 (k) are not simultaneously zero, then there is a unique solution to the problem (30). ote that our conditions depends on the number of reflections rather than the channels length. We can of course weaken the condition above, however the above condition for the array manifold typically holds, leaving us with conditions on the channels, which hold e.g., if the channel polynomials do not have common zeros on the unit circle. 6 imulation results In this section we present the results of several simulated experiments that demonstrate the performance of the algorithm. In all experiments the array consisted of two sensors, 2:5 apart and the number of reflections was 2, i.e., r 1 = r 2 =2. In the first experiment the directions of the signals were 1;1 = 1;2 = 0 o, and those of the multipath were 1;2 =15 o ; 2;2 =40 o (ote that this does not limit the generality of the simulations, since as explained earlier, we can align the direct paths of the two measurements, and only estimate the angles of the multipthas relative to the direct path. Typically after alignmentthemultipath will arrive with different AOA's, due to the fixed geometry of the reflectors). The reflection coefficients were ρ 1;1 = ρ 1;2 = 1;ρ 1;2 = 0:03 + 0:05j; ρ 2;2 = 0:13 + 0:19j, which corresponds to signal to multipath ratios of 25dB and 13dB respectively. At each R we have performed 25 Monte-Carlo trials. Figure (2) shows the array manifold error averaged over all DOA's as a function of the R. While the solid line represents the error after application of the algorithm, the dashed line presents the array manifold error in the first set of measurements. The array manifold estimation ME is computed by ME = 1 M MX X m=1 k=1 k^a( 2ßk 2ßk ;m) a( )k2 (50) where M is the number of trials and is the grid size. In all experiments M =25and =360 and ^a( 2ßk 2ßk ;m) is the m'th estimate of a( ). To gain further insight into the performance of the algorithm we present in figure (3) the results of a single experiment performed at R of 50dB. We can see that the error after the application of the algorithm is much smaller than the error at the raw set of measurements. Moreover we see that the error is about the same for all DOA's. In the second experiment the directions of the signals were 1;1 = 1;2 =0 o, and those of the multipath were 1;2 = 15 o ; 2;2 = 25 o. The reflection coefficients were ρ 1;1 = ρ 1;2 = 1;ρ 1;2 = 0:21 + 0:05j; ρ 2;2 =0:13 + 0:19j corresponding to signal to multipath ratios of 13dB and 12dB. At each Rwehave performed 25 trials. Figure (4) shows the array manifold error averaged over all DOA's as a function of the R. While the solid line represents the error after application of the algorithm, the dashed line presents the array manifold error in the first set of measurements. Figure (5) the results of a single experiment performedatr of 50dB. We can see that the error after 9

10 the application of the algorithm is much smaller than the error at the raw set of measurements. Moreover we see that the error is about the same for all DOA's. In these two experiments one clearly sees that the improvement is not only obvious but the error reduces to the level of the measurement noise. This demonstrates that the multipath is completely removed. In the third experiment, the relative angular separation between the reflections in the first set of measurements was held fixed at 15 o, while the relative angular separation in the second set of measurements varied from 19 o to 51 o in steps of 4 o. The reflection coefficients were ρ 1;1 =1;ρ 1;2 = 0:11 + 0:1j; ρ 2;1 = 0:7;ρ 2;2 = 0: :2j and the R was 30 db. The results are presented in figure 6. otice that the performance of the algorithm is essentially independent of the angular separation. 7 Concluding Remarks We have presented a novel method for the calibration of sensor arrays in the presence of multipath. The method is based on measuring the array manifold from two angularly separated locations, and involves a solution of a multidimensional optimization. The method does not depend on the relative angular locations of the reflections. 8 Acknowledgement The authors would like to thank the anonymous reviewers for their comments which greatly improved the exposition of this paper. References [1] J. Capon. High resolution frequency-wavenumber spectrum analysis. Proc. IEEE, pages , [2] Philip J. Davis. Circulant Matrices. John Wiley and ons, [3] A. Leshem and A. Y. Kasher. Maximum likelihood direction finding using clustering methods. In Proceedings of ICP Beijing93, pages , [4] R.O. chmidt. A ignal ubspace Approach to Multiple Emmiter Location and pectral Estimation. PhD thesis, tanford university, [5] C.M. ee. Method for array calibration in high resolution sensor array processing. IEE Proc. - Radar, onar, avig., pages 90 96, June [6] A. windlehurst and T. Kailath. A performance analysis of subspace-based methods in the presence of model errors - part 1: The music algorithm. IEEE Trans. on P, pages , July [7] A. windlehurst and T. Kailath. A performance analysis of subspace-based methods in the presence of model errors - part 2: Multidimensional algorithms. IEEE Trans. on P, pages , eptember

11 [8] M. Viberg and B. Ottersten. ensor array processing based on subspace fitting. IEEE Trans. on AP, pages , May [9] A.J. Weiss and B. Friedlander. Array shape calibration using sources in unknown locations - a maximum likelihood approach. IEEE Trans. on AP, pages , December [10] G. Xu, H. Liu, L. Tong, and T. Kailath. A least squares approach to blind channel identification. IEEE Trans. on P, pages , December [11] J. Yang and A. windlehurst. The effects of array calibration errors on DF-based signal copy performance. IEEE Trans. on P, pages , ovember [12] I. Ziskind and M. Wax. Maximum likelihood localization of multiple sources by alternating projections. IEEE Trans. on AP, pages , October

12 List of Figures 1 The calibration setup ( with one reflection) Array manifold errors vs. R. The multipath conditions: ρ 1;1 = 1;ρ 1;2 = 0:03 + 0:05j; ρ 2;1 = 1;ρ 2;2 = 0:13 + 0:19j. 11 = 21 = 0 o, 12 = 15 o, 22 = 40 o. M1 = 25dB; M 2 =13dB. olid line: Error after application of the algorithm. Dashed line: Error due to multipath (first set of measurements) Array manifold errors vs. DOA. The multipath conditions: ρ 1;1 =1;ρ 1;2 =0:03 + 0:05j; ρ 2;1 = 1;ρ 2;2 = 0:13 + 0:19j. 11 = 21 = 0 o, 12 = 15 o, 22 = 40 o. M1 = 24dB; M 2 = 13dB. Bottom line: Error after application of the algorithm. Upper lines: Error due to multipath (In two sets of measurements) Array manifold errors vs. R. The multipath conditions: ρ 1;1 = 1;ρ 1;2 = 0:21 + 0:05j; ρ 2;1 =1;ρ 2;2 =0:13+0:19j. 11 = 21 =0 o, 12 =15 o, 22 =25 o. M1 =13dB. M 2 =12dB. olid line: Error after application of the algorithm. Dashed line: Error due to multipath (first set of measurements) Array manifold errors vs. DOA. The multipath conditions: ρ 1;1 =1;ρ 1;2 =0:21 + 0:05j; ρ 2;1 =1;ρ 2;2 =0:13+0:19j. 11 = 21 =0 o, 12 =15 o, 22 =25 o. M1 =13dB. M 2 = 12dB. Bottom line: Error after application of the algorithm. Upper lines: Error due to multipath (In two sets of measurements) The error as a function of The multipath conditions: ρ 1;1 = 1;ρ 1;2 = 0:11 + 0:2j; ρ 2;1 =0:7;ρ 2;2 =0: :2j. R=30dB

13 θ Calibrating ource Antenna Array Figure 1: The calibration setup ( with one reflection). 13

14 5 Array Manifold ME vs. R M E R [db] Figure 2: Array manifold errors vs. R. The multipath conditions: ρ 1;1 = 1;ρ 1;2 = 0:03 + 0:05j; ρ 2;1 =1;ρ 2;2 =0:13 + 0:19j. 11 = 21 =0 o, 12 =15 o, 22 =40 o. M1 =25dB; M 2 =13dB. olid line: Error after application of the algorithm. Dashed line: Error due to multipath (first set of measurements). 14

15 10 Array manifolds ME before and after calibration ME [db] Angle of arrival [deg] Figure 3: Array manifold errors vs. DOA. The multipath conditions: ρ 1;1 = 1;ρ 1;2 = 0:03 + 0:05j; ρ 2;1 =1;ρ 2;2 =0:13 + 0:19j. 11 = 21 =0 o, 12 =15 o, 22 =40 o. M1 =24dB; M 2 =13dB. Bottom line: Error after application of the algorithm. Upper lines: Error due to multipath (In two sets of measurements). 15

16 10 Array Manifold ME vs. R M E R [db] Figure 4: Array manifold errors vs. R. The multipath conditions: ρ 1;1 = 1;ρ 1;2 = 0:21 + 0:05j; ρ 2;1 =1;ρ 2;2 =0:13 + 0:19j. 11 = 21 =0 o, 12 =15 o, 22 =25 o. M1 =13dB. M2 =12dB. olid line: Error after application of the algorithm. Dashed line: Error due to multipath (first set of measurements). 16

17 10 Array manifolds ME before and after calibration ME [db] Angle of arrival [deg] Figure 5: Array manifold errors vs. DOA. The multipath conditions: ρ 1;1 = 1;ρ 1;2 = 0:21 + 0:05j; ρ 2;1 =1;ρ 2;2 =0:13 + 0:19j. 11 = 21 =0 o, 12 =15 o, 22 =25 o. M1 =13dB. M2 =12dB. Bottom line: Error after application of the algorithm. Upper lines: Error due to multipath (In two sets of measurements). 17

18 16 Array Manifold ME vs. eparation M E separation [deg] Figure 6: The error as a function of The multipath conditions: ρ 1;1 = 1;ρ 1;2 = 0:11 + 0:2j; ρ 2;1 =0:7;ρ 2;2 =0: :2j. R=30dB. Dashed line: The error due to the multipath. olid line: The error after the application of the algorithm. 18