MULTI-CHANNEL PARAMETRIC ESTIMATOR FAST BLOCK MATRIX INVERSES

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1 MULTI-CANNEL ARAMETRIC ESTIMATOR FAST BLOCK MATRIX INVERSES S Lawrence Marle Jr School of Electrical Engineering and Comuter Science Oregon State University Corvallis, OR Marle@eecsoregonstateedu hilli M Corbell, Muralidhar Rangwamy Air Force Research Laboratory Sensors Directorate anscom AFB, MA hillicorbell@hanscomafmil, MuralidharRangwamy@hanscomafmil ABSTRACT The otimal adative linear combiner beamformer weights for a sensor array are exressed in terms of the inverse of the multi-channel MC covariance matrix Rather than form an estimate of the covariance matrix directly from the available data and inverting it, an alternative direct estimate of the inverse may be obtained by forming arametric MC linear rediction estimates and then exressing the inverse in terms of these arametric MC estimates The resulting arametric estimate of the inverse is tyically more accurate than inverting the estimate of the covariance matrix This aer reveals, for the rst time, the structure of the the inverse of the covariance matrix for the MC version of the covariance let squares linear rediction algorithm The inverse structure involves roducts of triangular block MC Toelitz matrices, which leads to ft comutational solutions for the otimal weights Index Terms Adative Arrays, Matrix Decomosition, Matrix Inversion, Radar Signal rocessing, Multichannel 1 INTRODUCTION The outut y w T x of a linear combiner sace-time lter oerating on an arbitrary geometry sensor array is exressed in terms of an inner roduct of a multi-channel MC weight vector w of dimension M 1 and a comarable dimension MC data vector from the array For a stationary array osition does not change with time, the MC data vector is a function of just satial dimension M For a non-stationary array osition changes with time, the MC data vector is a function of both satial dimension M and temoral dimension N An examle that motivates this aer is the radar sace-time adative rocessing STA roblem, summarized in [1] It is well known that the otimal adative weights that minimize the variance of the outut y while sing a signal from Dr Marle s research suorted by the Air Force Of ce of Scienti c Research AFOSR under the Summer Faculty Fellowshi rogram SFF administered by the American Society for Engineering Education ASEE Dr Rangwamy s research suorted by AFOSR under roject 2304 a referred steering vector direction reresented by the vector e is w R 1 e 1 for which the MC covariance matrix R is tyically estimated ˆR 1 R x[r]x [r] 2 R r1 over R meured realizations of the MC data vector x owever, we have found that imroved estimates of the inverse MC covariance matrix may be obtained using MC arametric estimation algorithms because the inverse covariance matrix can be exressed directly in terms of the estimated MC arameters This aer summarizes the relationshi of the inverse to the MC arameters Details of the actual erformance of the adative array using the MC arametric aroach will be found in comanion aers resented at the Asilomar 2006 and DAS 2006 conferences For reference, we rst illustrate the inversion formula in terms of the MC arameters in the known covariance ce, and show that the inverse can be exressed in terms of roducts of block triangular Toelitz structures Next, we reveal for the rst time the inverse obtained from MC data using the MC covariance method of linear rediction Surrisingly, the structure of the inverse can still be exressed in terms of roducts of block triangular Toelitz structures, which leads to ft comutational solutions for the otimal weight vectors covered in the other conference aers 2 MULTI-CANNEL LINEAR REDICTION ARAMETRIC MODEL AND INVERSE FOR KNOWN COVARIANCE The MC forward linear rediction error LE of model order for the MC signal x[n; r] of M channels at each realization r is de ned e a [n; r] x[n; r]+ A [k]x[n k; r] 3 k /07/$ IEEE II 1137 ICASS 2007

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3 with dimension M 1, inwhicha [k] are the MC forward linear rediction arameter matrices of dimension M M This may be exressed in vector inner roduct form e a [n; r] I a x [n; r] 4 for which a A [1] A [] is a M M block row vector of the MC forward L arameters and x 1 [n; r] x[n; r] x[n; r], x [n; r] x M [n; r] x[n ; r] 5 are M 1 and M +1 1 column vectors, resectively, of data samles Similarly, the MC backward linear rediction error LE is de ned e b [n; r] x[n ; r]+ b I x [n; r] B [k]x[n + k; r] k1 for which b B [] B [1] is a 1 block row vector scalar dimension: M M of the MC backward L arameters note that the backward L arameter vector h reversed time indexing relative to the forward L arameter vector The M M MC forward LE variance is both de ned and exressed a E { e a [n]e a [n] } I I a R 7 in which R is the M M MC block Toelitz autocovariance matrix [it is also hermitian symmetric with a scalar dimension of M +1 M +1] R R[0] R[1] R[ 1] R[] R [1] R[0] R[ 2] R[ 1] R [ 1] R [ 2] R[0] R[1] R [] R [ 1] R [1] R[0] 8 comosed of the stationary M M scalar-dimensioned MC autocovariance matrix elements r 11 [m] r 1M [m] R[m] 9 r M1 [m] r MM [m] in which r jk [m] is the scalar cross-covariance between channels j and k at time lag index m In a similar manner, the M M MC backward LE variance is given by b E { e b [n]e b [n] } b I b R I 10 a 6 The choices for the forward and backward L arameters a and b that minimize the variances a and b can be shown [3] to satisfy the air of MC Yule-Walker normal equations I a R a 0 11 b I R 0 b 12 in which 0 is a M M-array of all-zeros De ne the M M MC artial correlation coef cient Δ E { e a 1[n]e b 1[n 1] } 13 and the M M MC lattice lter arameters Γ a A [] and Γ b B []; note that A [] B [], in contrt to the one-dimensional ce where they were identical The ft MC Levinson algorithm [3] can solve the M +1 M +1 Eqs 11 and 12 in a number of comutational stes roortional to M 2, rather than roortional to the usual M 3, while avoiding the exlicit formation of the block matrix R Assuming the initialization a 0 b 0 R[0], the four stes of the MC Levinson-like recursion, that solves for the MC L arameters a k and b k and the MC LE variances a k and b k over orders k 1,,,are Δ k R[k] I a k 1 14 R[1] Γ a k Δ k b 1 k 1 15 Γ b k Δ k k 1 16 I a k I a k Γ a k 0 bk 1 I 17 I b k 0 b k 1 I + Γ b k I ak a k I Γ a kγ b k a k 1 19 b k I Γ b kγ a k b k 1 20 This algorithm reduces to 1-D Levinson recursion when M 1 one channel The referred form of the block Toelitz inverse [3] [2] which is useful to this roject, and which also deends only on the lt arameter order comuted, is R 1 A a A B b B 21 in which the block triangular Toelitz matrices are given I A [1] A [] A 0 A [1] 0 0 I 22 II 1138

4 0 B [] B [1] B 0 B [] a a a 1 b b b 1 3 MULTI-CANNEL COVARIANCE LEAST SQUARES LINEAR REDICTION ARAMETRIC MODEL AND INVERSE A full let squares minimization against all the MC linear rediction arameter matrices a and b simultaneously is the bis of the MC covariance method of let squares linear rediction To set u the roblem for a let squares solution, the MC forward LE and MC backward LE of order can be exressed M N R block row vectors when there are a nite number of N samles and R realizations for M channels e a e a [ +1;1]e a [N; R] I a X 26 e b e b [N;1]e b [ +1;R] b I X 27 in which the M+1 N R block rectangular Toelitz data matrix X and M R block data vector X[k] are de ned X X[ +1] X[N ] X[N] X[1] X[ +1] X[N ] 28 X[k] x[k;1] x[k; R] 29 The magnitude squared error sums [or variance estimates, if divided by N R] can then be exressed ˆ a e a ea 30 ˆ b e b eb 31 Minimizing the trace of each of these estimated variances [3] then leads to the following +1-block-dimensioned normal equations I a X X b I X X ˆ a 0 0 ˆ b In general, the MC forward and backward L arameters in the let squares ce are not comlex conjugates of the other, unlike the situation in the one-channel known autocovariance ce Comaring Eqs 32 and 33 to the recursions in the known covariance ce of the revious section, one can see that the block non-toelitz +1 +1let-squaresbed roduct matrix X X h relaced the role of the block Toelitz +1 +1autocovariance matrix R When the MC covariance let squares linear rediction algorithm is used, the inverse 1 X X will be used in lieu of R 1 in all the detection statistics discussed in [5] Desite the fact that let-squares normal Eqs 32 and 33 do not have a block Toelitz matrix, the non-toelitz X X matrix is the roduct of two rectangular-shaed block Toelitz data matrices X and X Thisissuf cient to generate an order-recursive ft comutational algorithm that requires a number of comutational oerations roortional to 2, rather than the usual solution roortional to 3, to solve simultaneously for both forward a and backward b MC L arameters The MC let-squares-bed ft algorithm requires the introduction of two additional gain adjustment vectors, de ned c X X x [N] x [N] x [N ] 34 d X X x [ +1]x [ +1]x [1] 35 in which the R +1M gain vectors are c c [0] c [] and d d [0] d [] We will also need to de ne the M M gain vector variances c I x [ +1] X X 1 x [ +1] 36 d I x [N] X X 1 x [N] 37 Note that trace tr{ c } 0and trace tr{ d } 0 The key MC Levinson-like order-udate recursions in the MC covariance let squares linear rediction ce are I a I a Γ a 0 b t adj 1 I b 0 b t adj 1 I I + Γ b I a 1 0 time-udate re- and the time-adjusted L arameter b t adj cursion h the form b t adj b + C 1 c 1 + C 2 d 1 40 for unseci ed M M matrix constants C 1 and C 2 Similar order and time udates for c, d, e c [n], ande d [n] areaart of the ft MC algorithm, but not shown here The inverse matrix 1 X X can be exlicitly exressed 1 X X A a A B b B + C 1 c C 1 D 41 1 d D 1 II 1139

5 in which the block triangular Toelitz matrices are A I A [1] I 0 0 A [ 1] A [ 2] I 0 A [] A [ 1] A [1] I B [] B B [2] B [3] 0 0 B [1] B [2] B [] 0 C 1 c 1 [0] c 1 [ 2] c 1 [ 3] 0 0 c 1 [ 1] c 1 [ 2] c 1 [0] 0 D 1 d 1 [0] d 1 [ 2] d 1 [ 3] 0 0 d 1 [ 1] d 1 [ 2] d 1 [0] are formed from the forward linear rediction arameters A [m], forward linear rediction squared error variance a, backward linear rediction arameters B [m], backward linear rediction squared error variance matrix b, gain adjustment block vector arameters c [m] and d [m], and matrix gain adjustment variances c and d These are de ned matrix equations sociated with the covariance linear rediction ce The structures of b, c,andd are similar to that shown for a 4 REFERENCES [1] M Rangwamy, M C Wicks, R Adve, and T B ale, Sace-time adative rocesing: a knowledge-bed ersective for airborne radar, IEEE Signal rocessing Mag, vol 23, 51 65, Jan 2006 [2] S L Marle, Jr, erformance of multichannel autoregressive sectral estimators, roc ICASS, vol1, , Ar 1986 [3] S L Marle, Jr, Digital Sectral Analysis with Alications, Englewood Cliff, NJ: rentice all, 1987 [4] S L Marle, Jr, Minimum variance sectral estimator ft algorithms bed on covariance and modi- ed covariance method of linear rediction, roc 35th Asilomar Conf Signals, Syst, Comut, , Nov 2001 [5] M Rangwamy, J R Roman, D W Davis, Q Zhang, B imed, and J Michels, arametric adative matched lter for airborne radar alications,, IEEE Trans Aeros Electron Syst, vol 36, , Ar 2000 a II 1140