On the Multidimensional Prediction and Filtering Problem and the Factorization of Spectral Matrices

Since the liuear prediction and filtering of nmltiple correlated stat ionary randonl signals involves the solution of a matr ix integral equation, several methods of solution of this equation have been developed. In this paper the direct solution by complex variable techniques is extended from the one-dimensional case. The problem reduces to the factorization of a spectral density matr ix and this factorization is the principal topic of the paper. Under very general conditions, a factorization procedure has been developed by Wiener and Masani (1) 3 for discrete t ime series. Their technique is applied here with some modifications to the factorization of the spectral density matrices of continuous processes. A simplified procedure is developed also for the case where the elements of the spectral density matrix are rational functions of frequency. Examples illust ra te the general technique and the simplified procedure. INTRODUCTION In the theory of the linear prediction and filtering of correlated stationary random signals, the form of the desired multidimensional network is specified by the solution of a matrix integral equation (2) f f t (r -= Rm(r) , _< r _< (1) ~7)fi (o-)d(7 0 where R(T -~) is a matrix containing as elements the covariance functions of the received waveforms ej(t), that is, Rjk(r) = E{e~(t)ek(t qr)}, (2) 1 Research staff member, IBM Research Center, Yorktown Heights, N. Y. 2 Associate Professor of Electrical Engineering, Princeton University, Princeton, N. J. 3 The boldface numbers in parentheses refer to the references appended to this paper. (Note-The Franklin Institute is not responsible for the statements and opinions advanced by contributors in the JOVRNAL.) 87 8 8 li~. W O N G ANI) J. B. ' I ' I I()M:xS [J. F. I. Further , if mj(t) are the desired outputs , then the ou tputs of the opt imum network are given by mj*(t) = ~ f ~ hjk(r)ek(t -r)dr, (3) and the elements of R~(r) by R. ,j~(r) = E{ej(t)mk(t + r) t. (4) Thus the matrices occurring in Eq. 1 are defined by Eqs. 2, 3, and 4, with the notat ion ~ denoting transpose in Eq. 1. The method of undetermined coefficients was proposed originally by Wiener (2) for the solution of Eq. 1. An alternate method of solution (3) is to reduce Eq. 1 to a system of simultaneous differential equations with constant coefficients, and represents a generalization of the work of Zadeh and Ragazzini (4). It is the purpose of this paper to call a t tent ion to a third technique which may not 1)e widely known. A MATRIX WIENER-HOPF APPROACH In a manner analogous to the one-dimensional case (2, S) Eq. 1 can be rewritten as f ~ R ( ~ ~ ) f ~ ( ~ ) d ~ R . , ( r ) = f ( r ) , ~ < < (5) T oD with f(r) = O, r > 0. (6) The Fourier transforms of R(r ) , h(r ) , R,,(r) and f(r) may be denoted by o(iw), I-I(i~0), . ~(ico) and F(ico), respectively, for example, . (ioa) =a f 2 e-~'R(r)dr" Taking transforms of both sides of Eq. 5 results in a, (%0)IrI (i~0) ,l,m (i~0) = F (i~0). (7) It follows from Eq. 6 that F(io)) is analytic and bounded in the upper half ~0-plane (6). Now, let o(i~0) be factored into the form . (¢~) = yvt (¢~) ~r (¢,o), (s) where ~v, (i~) ~ @ ( ¢~). (9) Aug., i96i . ] F A C T O R I Z A T I O N OF SPECTRAI~ ~ '{ATRICES 8 9 The factorization given by Eq. 8 is to l)e such that the matrix Re(ico) is element-by-element analytic and bounded in the lower half c0-plane, and its deter lninant Det Re(ico) is free of zeros in the lower half c0-plane. These two properties of ~le (ico) imply that ~ (ico) and its inverse ~te-, (ico) both represent transfer functions of physically realizable mult idimensional networks. 4 The problem of realizing the factorization of Eq. g will be deferred until the next section. Multiplying Eq. 7 from the left by [-~l(¢:co)]< yields '|'(ico)H(ico) [-q'~(ico)] '*,,,(ico) = [~(ico)]-'F(ico). (10) Now, since both w(ico) and ft(ico) are analytic and bounded in the lower half c0-plane, the time response of w (ico)ITI (ico) must vanish for negative time, that is, 2rr, ~ [',r(ico)f-I(ico)]e~'rdco = 0, r < 0. (11) Similarly, it follows that 2rr r > O. (12) Therefore, the time response of gd(ico)fI(ico) must be equal to the nonnegative-time portion of the time response of [-Ret(ico)]-lq~m(/w). This fact can be expressed by the relationship w(ico)ITI(ico) = 1 £ ~ e ~ * {f_~e'~'E~r*(i.)~ t.m(iu)du} dr. (13) From Eq. 13, the transposed transfer matrix ITI(ico) is given by which represents a generalization of the one-dimensional solution (5). THE FACTORIZATION OF THE SPECTRAL DENSITY MATRIX It can be seen that an explicit solution of Eq. 1 can be found provided tha t the factorization (Eq. 8) of the spectral density matrix ~(ico) can be effected. One factorization technique has been developed recently by Youla (~). Youla's technique is applicable whenever o(ic0) is element-by-element a rational function of co. An earlier factorization procedure was developed by Wiener and Masani (1) for discrete time T h u s I|e(ico) Illay be said to be the t rans fe r fum' t ion of a m i t l h n u m q f l m s e m l d t i d i m e m sional lint work. 90 E. WONG AND J. B. THOMAS [J. F. I. series. Their technique is applied here with some modifications to the factorization of the spectral densi ty matrices of continuous processes. The procedure will now be explained and illustrated. The matr ix ,I~(ic0) to be factored is assumed to satisfy the following conditions : (1) ,I~(i~0) is Hermi t ian on the real ~-axis, tha t is, ; for ~ real. (2) The de te rminan t Det o ( i~) satisfies the inequal i ty (15) f_~ I ln ]Det o(¢~) I I do0 oO k 2 + . ,2 (16) where ] [ denotes the absolute value, and k is any real finite positive constant (see Appendix). (3) Let the smallest and largest eigenvalues of q,(i00) on the real oa-axis be Xl and X2, respectively, then 0 < X l < X 2 < m. (17) Condit ions (1) and (2) are satisfied for all physical processes. Condition (3) is satisfied if the spectral densi ty matr ix o(i,0) is determined experimentally, as shown by Wiener and Masani (1), but need not be satisfied by theoret ical ly postulated spectral densi ty matrices. We note tha t the diagonal terms ¢jj(i~0) of the matrix ,~(i~o) represent auto-spectral densities, and can be factored easily into the form = (18) where the ~IG, (i~0) are analytic, bounded and free of zeros in the lower half c0-plane. Now let r (i~0) be a diagonal matrix with elements 1 lPik(ioa) ___a ~I'j~(ico) 6~k. (19) It is apparen t tha t the matrix ,l,'(i~0), defined by