784 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 46, NO. 3, MARCH 1998 An Efficient Algorithm for Solving General Periodic Toeplitz Systems Mrityunjoy Chakraborty Abstract— An efficient algorithm is presented for inverting matrices which are periodically Toeplitz, i.e., whose diagonal and subdiagonal entries exhibit periodic repetitions. Such matrices are not per symmetric and thus cannot be inverted by Trench’s method. An alternative approach based on appropriate matrix factorization and partitioning is suggested. The algorithm provides certain insight on the formation of the inverse matrix, is implementable on a set of circularly pipelined processors and, as a special case, can be used for inverting a set of block Toeplitz matrices without requiring any matrix operation. I. I . NTRODUCTION A linear system of equations of the form is said to be a periodic Toeplitz system with period if the matrix satisfies the periodicity condition: Such matrices typically represent the correlation structure of cyclostationary processes and have been studied in [1] and [2] in the context of periodic time series analysis, where the periodicity of the matrix elements along each diagonal and subdiagonal has been exploited to obtain efficient parameter estimation algorithms. In addition, such matrices also play a vital role in efficient realization of multichannel modeling and filtering algorithms, as shown in [1] and [2] and indirectly in [3] and [4]. An example of a set of such matrices with period, say, 3, can be constructed as where each matrix shown is periodic Toeplitz although, in each case, only the (1, 1)th element is seen to repeat itself since the period chosen is 3, whereas the dimension is . The matrices are, in fact, chosen carefully to represent the cross-correlation matrices of a cyclostationary process with period , measured at three successive indices of time. Such matrices overlap in a certain manner, as explained later in Section II. The algorithms presented in [1]–[4], however, apply only to certain specific sets of equations involving such matrices, namely, Yule–Walker (YW) equations [1] and modified Yule–Walker (MYW) equations [2]. This correspondence takes up the more general problem of solving arbitrary systems of the form , where is non-Hermitian periodic Toeplitz. An efficient algorithm to compute has been presented, which, apart from providing insight on the structure of , also has an interesting pipelined implementation. Since a periodic Toeplitz matrix acquires the so-called block Toeplitz structure when for an integer , the proposed algorithm can also be used for efficient inversion of block Toeplitz matrices with the additional advantage of not requiring any matrix computation. Manuscript received April 10, 1996; revised June 10, 1997. The associate editor coordinating the review of this paper and approving it for publication was Prof. Pierre Comon. The author is with the Department of Electronics and Electrical Communi- cations Engineering, Indian Institute of Technology, Kharagpur, India. Publisher Item Identifier S 1053-587X(98)01334-8. II. REVIEW OF PERIODIC TOEPLITZ SYSTEMS Consider a process , which is given to be periodically WSS with period , implying that , where denotes the autocorrelation function of , i.e., The general correlation matrix for the th index is then given by , where It is easy to verify that (1a) and (1b) Thus, for each order , there is a total of matrices: one each for the indices The objective is to compute ’s for each of these indices in an efficient manner. It will be useful in this context to consider the MYW equations for , solved in [2] and given, for the th index and the th order, by (2a) (2b) where , and The two unknowns and are given by the inner products of the first and the last rows of with the vectors and , respectively. In [2], an order-recursive procedure for solving (2a) and (2b) has been presented that exploits the periodicity along the diagonals of to obtain a pipelined realization, involving processors and computations. Interestingly, , like , is also periodically Toeplitz with period , and thus, the same algorithm can be used to solve the MYW equations for as well. We denote the corresponing solutions for (2a) and (2b), respectively, by and with and substituting for and The inversion algorithm proposed in this paper is based on the so- called “ ” and “ ” factorizations of , where the “ ” and the “ ” matrices are constructed using both and For the Hermitian case, however, only half of these solutions will be required as, in that case, , implying that and III. THE INVERSION ALGORITHM A. Derivation of the Inversion Formula We begin by considering the factorization of Con- struct (3) 1053–587X/98$10.00  1998 IEEE Authorized licensed use limited to: INDIAN INSTITUTE OF TECHNOLOGY KHARAGPUR. Downloaded on June 24, 2009 at 02:21 from IEEE Xplore. Restrictions apply.