Linear phase paraunitary filter banks: theory, factorizations and designs

-M channel maximally decimated filter banks have been used in the past to decompose signals into subbands. The theory of perfect-reconstruction filter banks has also been studied extensively. Nonparaunitary systems with linear phase filters have also been designed. In this paper, we study paraunitary systems in which each individual filter in the analysis synthesis banks has linear phase. Specific instances of this problem have been addressed by other authors, and linear phase paraunitary systems have been shown to exist. This property is often desirable for several applications, particularly in image processing. We begin by answering several theoretical questions pertaining to linear phase paraunitary systems. Next, we develop a minimal factorizdion for a large class of such systems. This factorization will be proved to be complete for even M . Further, we structurally impose the additional condition that the filters satisfy pairwise mirror-image symmetry in the frequency domain. This significantly reduces the number of parameters to be optimized in the design process. We then demonstrate the use of these filter banks in the generation of M-band orthonormal wavelets. Several design examples are also given to validate the theory. I . INTRODUCTION IGITAL filter banks have been used in the past to D decompose a signal into frequency subbands [1][ 121. The signals in different subbands are then coded and transmitted. Such schemes are popular for encoding data from speech and image signals. The process of decomposition and eventual reconstruction are done by what is termed as the “analysis-synthesis’’ filter bank system shown in Fig. 1. In this scheme, the Hi ( z ) are the analysis filters and Fi(z) are the synthesis filters. The boxes with 1M denote the decimators, or the subsampling devices, whereas the boxes with t M denote the expanders, which increase the sampling rate. Their definitions are as in [ 13, Fig. 2 is a representation of the subband coding scheme in terms of the polyphase matrices [3]. E ( z ) is the poly131. Manuscript received May 17, 1992; revised June 10, 1993. The Guest Editor coordinating the review of this paper and approving it for publication was Dr. Ahmed Tewfik. This was supported in part by the National Science Foundation under Grant MIP 8919196, in part by Tektronix Inc., and in part by Rockwell International. A. K. Soman and P. P. Vaidyanathan are with the Department of Electrical Engineering, California Institute of Technology, Pasadena, CA 91125. T. Q. Nguyen is with the Lincoln Lab, Massachusetts Institute of Technology, Lexington, MA 02 173. IEEE Log Number 9212190. Fig. 1 . A M-channel uniform filter bank 4 Blocking Mechamsm P Unbloclang Mechanism Fig. 2. A filter bank drawn in terms of the polyphase matrices phase matrix corresponding to the analysis filters, and R ( z ) is the polyphase matrix corresponding to the synthesis filters. The decimators and expanders have been moved across the polyphase matrices using the noble identities [3]. It has been shown that it is indeed possible to perfectly reconstruct the original signal using such analysissynthesis systems [5]-[ 121. In particular, this can be done by filters that have finite impulse response (FIR), and are hence guaranteed to be stable. One way to do this is to let R ( z ) = E ‘ ( z ) , and then choose the matrix E(z ) so that both matrices are FIR.