OPTIMUM COMPACTION FILTERS FOR CYCLOSTATIONARY SIGNALS Ashish Pandharipande and Soura Dasgupta Department of Electrical and Computer Engineering, The University of Iowa, Iowa City, IA-52242, USA. Email: pashish@engineering.uiowa.edu and dasgupta@eng.uiowa.edu ABSTRACT This paper considers the energy compaction problem for wide-sense cyclostationary (WSCS) signals. Periodic Nyquist- filters and optimum compaction filters are de- fined and their properties examined. A procedure for de- signing compaction filters is given, and application to sub- band coding of WSCS signals is explored. 1. INTRODUCTION The energy compaction concept plays an important role in subband coding theory, and energy compaction filters find applications in the design of orthonormal subband coders. Optimality of filter banks is in the sense of maximizing cod- ing gain, which is a measure of the distortion due to subband quantization. The optimal orthonormal subband coding of WSS and WSCS signals has been treated in [11], [12] and [1], [6] respectively. It has been shown [8], [12] that the op- timal orthonormal filter bank in the WSS case can be con- structed by designing the analysis filters one at a time by choosing them to be optimal compaction filters for appro- priate power spectral densities (psd’s) derived from the in- put signal psd. Compaction filters thus are of interest due to their connection with optimal subband coding and prin- cipal component filter banks. Compaction filters have been treated in some detail for the WSS case in [8], [9], [12]. The compaction filter was formulated as an eigen problem in [8] and given a principal component approach. In [9], com- paction filters were derived by an energy analysis. Proper- ties of compaction filters have been further studied in [12]. In this paper, we develop the compaction filter concept in the context of wide-sense cyclostationary (WSCS) sig- nals whose importance has been described in [2], [3]. A signal is WSCS with period if for all : where denotes the expectation operator. Our goal is to develop the concept of an -periodic compaction filter Supported in part by US Army contract, DAAAD19-00-1-0534, and NSF grants ECS-9970105 and CCR-9973133. for such an -periodic WSCS signal. A linear filter with impulse response is called -periodic, if for all : (1.1) The definition of compaction filters in the WSS case relies on the notion of Nyquist-N or -band filters. Thus en route to the development of the compaction concept we will also formulate the notion of Nyquist-N filters for WSCS signals. We are motivated by the design of optimum subband coders for WSCS signals, and their relation to the com- paction process. Specifically the subband coders of interest are -channel uniform maximally decimated filter banks depicted in fig. 1. Here is WSCS with period and at time , is a -bit quantizer. When is Wide Sense Stationary (WSS) one selects the analysis fil- ters and the synthesis filters to be linear time invariant (LTI). Since here is WSCS with pe- riod , we assume that these filters are Linear Periodically Time Varying (LPTV) with period . The time index in and recognizes their lack of time invari- ance. . . . . . . . . . . . . Fig. 1. An -channel filter bank as subband coder In Section 2, we recapitulate the notions of Nyquist- filters and optimum compaction filters for WSS signals. Section 3 defines an -periodic Nyquist- filter, and de- rives some properties. Section 4 then introduces the notion of an optimum compaction filter for WSCS signals, relates it to the eigenvalues of certain psd matrices associated with the corresponding signal, proposes a design approach, and relates the compaction process to subband coding. Section 5 is the conclusion. II - 1201 0-7803-7402-9/02/$17.00 ©2002 IEEE