Vector Filter Banks and Multirate Filter Banks with Block Sampling z * z Xiang-Gen Xia and Bruce W. Suter Department of Electrical and Computer Engineering Air Force Institute of Technology Wright-Patterson AFB, OH 45433-7765 Abstract In this paper, we study general vector filter banks where the input signals and transfer functions in conventional multirate filter banks are replaced by vector signals and transfer matrices, respectively. We show that multirate fil- ter banks with block sampling and linear time invariant transfer functions studied by Khansari and Leon-Garcia are special vector filter banks where the transfer matrices are pseudo-circulant. We then present necessary and suf- ficient conditions for the alias free property, zyxwvutsrq FIR systems with zyxwvutsrqpon FIR inverses, paraunitariness for vector filter banks. We also present a necessary and suficient condition for paraunitary multirate filter banks with block sampling. 1 Introduction Multirate filter banks have been recently studied exten- sively, and found many applications in data compression, adaptive signal processing, numerical analysis and many other fields. As a natural generalization of multirate fil- ter bank theory, multirate filter banks with block sampling were recently discussed by Khansari and Leon-Garcia [l], where the traditional uniform down/up sampling in mul- tirate filter banks is replaced by block down/up sampling. More specifically, in traditional multirate filters, down- sampling by a factor M corresponds to taking the first sample in each block of M samples, while in multirate fil- ter banks with block sampling, the down-sampling by a factor M (called block down-sampling) involves taking the first N samples of each block of MN samples. Block up- sampling can be defined in an analogous manner. In block sampling, there are two parameters: zyxwvuts M, the sampling rate and N, the block size. In this paper, we will consider more general filter banks so called vector filter banks where the input signals are replaced by vector signals and the transfer functions are replaced by transfer matrices in conventional multirate filter banks. We will show that multirate filter banks with block sampling are a special case of vector filter banks when the transfer matrices are pseudo-circulant. zyxwvu 'This research was supported in part by the Air Force Of- fice of Scientific Research under Grants AFOSR-616-93-0019, AFOSR-PO-94-0004, AFOSR-616-94-0001, and AFOSR-616- zyxwvu 94-0025. In this paper, we focus on multirate filter banks with linear time invariant transfer functions and matrices. We show that multirate filter banks with block sampling and linear time invariant transfer functions studied by Khansari and Leon-Garcia [l] are special vector filter banks where the transfer matrices are pseudo-circulant. We present polyphase representations for vector filter banks. With the polyphase representations we present a necessary and sufficient condition for alias free vector fil- ter banks, which is that the product of analysis and syn- thesis polyphase matrices is generalized pseudo-circulant matrix. We then study necessary and sufficient conditions for perfect reconstruction vector filter banks. With these developments we present paraunitariness for a vector fil- ter bank for FIR paraunitary vector filter banks. We then apply the theory for vector filter banks to multirate filter banks with block sampling and present a necessary and sufficient condition for paraunitary multirate filter banks with block sampling. 2 Some Results on Vector Filter Banks We begin with some basic definitions. Throughout this paper, lowercase letters denote discrete-time signals, lower- case boldface letters denote N dimensional vectors blocked from the corresponding lowercase letters, captial letters denote z-transforms of the corresponding lowercase letter signals, and captial boldfwe letters denote matrices. All transfer functions and transfer matrices are assumed as linear time invariant (LTI). The block down/up sampling (or decimator/expander) by a factor M and a block size N is defined by (a) (M.N)-iolder decimator (b) (M.N)-folder expander Figure 1: Block down/up sampling 1020 U.S. Government Work Not Protected by US. Copyright