CAUSAL FIR FILTER BANKS WITH ARBITRARY SYSTEM DELAY Gerald Schuller Bell-Laboratories 600 Mountain Ave. Murray Hill, NJ07974 email: schuller@lucent.com Tanja Karp University of Mannheim B6, 26 68131 Mannheim, Germany email: karp@rumms.uni-mannheim.de ABSTRACT A design method for causal bi-orthogonal PR FIR -band filter banks is described, which allows an explicit control over system delay, independent of the filter length, with the lowest possible delay equal to the blocking delay of samples. The design method is very general and can be applied to non-uniform filter banks but also treats uniform modulated filter banks as a special case. 1. INTRODUCTION Filter banks are used in much of the same way as block transforms, for producing a short time frequency domain representation with frequency bands of a signal (analy- sis), and reconstructing the signal from this representation (synthesis), but they are a more general approach. For time-domain data the cascade of the analysis and synthesis filter bank introduces a certain delay, called the system delay. For spatial domain data this is a spatial shift. The system delay is an important property of filter banks. It has not received some attention but recently [1, 2, 3]. Tra- ditionally, filter banks were designed to be orthogonal. In this case the system delay is connected to the filter length: if is the length of each filter in the analysis and synthe- sis filter bank, the standard system delay of orthogonal filter banks is samples. However, many applications re- quire analysis filters with a high stopband attenuation and a small transition bandwidth and thus long analysis filters (if we restrict ourselves to the case of FIR filters) as well as a short system delay. This problem has been partly overcome with the design of bi-orthogonal filter banks [1, 2, 3] where analysis and synthesis filters do not need to be time reversed versions of each other. In this paper we give a formulation that allows for the design of general -band bi-orthogonal filter banks with perfect reconstruction (PR) as well as bi- orthogonal modulated filter banks which are known for their Work was mainly done while with the University of Hanover, Hanover, Germany low implementation cost, where the system delay can be chosen independently of the filter length. 2. DEFINITIONS For an -band analysis/synthesis filter bank, the input is represented by an -dimensional vector composed of the downsampled input components Its -transform is the vector . The polyphase represen- tation for an -band filter bank with input signal , the subband signal , and the reconstructed signal is for the analysis and for the synthesis [4]. is the analysis polyphase matrix, the synthesis polyphase matrix. Causal filters have no taps at times before zero. This means that and contain no positive powers of . The filter bank is PR if where is a Shift Matrix, which circularly shifts the elements of a vector or matrix by one sample [5, 9], . . . . . . . . . . . . . . . The delay here consists of blocks of length mi- nus shifts of single samples. The system delay contains an additional so-called blocking delay of samples, which results from blocking the input samples into blocks of length in the polyphase formulation [4]. Hence the