Time-Varying Perfect Reconstruction Filter Banks for Finite Length Signals Vitor Silva and Luís de Sá Instituto de Telecomunicações, Departamento de Engenharia Electrotécnica, Pólo II da Universidade de Coimbra, 3030 Coimbra, Portugal Tel: +351 39 7006255, Fax: +351 39 7006247, Email: vitor@it.uc.pt Abstract: A new approach to the non-expansive two band decomposition of finite length signals is introduced. The technique is based on the time-varying perfect reconstruction filter banks concept. In this approach the main analysis filter bank is instantaneously switched, at the signal boundaries, to a shorter filter bank: the D 1 or Haar filter bank. The technique can be applied to all types of two band perfect reconstruction filter banks. A low complexity design method for the required reconstruction transition filters is presented. In opposition to other types of time-varying filter banks, the proposed analysis filter bank has good low pass and high pass characteristics. 1 Introduction One key problem in subband and wavelet coding systems, is the decomposition of limited size signals. If we consider the case of a two-channel filter bank and an input signal x[n] of even length N (non-zero samples for n N = - 01 1 ,, , ! ), we are interested in representing each of the two subbands components of x[n] by N 2 samples. However, the decimated outputs of the analysis filters have more than N 2 non-zero samples since the linear convolution of the input signal with the filter bank is an expansive operation. Several approaches have been developed in order to solve this problem. Some of these are based on linear signal extensions, namely, periodic (and symmetric periodic) extensions [1]-[3], zero padding [4] and replication of boundary values [3]. In the periodic extension approach the input signal is made periodic, with period N, so that the subbands are also periodic with period N 2 for all types of filters. The problem with this technique is the introduction of high frequency coefficients due to the artificial jumps at the signal boundaries. This is avoided if we use for example an half sample symmetric periodic extension where the input signal is made periodic with period 2N (other symmetric periodic extensions are possible, see [3]). As a consequence the subbands have a period of N but only N/2 samples are significant and need to be retained for reconstruction. This technique has been widely used with linear phase filters since, in this case, the subbands are also symmetric periodic. With non-linear phase filters the linear relations to obtain the complete period of N samples for each subband are not simple symmetrical relations. It is important to note that there is no need to make the signal periodic, e.g., of infinite duration. Actually in both the periodic and symmetric periodic extensions we only need to extend the signal some samples to its left and some samples to its right (the size of the lateral extensions depends on the size of the analysis and synthesis filters) so that the reconstructed signal equals the input signal for n N = - 01 1 ,, , ! . As can be easily shown given the linear relations defining the lateral extensions of the input signal it is possible to obtain linear relations to laterally extend the sets of N 2 samples of each subband [5]. Zero padding and replication of boundary values are just different examples of lateral extensions of the input signal. A different approach to the decomposition of finite length signals, without assuming signal extensions, has been recently introduced using time-varying boundary filters [6]-[9]. The main idea is to switch the main analysis and synthesis filters to different ones (so we have time-varying filters) when they operate near the boundaries of the input signal (in the analysis) or near the boundaries of the up-sampled subbands (in the synthesis). Some boundary filters solutions have been developed in a discrete time signal processing perspective for the two and M band orthogonal filter bank cases [6]-[9]. Similar developments have been introduced in the context of the wavelet theory [10]. It is also important to note that all signals extensions can be interpreted as special cases of time-varying boundary filters solutions. One problem with time-varying boundary filters is that, in general, they show very poor low and high pass characteristics, even when they are optimized in frequency [8], [9].