Some algebraic aspects of signal processing Original Research Article

It has recently been shown in (M. Barnabei, L.B. Montefusco, Linear Algebra and applications 274 (1998) 367–388) that the algebraic-combinatorial notion of recursive matrix can fruitfully be used to represent and easily handle the basic operations of filter theory, such as convolution, up-sampling, and down-sampling. In this paper we show how the recursive matrix reinterpretation of two-channel FIR filter bank theory leads to a notable simplification in language and proofs, together with an easy and immediate generalization to the M-channel case. For example, in both 2-channel and M-channel cases, perfect reconstruction and alias concelation conditions can be restated in an algebraic language, thereby obtaining an easy and constructive proof using the fundamental properties of recursive matrices.