Analyse, synthèse et complexité de calcul de bancs de filtres numériques
Digital signal processing with filter banks, that is filtering a single signal with several filters and subsequent subsampling (or the converse, that is multiplexing), is an important subject with numerous applications (sub-band coding, transmultiplexing). This work first develops an analysis framework for signal processing by means of filter banks. The framework is based on matrix notation and a generalized polyphase representation of filter banks. This powerful formalism is used to derive fundamental results on filter banks. It is shown that signals can always be reconstructed without aliasing (or crosstalk) and in which cases the reconstruction can be perfect. The duality between sub-band coders and transmultiplexers is also demonstrated, thus unifying the analysis of these two cases. The synthesis of filter banks is considered next. Analytical and optimization methods are presented, together with examples of perfect FIR reconstruction for arbitrary N. Modulated filter banks (among them, pseudo-QMF banks) and generalizations (complex QMF and the bidimensional case) are then introduced. The computational complexity of filter banks is then considered, and it is shown how to substantially reduce the number of operations in some important cases (filter trees, modulated filter banks). This highlights the importance of fast transforms for the evaluation of filter banks. The last chapter is concerned with the computation of fast transforms (Fourier, cosine) and introduces a new algorithm for length N=2m transforms. This algorithm is generalized to a whole set of problems and always achieves the minimum known number of operations. Two implementations of this algorithm, one in hardware (a VLSI chip for a real-time video coder) and the other in software (efficient code for a signal processor) are described. In short, this work shows that filter bank problems are radically different from classical single filter problems. With the obtained results, a number of new perspectives on signal processing by filter banks have been opened.