Digital Signal Processing, 10th Edition

12Filter Banks

The signal decomposition and reconstruction techniques presented in the previous chapter can be generalized to any number of sub-bands, using banks of more than two filters. In that case, in principle, decimation can be performed at the output of the analysis filters, but, in general, it is preferable to decimate at system input in order to benefit from the combination of a polyphase network and DFT, thus minimizing the computational complexity.

12.1 Decomposition and Reconstruction

In the realization of filter banks using polyphase networks and DFT, as explained in Section 10.7, the operations involved are reversible and this leads to the arrangement shown in Figure 12.1 for the decomposition and reconstruction of a signal [1, 2].

The difficulty, in practice, consists of applying the operations associated with the functions .

The filter H(Z) which serves as a basis for the process, sometimes called the prototype filter, has a polyphase decomposition whose elements satisfy relation (10.8):

(12.1)

If the prototype filter H(Z) has a cutoff frequency of less than f_s/2N and infinite attenuation for frequencies greater than or equal to f_s/2N – that is, if aliasing of the spectrum due to sampling at the rate 1/NT is negligible – one can write:

(12.2)

Under these ...