4.8 Filterbanks with > 2 branches

Filterbanks with>2 branches

Thus far the previous discussion on filterbanks has concentrated on "modern" filterbanks with only two branches.There are two standard ways by which the number of branches can be increased.

M-band filterbanks

The ideas used to construct two-branch PR-FIR filterbanks can be directly extended to the $M$ -branch case. (See Vaidyanathan and Mitra ) This yields, for example, a polynomial matrix $H(z)$ with $M$ rows and $M$ columns. For these $M$ -band filterbanks, the sub-bands will have uniform widths $\frac{2\pi }{L}$ radians (in the ideal case) .

Multi-level (cascade) filterbanks

The two-branch PR-FIR filterbanks can be cascaded to yield PR-FIR filterbanks whose sub-band widths equal $2^{-k}\pi$ for non-negative integers $k$ (in the ideal case). If the magnitude responses of the filters are not well behaved,however, the cascading will result in poor effective frequency-selectivity. Below we show a filterbank in which the low-frequency sub-bands are narrower than the high-frequencysub-band. Note that the number of input samples equals the total number of sub-band samples.

We shall see these structures in the context of the discrete wavelet transform.