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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.
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) .
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.
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