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Course by: Phil Schniter. E-mail the author

# Filterbanks with >2 Branches

Module by: Phil Schniter. E-mail the author

Summary: This module discusses two commons ways to design "modern" filterbanks with more than two 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 MM-branch case. (See Vaidyanathan and Mitra) This yields, for example, a polynomial matrix Hz H z with MM rows and MM columns. For these MM-band filterbanks, the sub-bands will have uniform widths 2πL 2 L radians (in the ideal case) Figure 1.

### Multi-Level (Cascade) Filterbanks

The two-branch PR-FIR filterbanks can be cascaded to yield PR-FIR filterbanks whose sub-band widths equal 2kπ 2 k for non-negative integers kk (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-frequency sub-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.

## References

1. P.P. Vaidyanathan. (1993). Multirate Systems and Filterbanks. Englewood Cliffs, NJ: Prentice Hall.
2. S.K. Mitra. (2001). Digital Signal Processing. (2nd edition). New York: McGraw-Hill.

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