Multi-Band Filter Design

Multi-band filters are used to separate the frequency components of a signal and pass certain selected frequency ranges while filtering out other frequency ranges. This divides the frequency spectrum into a series of pass bands and stop bands separated by transition bands. A multi-band filter is characterized by the limits and characteristics of each band and by a few overall characteristics.

Pass bands are characterized by their desired gain values (usually +0 dB or x1) and their ripple values which measure the maximum desired deviation from the gain values. Stop bands are characterized by their rejection levels, which represent the minimum acceptable attenuation. Transition bands cover the portions of the spectrum between the stop band and the pass bands; they represent unspecified regions within which no constraints are placed on the filter. A narrow transition band indicates a sharp transition between bands, and will, in general, force either a poorer fit or a longer filter.

In addition to the limits and characteristics of each band, an FIR filter design depends on the sampling frequency of the input data. In fact, the usable frequency range is between DC (0 Hz) and half the sampling frequency, which is called the folding frequency. A filter designed for data sampled at one frequency will not behave the same with data sampled at a different rate: the band characteristics remain the same, but the band limits change proportionally with the change in sampling frequency.

Two additional characteristics of an FIR filter are its length and symmetry. The length refers to the number of “taps” or coefficients; increasing the length generally improves the fit at the expense of design speed, execution speed and sample delay. Even length and odd length filters display significantly different behavior. Filter symmetry (even, odd or none) refers to the symmetry of the coefficients and is independent of the evenness or oddness of filter length, although there is some interplay between the two characteristics.

Multi-band filters are typically designed to have even symmetry and odd length, although other choices may be appropriate to guarantee zero response at DC or at the folding frequency or to provide phase shifting effects or non-integer sample delays.

As an example, consider the following multi-band audio filter designed with even symmetry and a sampling frequency of 10 KHz:

When this filter is generated with 33 taps, the coefficients and frequency response graph (in decibels) are as follows:

Taps: -0.0682469 -0.0214445 0.000569996 -0.0207654 .079821 0.021396 0.00299209 -0.051855 0.16201 -0.0871818 0.00290553 0.183798 -0.581661 0.200654 -0.000530207 -0.262153 0.80428 -0.262153 -0.000530207 0.200654 -0.581661 0.183798 0.00290553 -0.0871818 0.16201 -0.051855 0.00299209 0.021396 0.079821 -0.0207654 0.000569996 -0.0214445 -0.0682469

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