DSPStore

Half Band Digital Filters

by Andy Bateman (DSPStore editor)

Whenever I am giving a training course on DSP algorithms, I always enjoy talking about Half-band filters. The simple act of choosing your filter response to be symmetrical about ? of the sample rate and as a result almost halving the complexity of your filter is such a simple yet valuable thing to do. I would encourage every designer to seek out half-band filter solutions wherever possible.

Half-band filters are in fact a special case of a more general class of filters often termed Mth-band filters, which exhibit symmetry about fs/2M.

An Mth band FIR filter has the property that every Mth coefficient of the filter (i.e. every Mth component of the filter's impulse response) is zero – apart from the centre coefficient (tap). This property is illustrated in Figure 1 for a third band filter, M=3.

The obvious benefit of using Mth band filters is that there is no need to perform the multiplication for the filter tap when the filter coefficient is zero! This of course significantly reduced the processing load, silicon area, or throughput of the filter depending on the implementation used.

This saving on filter complexity is greatest when M= 2, giving rise to a half-band filter. For a half-band filter, the frequency response of the filter must be symmetrical about fs/4, as shown in Figure 2. Strictly speaking, the ripple in the pass-band must equal the ripple in the stop-band and the transition band from pass-band to stop-band must have a symmetry about fs/4.

The mathematical description for the frequency response for a half-band filter is:-

H(e jw ) + H(e j(0.5.fs-w) ) = 1

This is also the definition of a Nyquist filter (i.e. a half-band filter is an example of a Nyquist filter). Nyquist filters are a very important pulse shaping filter in digital communications, often implemented as a ‘Raised Cosine Filter'. We shall look more closely at RRC filters in a later edition.

For a half-band filter, every other coefficient is zero – apart from the center tap, Figure 3. This leads to a very efficient FIR filter realisation, Figure 4, particularly when employed in interpolation and decimation applications.

It is most common to implement half-band filters (or more generally Mth-band filters) as FIR filters, but we can also benefit from reduced complexity by adopting half-band IIR filter solutions.

The symmetry of the coefficients in most FIR Mth-band filter designs means that they will exhibit linear phase (constant group delay response) with the only condition being that an ODD number of coefficients (taps) is used.

Note:- Use of half-band filters can seriously reduce your processing load!