Discussion: Interpolation Can of Worms

Erik de Castro Lopo wrote:

> 
> If you have a true half band filter of the form:
> 
>     h[n] = {..., g[-2], 0, g[-1], 0, g[0], 1, g[1], 0, g[2], 0, g[3], ...}
> 
> then an efficient polyphase implementation is possible.
> 
> Erik


Even if the filter is not a half band filter, a polyphase implementation 
buys you the efficiency.  What the half band does, is it reduces the odd 
branch to a simple delay and the even branch is a symmetric FIR filter. 
  The more general case (non-half-band) gets you two sub filters that 
you need to sum the outputs for decimation or commutate for interpolation.

Andor wrote:


> 
> Some while back, Fred posted a link to a program that computes
> equiripple halfband filters
> (http://groups.google.ch/group/comp.dsp/msg/5476b541eb7c55ce?dmode=source)
> - that should beat the pants off any filter for factor 2 upsampling
> (w.r.t. computational efficiency). BTW, we've had this (or similar)
> discussion before:
> 
> http://groups.google.ch/group/comp.dsp/browse_frm/thread/8de91e30641d10e8/4e69c0ef1c9f4edf?#4e69c0ef1c9f4edf
> 
> My 0.02$ is that the appropriate filter (either halfband or Erik's
> proper downsampler) depends on the application. For audio sample rate
> conversion, halfband filters usually are critical. However, for
> sidechain upsampling or metering, halfband filters are very useful.
> 
> Regards,
> Andor
> 

Goodman and Carey wrote a paper on efficient decimation/interpolation 
back in the 70's I think that addresses this.  They  presented a set of 
half band filters and a methodology for selecting which filters to use 
in a chain of filters for a decimating / interpolating multi-rate 
filter.  The longest of their halfband filters was IIRC 19 taps. 
Basically, the higher frequency filters can be shorter filters since the 
more gradual cut-off is filtered out by subsequent filters anyway.