Stringent filter requirements

Question

I am trying to resample a signal using Matlab's resample function. The documentation explains that conceptually, it is upsampling by p, then applying an interpolation filter, then downsampling by q, although it implements it using polyphase filters. If my original sampling rate is $f_s$, the upsampled signal is at rate $pf_s$ and the downsampled signal is at rate $\frac{p}{q}f_s$. I want my interpolation filter to have a cutoff at $\frac{p}{2q}f_s$ (I am downsampling, so q>p), but it is applied to the upsampled signal at rate $pf_s$, so the normalized cutoff frequency is then $\frac{1}{2q}$. I have a large q value (of 35) so my filter requirements are pretty stringent. I am using firls (Matlab's least-squares FIR generator) to try to generate a filter, but even an order 127 filter gives me too high of a cutoff value when I plot using freqz. I am seeing aliasing in the resampled data. I have also tried simply not specifying a filter and using resample with only 3 arguments (Matlab applies a default filter), but it aliases as well. I could always just increase the filter order, but processing time is prohibitive (it seems to be the case that doubling the number of taps roughly doubles the processing time). One of my coworkers suggested using polyphase filters as a way around this problem. I have been working through this document to try to understand how to implement one. From what I understand so far, I see how polyphase filters can be used to make the upsample/filter/downsample process more computationally efficient (and Matlab's resample already uses this), but it seems that polyphase filters use a filter bank based on a "master" filter that would still need a cutoff at $\frac{1}{2q}$. I was hoping maybe there is some simple solution, or perhaps somebody could explain to me how to use polyphase filters to solve this.

Edit: I suppose I should clarify that my cutoff is not right at $\frac{1}{2q}$, but my transition band/cutoff is somewhat lower so this only makes it even more stringent.

Answer 1

The polyphase structure allows for running all the filter banks at the lower sampling rate (the input rate for interpolation, and the output rate for decimation). Regardless the fundamental operation and performance of the filter is identical to the direct approach of upsample - filter - downsample. Whatever requirements are needed for the filter in that case result in the required coefficients for that filter, and those coefficients are then used as the coefficients for the bank of filters in the polyphase approach. So if you have $N$ coefficients, the $M$ polyphase banks will each have $N/M$ coefficients, more filters but all smaller and running at the lower rate which is the efficiency provided.

The filter coefficients map to the polyphase filter banks in row to column form as detailed further in this post: How to implement Polyphase filter?

Further, as detailed here, and not specific to polyphase implementation but the ideal interpolation filter or decimation filter in general, is to use multiband filters which concentrate the rejection at the image locations only (for interpolation) or alias locations only (for decimation) which can lead to significant efficiency improvements if the bandwidth occupied is sufficiently less than Nyquist at the final output rate. For fractional resampling, the multiband filter is designed with the combined rejection considering both the image and alias bands.