I am implementing a fractional delay for resampling purposes, using the Farrow structure to enable continuously variable delays. For now I am simply using windowed sinc functions as my low-pass filters. My method was originally as follows:

  1. Create a series of $K$ $N$-sample windowed sinc functions covering the delay range: $N/2-1$ to $N/2$, and all the fractional delay values ($d$) in between.
  2. For each sample across all the filters, create a $K-1^{th}$ order polynomial in d to replace the coefficients.
  3. Stick the poynomial coefficients into a Farrow structure filter, and Bob's your uncle.

However, on reading some papers this for example, it became apparent that I need to consider the "overall filter"; all the separate polyphase branch filters from the farrow structure lumped together. Now, taking this into account, if I crate my individual filters as above, ranging from $N/2-1$ to $N/2$, then I have a problem, as shown in the image. The "overall filter" is not a nice lowpass function anymore. Essentially it the two filters at $N/2-1$ and $N/2$ are the same, just delayed 1 sample, so we get the reptition of values shown in the plot.

Bad sinc

My attempt at solving this is instead to create my initial filters over the range N/2-1+0.5/K : N/2-0.5/K. This way, they are still evenly distributed in terms of delay and now the "overall filter" looks great:

Good sinc (yes I know it's not windowed :)

BUT... now my delays do not cover a full sample of fractional delay, but instead $1-1/K$ samples. The only way to approach a full 1-sample range is to increase $K$, the number of filters created at the start, and this is not ideal as it just increases computation and I am VERY restricted in that sense.

I also tried a second approach, creating $K$ original filters, then approximating with an $L^{th}$ order polynomial, where $L < K$. This allows me to create a large ($K$) number of filters to start with, so I can get very close to the full N/2-1 : N/2 range, then approximate with a low order polynomial. This is OK, but it does introduce some unwanted artefacts into the "overall filter" response.

So, my questions is this: Is there a way to have it all; a low-order farrow fractional delay filter, that covers the full range of a sample delay, without introducing any nasties into the frequency response of the "overall filter"?

Any thoughts welcome :)

P.S. I know the sinc functions in the plots are rectangular windowed and thus have a high sidelobes, but this is not my problem.


2 Answers 2


Do you think this link could help you?


I'm not sure I understand 100% of your problem, but IIRC when implementing fractional delays with say a FIR filter of 13 coefficients, you should stick with a group delay between 6 and 7.


Use a lower order polynomial than K-1 and feed the polynomial fitter more sample points than K from your windowed Sinc, including below and above +-N/2. Otherwise the lobe interpolators within the Farrow structure won't splice together nicely without larger discontinuities at the splice points.

  • $\begingroup$ Ahhh, so you go beyond the required delay range. That didn't occur to me! Will give it a shot, thanks $\endgroup$ Jan 14, 2016 at 8:51

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.