DSPRelated.com
Free Books
Online · Live + on demand
February 9-11, 2027
Price goes up Nov 1: register now to lock in today's rate
Register Now →
   Free Books    Physical Audio Signal Processing  

Fractional Delay Filters

In fractional-delay filtering applications, the interpolator typically slides forward through time to produce a time series of interpolated values, thereby implementing a non-integer signal delay:

$\displaystyle \hat{y}\left(n-\frac{N}{2}-\eta\right) = h(0)\,y(n) + h(1)\,y(n-1) + \cdots h(N)\,y(0) $

where $ \eta\in[-1/2,1/2]$ spans the central one-sample range of the interpolator. Equivalently, the interpolator may be viewed as an FIR filter having a linear phase response corresponding to a delay of $ N/2 + \eta$ samples. Such filters are often used in series with a delay line in order to implement an interpolated delay line (§4.1) that effectively provides a continuously variable delay for discrete-time signals.

The frequency response [449] of the fractional-delay FIR filter $ h(n)$ is

$\displaystyle H(e^{j\omega}) \eqsp \sum_{n=0}^N h(n)e^{-j\omega n}. $

For an ideal fractional-delay filter, the frequency response should be equal to that of an ideal delay

$\displaystyle H^\ast(e^{j\omega}) \eqsp e^{-j\omega\Delta} $

where $ \Delta\isdeftext N/2 + \eta$ denotes the total desired delay of the filter. Thus, the ideal desired frequency response is a linear phase term corresponding to a delay of $ \Delta$ samples.

Next Section:
Lagrange Interpolation Optimality
Previous Section:
Interpolation of Uniformly Spaced Samples