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:

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

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 [452] of the fractional-delay FIR filter $h(n)$ is

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

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

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

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.