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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 line4.1) 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.
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Lagrange Interpolation Optimality
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