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Convolution Interpretation

Linearly interpolated fractional delay is equivalent to filtering and resampling a weighted impulse train (the input signal samples) with a continuous-time filter having the simple triangular impulse response

$\displaystyle h_l(t) = \left\{\begin{array}{ll} 1-\left\vert t/T\right\vert, & \left\vert t\right\vert\leq T, \\ [5pt] 0, & \hbox{otherwise}. \\ \end{array} \right. \protect$ (5.4)

Convolution of the weighted impulse train with $ h_l(t)$ produces a continuous-time linearly interpolated signal

$\displaystyle x(t) = \sum_{n=-\infty}^{\infty} x(nT) h_l(t-nT). \protect$ (5.5)

This continuous result can then be resampled at the desired fractional delay.

In discrete time processing, the operation Eq.(4.5) can be approximated arbitrarily closely by digital upsampling by a large integer factor $ M$ , delaying by $ L$ samples (an integer), then finally downsampling by $ M$ , as depicted in Fig.4.7 [96]. The integers $ L$ and $ M$ are chosen so that $ \eta \approx L/M$ , where $ \eta$ the desired fractional delay.

Figure 4.7: Linear interpolation as a convolution.
\includegraphics[width=0.8\twidth]{eps/polyphaseli}

The convolution interpretation of linear interpolation, Lagrange interpolation, and others, is discussed in [410].

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``Physical Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2010, ISBN 978-0-9745607-2-4
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Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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