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

Convolution of the weighted impulse train with produces a continuous-time linearly interpolated signal

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 , delaying by samples (an integer), then finally downsampling by , as depicted in Fig.4.7 [96]. The integers and are chosen so that , where the desired fractional delay.

The convolution interpretation of linear interpolation, Lagrange interpolation, and others, is discussed in [407].

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About the Author: Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.