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 [410].

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Center for Computer Research in Music and Acoustics (CCRMA), Stanford University