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