#### 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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Frequency Response of Linear Interpolation

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Minimizing First-Order Allpass Transient Response