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
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 . 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 .