Continuous time-delay filters are analog all-pass filters whose
Laplace transform is 
. Extension of time-delay filters to
digital filters leads for integer delay d=p to a basic delay line
whose z-transform is 
. For any other rational value of the
delay d, fractional delay digital filters (FDDF) shall be defined by
reference to analog time-delay filters
[Crochiere and Rabiner, 1983]: from a time sequence 
we rebuild the original analog signal, then we delay it from the
proper delay and finally we re-sample it in order to get
. Notice that this process
makes sense if and only if the original time sequence
corresponds to the sampling of a
band limited analog signal. This means that 
has no component at the Nyquist frequency.
The Fourier transform 
of these filters exists and
is equal to 
. Notice that the z-transform of the
impulse response doesn't exist but we shall use the analytic extension
of 
which plays the same role as the usual transfer
function. This analytic extension is 
.