The ideal FDDF may be defined by reference to continuous time-delay
filters as described (see [1]) in equation
1, where we assume that the sampling rate
is 1. From a time sequence
we rebuild the original band
limited continuous signal, then we delay it and
finally we re-sample it in order to get 
.
This definition is no longer consistent when residual power remains
at the Nyquist frequency. This can be understood since the phase at
Nyquist frequency for any real time series is to be null.
For instance, the sequence
can not be time-shifted because it
doesn't define a unique continuous signal. Notice that, in this case,
the impulse response, the shifted cardinal sine
, is not
absolutely summable. Consequently FDDFs are not BIBO
filters. We limit the acceptable
input sequences to those without power at the Nyquist frequency. In
this subspace of time sequences, FDDFs are consistent BIBO filters.
