next up previous
Next: An Ideal Transfer Function Up: Ideal Fractional Delay Previous: Ideal Fractional Delay

From Continuous to Discrete

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 tex2html_wrap_inline723 we rebuild the original band limited continuous signal, then we delay it and finally we re-sample it in order to get tex2html_wrap_inline725 .

  figure25

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 tex2html_wrap_inline727 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 tex2html_wrap_inline729 , is not absolutely summable. Consequently FDDFs are not BIBOgif 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.



Stephan Tassart
Wed May 21 17:49:28 MET DST 1997