More General Allpass Filters
- The series combination of feedback and feedforward comb-filters is allpass when their delay lines are the same length and their feedback and feedforward coefficents are the same. An example is shown in Fig.2.30.
- Any delay element in an allpass filter can be replaced by an allpass filter to obtain a new (typically higher order) allpass filter. The special case of nested first-order allpass filters yielded the lattice digital filter structure of Fig.2.32.
Notice that only stable filters can be lossless since, otherwise, is generally infinite, even when is finite. We further assume all filters are causal3.14 for simplicity. It is straightforward to show the following:
Thus, ``lossless'' and ``unity-gain allpass'' are synonymous. For an allpass filter with gain at each frequency, the energy gain of the filter is for every input signal . Since we can describe such a filter as an allpass times a constant gain, the term ``allpass'' will refer here to the case .
Example Allpass Filters
Nested Allpass Filters