### More General Allpass Filters

We have so far seen two types of 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.
We now discuss allpass filters more generally in the SISO case. (See Appendix D of [449] for the MIMO case.)

Definition: A linear, time-invariant filter is said to be lossless if it preserves signal energy for every input signal. That is, if the input signal is , and the output signal is , we must have

In terms of the signal norm , this can be expressed more succinctly as

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:

It can be shown [449, Appendix C] that stable, linear, time-invariant (LTI) filter transfer function is lossless if and only if

That is, the frequency response must have magnitude 1 everywhere over the unit circle in the complex plane.

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 .

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Example Allpass Filters
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Nested Allpass Filters