We have so far seen two types of allpass filters:
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 [452, 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 .