To motivate the idea of paraunitary filters, let's first review some properties of lossless filters, progressing from the simplest cases up to paraunitary filter banks:
A linear, time-invariant filter is said to be lossless (or allpass) if it preserves signal energy. That is, if the input signal is , and the output signal is , then we have
It is straightforward to show that losslessness implies
The paraconjugate of a transfer function may be defined as the analytic continuation of the complex conjugate from the unit circle to the whole plane:
We refrain from conjugating in the definition of the paraconjugate because is not analytic in the complex-variables sense. Instead, we invert , which is analytic, and which reduces to complex conjugation on the unit circle.
The paraconjugate may be used to characterize allpass filters as follows:
A causal, stable, filter is allpass if and only if
To generalize lossless filters to the multi-input, multi-output (MIMO) case, we must generalize conjugation to MIMO transfer function matrices.
A transfer function matrix is said to be lossless if it is stable and its frequency-response matrix is unitary. That is,
Note that is a matrix product of a times a matrix. If , then the rank must be deficient. Therefore, we must have . (There must be at least as many outputs as there are inputs, but it's ok to have extra outputs.)
A lossless transfer function matrix is paraunitary, i.e.,