An
-channel analysis filter bank can be viewed as an
MIMO filter
A paraunitary filter bank must therefore obey
More generally, we allow paraunitary filter banks to scale and/or delay the input signal:
where
We can note the following properties of paraunitary filter banks:
That is, since the paraconjugate is the inverse of a paraunitary filter matrix, it is exactly what we need for perfect reconstruction.
This follows immediately from looking at the paraunitary property on the unit circle.
where
This follows from the fact that paraconjugating an FIR filter amounts to simply flipping (and conjugating) its coefficients.
Note that only trivial FIR filters can be paraunitary in the single-input, single-output (SISO) case. In the MIMO case, on the other hand, paraunitary systems can be composed of FIR filters of any order.
This follows from the fact that
, i.e., flipping an FIR filter impulse response
conjugates the frequency response, which does not affect its amplitude
response
.
is unimodular. See Vaidyanathan for further details (p. 663).