An -channel 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 is some nonnegative integer and .
We can note the following properties of paraunitary filter banks:
Clearly, not every filter bank will be invertible in this way. When it is, it may be called a perfect reconstruction filter bank. When a filter bank transfer function is paraunitary, its corresponding synthesis filter bank is simply the paraconjugate filter bank , or
This follows immediately from looking at the paraunitary property on the unit circle.
where is the filter length. (When the filter coefficients are complex, includes a complex conjugation as well.)
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.