Paraunitary Filter Banks
Paraunitary filter banks form an interesting subset of perfect reconstruction (PR) filter banks. We saw above that we get a PR filter bank whenever the synthesis polyphase matrix times the analysis polyphase matrix is the identity matrix , i.e., when
In particular, if is the paraconjugate of , we say the filter bank is paraunitary.
Paraconjugation is the generalization of the complex conjugate transpose operation from the unit circle to the entire plane. A paraunitary filter bank is therefore a generalization of an orthogonal filter bank. Recall that an orthogonal filter bank is one in which is an orthogonal (or unitary) matrix, to within a constant scale factor, and is its transpose (or Hermitian transpose).
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:
In terms of the signal norm (§4.10.1), this can be expressed more succinctly as
Notice that only stable filters can be lossless since, otherwise, . We further assume all filters are causal for simplicity.
It is straightforward to show that losslessness implies
That is, the frequency response must have magnitude 1 everywhere on the unit circle in the plane. Another way to express this is to write
and this form generalizes to over the entire the plane.
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:
where denotes complex conjugation of the coefficients only of and not the powers of . For example, if , then . We can write, for example,
in which the conjugation of serves to cancel the outer conjugation.
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
Note that this is equivalent to the previous result on the unit circle since
To generalize lossless filters to the multi-input, multi-output (MIMO) case, we must generalize conjugation to MIMO transfer function matrices.
for all , where denotes the identity matrix, and denotes the Hermitian transpose (complex-conjugate transpose) of :
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.,
Thus, every paraunitary matrix transfer function is unitary on the unit circle for all . Away from the unit circle, paraunitary is the unique analytic continuation of unitary .
Lossless Filter Examples
The simplest lossless filter is a unit-modulus gain
where can be any phase value. In the real case can only be 0 or , hence .
A lossless FIR filter can only consist of a single nonzero tap:
for some fixed integer , where is again some constant phase, constrained to be 0 or in the real-filter case. We consider only causal filters here, so .
where , , and . The polynomial can be obtained by reversing the order of the coefficients in , conjugating them, and multiplying by . (The factor above serves to restore negative powers of and hence causality.) Such filters are generally called allpass filters.
Paraunitary systems are essentially multi-input, multi-output (MIMO) allpass filters. Let denote the matrix transfer function of a paraunitary system. In the square case ( ), the matrix determinant, , is an allpass filter. Therefore, if a square contains FIR elements, its determinant is a simple delay: for some integer .
An -channel analysis filter bank can be viewed as an MIMO filter:
A paraunitary filter bank must therefore satisfy
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:
- The synthesis filter bank is simply the paraconjugate of the
analysis filter bank:
That is, since the paraconjugate is the inverse of a paraunitary filter matrix, it is exactly what we need for perfect reconstruction.
- The channel filters
are power complementary:
This follows immediately from looking at the paraunitary property on the unit circle.
is FIR, the corresponding synthesis filter matrix
is also FIR.
is FIR, each synthesis filter,
, is simply the
of its corresponding
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. As we observed in (11.83) above (§11.5.2), only trivial FIR filters of the form 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.
- FIR analysis and synthesis filters in paraunitary filter banks
have the same amplitude response. This follows from the fact
, i.e., flipping an FIR filter
conjugates the frequency response, which does
not affect its amplitude response
- The polyphase matrix
for any FIR paraunitary perfect
reconstruction filter bank can be written as the product of a
paraunitary and a unimodular matrix, where a
unimodular polynomial matrix
is any square
polynomial matrix having a constant nonzero
determinant. For example,
The paraconjugate of is
Thus, the Haar filter bank is paraunitary. This is true for any power-complementary filter bank, since when is , power-complementary and paraunitary are the same property. For more about paraunitary filter banks, see Chapter 6 of .
Filter Banks Equivalent to STFTs
Perfect Reconstruction Filter Banks