Properties of Paraunitary Filter Banks

An $N$-channel analysis filter bank can be viewed as an $N\times 1$ MIMO filter:

$$\bold{H}(z) = \left[\begin{array}{c} H_1(z) \\ [2pt] H_2(z) \\ [2pt] \vdots \\ [2pt] H_N(z)\end{array}\right]$$

A paraunitary filter bank must therefore satisfy

$${\tilde {\bold{H}}}(z)\bold{H}(z) = 1.$$

More generally, we allow paraunitary filter banks to scale and/or delay the input signal:

$${\tilde {\bold{H}}}(z)\bold{H}(z) = c_K z^{-K}$$

where $K$ is some nonnegative integer and $c_K\neq 0$.

We can note the following properties of paraunitary filter banks:

• The synthesis filter bank is simply the paraconjugate of the analysis filter bank:
  $$\bold{F}(z) = {\tilde {\bold{H}}}(z)$$
  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 $H_k(z)$ are power complementary:
  $$\left\vert H_1(e^{j\omega})\right\vert^2 + \left\vert H_2(e^{j\omega})\right\vert^2 + \cdots + \left\vert H_N(e^{j\omega})\right\vert^2 = 1$$
  This follows immediately from looking at the paraunitary property on the unit circle.

• When $\bold{H}(z)$ is FIR, the corresponding synthesis filter matrix ${\tilde {\bold{H}}}(z)$ is also FIR.

• When $\bold{H}(z)$ is FIR, each synthesis filter, $F_k(z) = {\tilde {\bold{H}}}_k(z),\, k=1,\ldots,N$, is simply the $\hbox{\sc Flip}$ of its corresponding analysis filter $H_k(z)=\bold{H}_k(z)$:
  $$f_k(n) = h_k(L-n)$$
  where $L$ is the filter length. (When the filter coefficients are complex, $\hbox{\sc Flip}$ includes a complex conjugation as well.) This result follows from the fact that paraconjugating an FIR filter amounts to simply flipping (and conjugating) its coefficients. As we observed in Example 2 of §10.5.3 above, only trivial FIR filters of the form $H(z) = e^{j\phi} z^{-K}$ 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 that $\hbox{\sc Flip}(h) \;\leftrightarrow\;\overline{H}$, i.e., flipping an FIR filter impulse response $h(n)$ conjugates the frequency response, which does not affect its amplitude response $\vert H(e^{j\omega})\vert$.

• The polyphase matrix $\bold{E}(z)$ 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 $\bold{U}(z)$ is any square polynomial matrix having a constant nonzero determinant. For example,
  $$\bold{U}(z) = \left[\begin{array}{cc} 1+z^3 & z^2 \\ [2pt] z & 1 \end{array}\right]$$
  is unimodular. See [252, p. 663] for further details.