Properties of Paraunitary Filter Banks

Paraunitary systems are essentially multi-input, multi-output (MIMO) allpass filters. Let $\bold{H}(z)$ denote the $p\times q$ matrix transfer function of a paraunitary system. In the square case ($p=q$ ), the matrix determinant, $\det[\bold{H}(z)]$ , is an allpass filter. Therefore, if a square $\bold{H}(z)$ contains FIR elements, its determinant is a simple delay: $\det[\bold{H}(z)]=z^{-K}$ for some integer $K$ .

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

$$\bold{H}(z) \eqsp \left[\begin{array}{c} H_1(z) \\ [2pt] H_2(z) \\ [2pt] \vdots \\ [2pt] H_N(z)\end{array}\right]$$ (12.86)

A paraunitary filter bank must therefore satisfy

$${\tilde {\bold{H}}}(z)\bold{H}(z) \eqsp 1.$$ (12.87)

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

$${\tilde {\bold{H}}}(z)\bold{H}(z) \eqsp c_K z^{-K}$$ (12.88)

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:

  $$\bold{F}(z) \eqsp {\tilde {\bold{H}}}(z)$$ (12.89)

  
  
  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:

  $$\left\vert H_1(\ejo )\right\vert^2 + \left\vert H_2(\ejo )\right\vert^2 + \cdots + \left\vert H_N(\ejo )\right\vert^2 \eqsp 1$$ (12.90)

  
  
  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 $H_k(z)=\bold{H}_k(z)$ :

  $$f_k(n) \eqsp h_k(L-n)$$ (12.91)

  
  
  where $L$ is the filter length. (When the filter coefficients are complex, $\hbox{\sc Flip}$ 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 $H(z) = e^{j\phi} z^{-K}$ 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 $\vert H(\ejo )\vert$ .

• 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,
  $$\bold{U}(z) \eqsp \left[\begin{array}{cc} 1+z^3 & z^2 \\ [2pt] z & 1 \end{array}\right]$$
  is unimodular. See [287, p. 663] for further details.