Orthogonal Two-Channel Filter Banks

Recall the reconstruction equation for the two-channel, critically sampled, perfect-reconstruction filter-bank:

$$\begin{eqnarray*} \hat{X}(z) &=& \frac{1}{2}[H_0(z)F_0(z) + H_1(z)F_1(z)]X(z) \nonumber\\ [0.1in] &+& \frac{1}{2}[H_0(-z)F_0(z) + H_1(-z)F_1(z)]X(-z) \end{eqnarray*}$$

This can be written in matrix form as

$$\hat{X}(z) = \frac{1}{2} \left[\begin{array}{c} F_0(z) \\ [2pt] F_1(z) \end{array}\right]^{T} \left[\begin{array}{cc} H_0(z) & H_0(-z) \\ [2pt] H_1(z) & H_1(-z) \end{array}\right] \left[\begin{array}{c} X(z) \\ [2pt] X(-z) \end{array}\right]$$

where the above $2 \times 2$ matrix, $\bold{H}_m(z)$ , is called the alias component matrix (or analysis modulation matrix). If

$${\tilde {\bold{H}}}_m(z)\bold{H}_m(z) = 2\bold{I}$$

where ${\tilde {\bold{H}}}_m(z)\mathrel{\stackrel{\mathrm{\Delta}}{=}}\bold{H}_m^T(z^{-1})$ denotes the paraconjugate of $\bold{H}_m(z)$ , then the alias component (AC) matrix is lossless, and the (real) filter bank is orthogonal.

It turns out orthogonal filter banks give perfect reconstruction filter banks for any number of channels.

Orthogonal filter banks are also called paraunitary filter banks, which we'll study shortly in polyphase form.

The AC matrix is paraunitary if and only if the polyphase matrix is paraunitary. (See Vaidyanathan.)