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Orthogonal Two-Channel Filter Banks

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

\hat{X}(z) &=& \frac{1}{2}[H_0(z)F_0(z) + H_1(z)F_1(z)]X(z)
...\\ [0.1in]
&+& \frac{1}{2}[H_0(-z)F_0(z) + H_1(-z)F_1(z)]X(-z)

This can be written in matrix form as

$\displaystyle \hat{X}(z) = \frac{1}{2} \left[\begin{array}{c} F_0(z) \\ [2pt] F...
\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

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

where $ {\tilde {\bold{H}}}_m(z)\isdef \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 in polyphase form in §10.5 below. The AC matrix is paraunitary if and only if the polyphase matrix (defined in the next section) is paraunitary [266].

Previous: Conjugate Quadrature Filters (CQF)
Next: Perfect Reconstruction Filter Banks

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About the Author: Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See for details.


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