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### Orthogonal Two-Channel Filter Banks

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

This can be written in matrix form as

where the above

matrix,

, is called the

*alias component matrix* (or analysis modulation matrix

).
If

where

denotes the

*paraconjugate* of

, 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

**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

http://ccrma.stanford.edu/~jos/ for details.