### 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

(12.47) |

where the above matrix, , is called the

*alias component matrix*(or analysis modulation matrix). If

(12.48) |

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 §11.5 below. The AC matrix is paraunitary if
and only if the *polyphase matrix* (defined in the next section)
is paraunitary [287].

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Simple Examples of Perfect Reconstruction

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Conjugate Quadrature Filters (CQF)