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This module introduces the ideas behind and design issues of Orthogonal Perfect Reconstruction of FIR filterbanks.

Orthogonal pr filterbanks

The FIR perfect-reconstruction (PR) conditions leave some freedom in the choice of H 0 z and H 1 z . Orthogonal PR filterbanks are defined by causal real-coefficienteven-length- N analysis filters that satisfy the following two equations:

1 H 0 z H 0 z H 0 -z H 0 z
H 1 z ± z N 1 H 0 z
To verify that these design choices satisfy the FIR-PR requirements for H 0 z and H 1 z , we evaluate H z under the second condition above. This yields
H z ± H 0 z H 1 z H 0 -z H 1 z z N 1 H 0 z H 0 z H 0 -z H 0 z z N 1
which corresponds to c -1 and l N 1 in the FIR-PR determinant condition H z c z l . The remaining FIR-PR conditions then imply that the synthesis filters are given by
G 0 z -2 H 1 z 2 z N 1 H 0 z
G 1 z 2 H 0 z 2 z N 1 H 1 z
The orthogonal PR design rules imply that H 0 ω is "power symmetric" and that H 0 ω H 1 ω form a "power complementary" pair. To see the power symmetry, we rewrite the first design rule using z ω and -1 ± π , which gives
1 H 0 ω H 0 ω H 0 ω H 0 ω H 0 ω 2 H 0 ω 2 H 0 ω 2 H 0 ω 2
The last two steps leveraged the fact that the DTFT of a real-coefficient filter is conjugate-symmetric. Thepower-symmetry property is illustrated in :

Power complementarity follows from the second orthogonal PR design rule, which implies H 1 ω H 0 ω . Plugging this into the previous equation, we find

1 H 0 ω 2 H 1 ω 2
The power-complimentary property is illustrated in :

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Source:  OpenStax, Digital signal processing (ohio state ee700). OpenStax CNX. Jan 22, 2004 Download for free at http://cnx.org/content/col10144/1.8
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