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Perfect Reconstruction FIR Filter Banks

Module by: Phil Schniter. E-mail the author

Summary: This module will drop the restrictive QMF conditions and focus on using FIR filters to achieve perfect reconstruction from filterbanks.

FIR Perfect-Reconstruction Conditions

The QMF design choices prevented the design of a useful (i.e., frequency selective) perfect-reconstruction (PR) FIR filterbank. This motivates us to re-examine PR filterbank design without the overly-restrictive QMF conditions. However, we will still require causal FIR filters with real coefficients.

Recalling that the two-channel filterbank (Figure 1),

Figure 1
Figure 1 (PR_f1.png)
has the input/output relation:
Yz=12( XzX-z )( H 0 z H 1 z H 0 z H 1 z )( G 0 z G 1 z ) Y z 1 2 X z X -z H 0 z H 1 z H 0 z H 1 z G 0 z G 1 z
we see that the delay-ll perfect reconstruction requires
( 2zl 0 )=( H 0 z H 1 z H 0 z H 1 z )( G 0 z G 1 z ) 2 z l 0 H 0 z H 1 z H 0 z H 1 z G 0 z G 1 z
where Hz=( H 0 z H 1 z H 0 z H 1 z ) H z H 0 z H 1 z H 0 z H 1 z or, equivalently, that
( G 0 z G 1 z )=H -1z( 2zl 0 )=1detHz( H 1 z H 1 z H 0 z H 0 z )( 2zl 0 )=2detHz( zl H 1 (z) (zl H 0 z) ) G 0 z G 1 z H z -1 2 z l 0 1 H z H 1 z H 1 z H 0 z H 0 z 2 z l 0 2 H z z l H 1 z z l H 0 z
detHz= H 0 z H 1 z H 0 z H 1 z H z H 0 z H 1 z H 0 z H 1 z
For FIR G 0 z G 0 z and G 1 z G 1 z , we require 1 that
detHz=czk H z c z k
for cR c and kZ k . Under this determinant condition, we find that
G 0 z G 1 z=2z(lk)c H 1 z H 0 z G 0 z G 1 z 2 z l k c H 1 z H 0 z
Assuming that H 0 z H 0 z and H 1 z H 1 z are causal with non-zero initial coefficient, we choose k=l k l to keep G 0 z G 0 z and G 1 z G 1 z causal and free of unnecessary delay.

Summary of Two-Channel FIR-PR Conditions

Summarizing the two-channel FIR-PR conditions: H 0 z H 1 z=  causal real-coefficient  FIR H 0 z H 1 z   causal real-coefficient  FIR c ,cRlZ:detHz=czl c c l H z c z l G 0 z=2c H 1 z G 0 z 2 c H 1 z G 1 z=-2c H 0 z G 1 z -2 c H 0 z


  1. Since we cannot assume that FIR H 0 z H 0 z and H 1 z H 1 z share a common root.

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