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Perfect Reconstruction FIR Filter Banks

Module by: Phil Schniter

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),
PR_f1.png
Figure 1
has the input/output relation:
Yz=12XzX-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 (1)
we see that the delay-ll perfect reconstruction requires
2z-l0= 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 (2)
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 -1z2z-l0=1detHz H 1 -z- H 1 z- H 0 -z H 0 z2z-l0=2detHzz-l H 1 -z-z-l 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 (3)
where
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 (4)
For FIR G 0 z G 0 z and G 1 z G 1 z , we require 1 that
detHz=cz-k H z c z k (5)
for c∈ℝ c and k∈ℤ k . Under this determinant condition, we find that
G 0 z G 1 z =2z-l-kc H 1 -z- H 0 -z G 0 z G 1 z 2 z l k c H 1 z H 0 z (6)
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,c∈ℝ∧l∈ℤ:detHz=cz-l 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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