Perfect Reconstruction FIR Filter Banks 2.14 2001/12/25 2005/10/04 15:05:24.962 GMT-5 Phil Schniter schniter@ee.eng.ohio-state.edu Phil Schniter schniter@ee.eng.ohio-state.edu Filter bank filterbanks FIR reconstruction two-channel 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 (),

has the input/output relation: 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-l perfect reconstruction requires 2 z l 0 H 0 z H 1 z H 0 z H 1 z G 0 z G 1 z where H z H 0 z H 1 z H 0 z H 1 z or, equivalently, that 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 where H z H 0 z H 1 z H 0 z H 1 z For FIR G 0 z and G 1 z , we require Since we cannot assume that FIR H 0 z and H 1 z share a common root. that H z c z k for c and k . Under this determinant condition, we find that G 0 z G 1 z 2 z l k c H 1 z H 0 z Assuming that H 0 z and H 1 z are causal with non-zero initial coefficient, we choose k l to keep G 0 z and 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 c c l H z c z l G 0 z 2 c H 1 z G 1 z -2 c H 0 z