0.8 Filter banks and transmultiplexers  (Page 9/23)

By changing the sign of the filter $h_1\left(n\right)$ , if necessary, one can always write $H_p\left(z\right)$ in the form

Now, if $H_{0,j}^R\left(z\right)$ is the reflection of $H_{0,j}\left(z\right)$ (i.e., $H_{0,j}\left(z\right)=z^{-K+1}{H}_{0,j}\left(z^{-1}\right)$ ), then (from the algebraic form of $R\left(\theta \right)$ )

With these parameterizations, filter banks can be designed as an unconstrained optimization problem. The parameterizations described are important foranother reason. It turns out that the most efficient (from the number of arithmetic operations) implementation of unitary filter banks is usingthe Householder parameterization. With arbitrary filter banks, one can organize the computations so as capitalize on the rate-change operationsof upsampling and downsampling. For example, one need not compute values that are thrown away by downsampling. The gain from using the parameterizationof unitary filter banks is over and above this obvious gain (for example, see pages 330-331 and 386-387 in [link] ). Besides, with small modifications these parameterizations allowfor unitariness to be preserved, even under filter coefficient quantization—with this having implications for fixed-point implementation of thesefilter banks in hardware digital signal processors [link] .

Unitary filter banks—some illustrative examples

A few concrete examples of $M$ -band unitary filter banks and their parameterizations should clarify our discussion.

First consider the two-band filter bank associated with Daubechies' four-coefficient scaling function and wavelet that we saw in Section: Parameterization of the Scaling Coefficients . Recall that the lowpass filter (the scaling filter) is given by

The highpass filter (wavelet filter) is given by $h_1\left(n\right)={(-1)}^n{h}_0(3-n)$ , and both [link] and [link] are satisfied with $g_i\left(n\right)=h_i(-n)$ . The matrix representation of the analysis bank of this filter bank is given by

One readily verifies that ${\mathbf{H}}^T\mathbf{H}=\mathbf{I}$ and $\mathbf{H}{\mathbf{H}}^T=\mathbf{I}$ . The polyphase representation of this filter bank is given by

and one can show that $H_p^T\left(z^{-1}\right)H_p\left(z\right)=I$ and $H_p\left(z\right)H_p^T\left(z^{-1}\right)=I$ . The Householder factorization of $H_p\left(z\right)$ is given by

where

Incidentally, all two-band unitary filter banks associated with wavelet tight frames have the same value of $V_0$ . Therefore, all filter banks associated with two-band wavelet tight frames arecompletely specified by a set of orthogonal vectors $v_i$ , $K-1$ of them if the $h_0$ is of length $2K$ . Indeed, for the six-coefficient Daubechies wavelets (see Section: Parameterization of the Scaling Coefficients ), the parameterization of $H_p\left(z\right)$ is associated with the following two unitary vectors (since $K=3$ ): $v_1^T=\left[-,.,3842,.,9232\right]$ and $v_2^T=\left[-,.,1053,-,.,9944\right]$ .

The Givens' rotation based factorization of $H_p\left(z\right)$ for the 4-coefficient Daubechies filters given by: