Amplitude-Complementary 2-Channel Filter Bank

Perhaps the most natural choice of analysis filters for our two-channel, critically sampled filter bank, is an amplitude-complementary lowpass/highpass pair, i.e.,

$$H_1(z) = 1-H_0(z)$$

where we impose the unity dc gain constraint $H_0(1)=1$ .

Amplitude-complementary thus means constant overlap-add (COLA) on the unit circle in the $z$ plane.

Plugging the COLA constraint into the Filtering and Aliasing Cancellation constraint () gives

$$\begin{eqnarray*} c &=& H_0(z)[1-H_0(-z)] - [1-H_0(z)]H_0(-z) \\ &=& H_0(z) - H_0(-z) \quad\longleftrightarrow \\ A\delta(n-D) &=& h_0(n) - (-1)^n h_0(n) \\ &=& \left\{\begin{array}{ll} 0, & \hbox{$n$\ even} \\ [5pt] 2h_0(n), & \hbox{$n$\ odd} \\ \end{array} \right. \end{eqnarray*}$$

• Even-indexed terms of the impulse response are unconstrained, since they subtract out in the constraint.

• For perfect reconstruction, exactly one odd-indexed term must be nonzero in the lowpass impulse response $h_0(n)$ . The simplest choice is $h_0(1)\neq 0$ .

• Thus, the lowpass-filter impulse response can be anything of the form
  $$h_0 = [h_0(0), \bold{h_0(1)}, h_0(2), 0, h_0(4), 0, h_0(6), 0, \ldots]$$
  or
  $$h_0 = [h_0(0), 0, h_0(2), \bold{h_0(3)}, h_0(4), 0, h_0(6), 0, \ldots]$$
  etc.

• The corresponding highpass-filter impulse response is
  $$h_1(n) = \delta(n) - h_0(n).$$

• The first example above corresponds to the highpass-filter
  $$h_1 = [1-h_0(0), -h_0(1), -h_0(2), 0, -h_0(4), 0, -h_0(6), 0, \ldots]$$
  etc.

The above class of amplitude-complementary filters can be characterized as follows:

$$\begin{eqnarray*} H_0(z) &=& E_0(z^2) + h_0(o) z^{-o}, \quad E_0(1)+h_0(o)=1, \, \hbox{$o$\ odd}\\ H_1(z) &=& 1-H_0(z) = 1 - E_0(z^2) - h_0(o) z^{-o} \end{eqnarray*}$$

In summary, we have shown that an amplitude-complementary lowpass/highpass analysis filter pair yields perfect reconstruction (aliasing and filtering cancellation) when there is exactly one odd-indexed term in the impulse response of $h_0(n)$ .

Problem:

• $E_0(z^2)$ repeats twice around the unit circle.

• Since we assume real coefficients, the frequency response, $E_0(e^{j2\omega})$ is magnitude-symmetric about $\omega=\pi/2$ as well as $\pi$ .

• This is not good since we only have one degree of freedom, $h_0(o) z^{-o}$ , with which we can break the $\pi/2$ symmetry to reduce the high-frequency gain and/or boost the low-frequency gain.

• In other words, this class of filters cannot be expected to give us high quality lowpass or highpass behavior.

To enable the use of high-quality lowpass and highpass channel filters, we must relax the amplitude-complementary constraint (and/or filtering cancellation and/or aliasing cancellation) and find another approach.