Quadrature Mirror Filters (QMF)

The well studied subject of Quadrature Mirror Filters (QMF) is entered by imposing the following symmetry constraint on the analysis filters:

$$H_1(z) = H_0(-z)\quad \hbox{(QMF Symmetry Constraint)} \protect$$ (11.6)

That is, the filter for channel 1 is constrained to be a $\pi$ -rotation of filter 0 along the unit circle. In the time domain, $h_1(n) = (-1)^n h_0(n)$ , i.e., all odd-index coefficients are negated. If $H_0$ is a lowpass filter cutting off near $\omega=\pi/2$ (as is typical), then $H_1$ is a complementary highpass filter. The exact cut-off frequency can be adjusted along with the roll-off rate to provide a maximally constant frequency-response sum.

Two-channel QMFs have been around since at least 1976 [45], and appear to be the first critically sampled perfect reconstruction filter banks. Historically, the term QMF applied only to two-channel filter banks having the QMF symmetry constraint (10.6). Today, the term ``QMF filter bank'' may refer to more general PR filter banks with any number of channels and not obeying (10.6) [249].

Combining the QMF symmetry constraint with the aliasing-cancellation constraints, given by

$$\begin{eqnarray*} F_0(z) &=& \quad\! H_1(-z) = \quad\! H_0(z)\\ [0.1in] F_1(z) &=& -H_0(-z) = -H_1(z), \end{eqnarray*}$$

the perfect reconstruction requirement reduces to

$$\hbox{constant} = H_0(z)F_0(z) + H_1(z)F_1(z) = H_0^2(z) - H_0^2(-z)$$

$$\hbox{(QMF Perfect Reconstruction Constraint)} \protect$$ (11.7)

Now, all four filters are determined by $H_0(z)$ .

It is easy to show using the polyphase representation of $H_0(z)$ (see [249]) that the only causal FIR QMF analysis filters yielding exact perfect reconstruction are two-tap FIR filters of the form

$$\begin{eqnarray*} H_0(z) &=& c_0 z^{-2n_0} + c_1 z^{-(2n_1+1)}\\ H_1(z) &=& c_0 z^{-2n_0} - c_1 z^{-(2n_1+1)} \end{eqnarray*}$$

where $c_0$ and $c_1$ are constants, and $n_0$ and $n_1$ are integers. Therefore, only weak channel filters are available in the QMF case ( $H_1(z)=H_0(-z)$ ), as we saw in the amplitude-complementary case. On the other hand, very high quality IIR solutions are possible. See [249, pp. 201-204] for details. In practice, approximate ``pseudo QMF'' filters are more practical, which only give approximate perfect reconstruction. We'll return to this topic in §10.7.1.

The Haar filters, which we saw gave perfect reconstruction in the amplitude-complementary case, are also examples of a QMF filter bank:

$$\begin{eqnarray*} H_0(z) &=& 1 + z^{-1}\\ [0.1in] H_1(z) &=& 1 - z^{-1} \end{eqnarray*}$$

In this example, $c_0=c_1=1$ , and $n_0=n_1=0$ .