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:

$$H_1(z) \eqsp H_0(-z)\quad \hbox{(QMF Symmetry Constraint)} \protect$$ (12.33)

That is, the filter for channel 1 is constrained to be a $\pi$ -rotation of filter 0 along the unit circle. This of course makes perfect sense for a two-channel band-splitting filter bank, and can form the basis of a dyadic tree band splitting, as we'll look at in §11.9.1 below.

In the time domain, the QMF constraint (11.33) becomes $h_1(n) = (-1)^n h_0(n)$ , 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 [51], and appear to be the first critically sampled perfect reconstruction filter banks. Moreover, the Princen-Bradley filter bank, the initial foundation of MPEG audio as we now know it, was conceived as the Fourier dual of QMFs [214]. Historically, the term QMF applied only to two-channel filter banks having the QMF symmetry constraint (11.33). Today, the term ``QMF filter bank'' may refer to more general PR filter banks with any number of channels and not obeying (11.33) [287].

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

$$\begin{eqnarray*} F_0(z) &=& \quad\! H_1(-z) \eqsp \quad\! H_0(z)\\ [5pt] F_1(z) &=& -H_0(-z) \eqsp -H_1(z), \end{eqnarray*}$$

the perfect reconstruction requirement reduces to

$$g\,z^{-d} \eqsp H_0(z)F_0(z) + H_1(z)F_1(z) \eqsp H_0^2(z) - H_0^2(-z). \protect$$ (12.34)

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

It is easy to show using the polyphase representation of $H_0(z)$ (see [287]) 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 above. On the other hand, very high quality IIR solutions are possible. See [287, 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 §11.7.1.

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

$$\begin{eqnarray*} H_0(z) &=& 1 + z^{-1}\\ [5pt] H_1(z) &=& 1 - z^{-1} \end{eqnarray*}$$

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