Quadrature Mirror Filterbanks (QMF)

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

$$H_1(z) = H_0(-z)\quad \hbox{(QMF Symmetry Constraint)} \protect$$ (5)

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.

Two-channel QMFs have been around since at least 1976 (see Croisier et al. in Music 421 Citations), and appear to be the first critically sampled perfect reconstruction filter banks.

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 to provide a maximally constant frequency-response sum.

Historically, the term QMF applied only to two-channel filter banks having the QMF symmetry constraint (). Today, the term ``QMF filter bank'' may refer to more general PR filter banks with any number of channels and not obeying ().

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

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

$$\hbox{(QMF Perfect Reconstruction Constraint)} \protect$$ (6)

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

It is easy to show using the polyphase representation of $H_0(z)$ (see Vaidyanathan) 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.

• 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 Vaidyanathan for details (pp. 201-204).

• In practice, approximate ``pseudo QMF'' filters are common, which only give approximate perfect reconstruction. We'll return to this topic later.

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 $n_0=n_1=0$ .