9.5 QMF Filter Banks

One of the earliest proposed approaches for the design of 2-band filter banks is the so-called quadrature mirror filter (QMF) bank (Croisier et al., 1976), where the analysis highpass filter is designed to alternate the signs of the impulse-response samples of the lowpass filter, that is

(9.53)

Please note that it is assumed the filters have real coefficients. For this choice of the analysis filter bank, the magnitude response of the highpass filter, H ₁( e ^j?), is the mirror image of the lowpass filter magnitude response, H ₀(e ^{j ?}), with respect to the quadrature frequency . Hence the QMF nomenclature.

The 2-band filter bank illustrated in Figure 9.19 and discussed in Section 9.4 can also be analyzed in the following way. The signals after the analysis filter are described by

(9.54)

for k = 0, 1. The decimated signals are

(9.55)

for k = 0, 1, whereas the signals after the interpolator are

(9.56)

Then, the reconstructed signal is represented as

(9.57)

The last equality is the so-called modulation-matrix representation of a 2-band filter bank. The aliasing effect is represented by the terms containing X(- z). A possible solution to avoid aliasing is to choose the synthesis filters such that

(9.58)

(9.59)

This choice keeps the filters G ₀( z) and G ₁( z) as lowpass and highpass filters, respectively, as desired. Also, the aliasing is now canceled by the...