Digital Signal Processing, 10th Edition

11QMF Filters and Wavelets

The compression of certain signals, such as speech, sound, or images, involves sub-band decomposition combined with sampling rate reduction and reconstruction from sub-bands after storage or transmission. The simplest approach to these operations is to use banks of two filters [1].

11.1 Decomposition into Two Sub-Bands and Reconstruction

The block diagram of the system is given in Figure 11.1. The signal x(n) to be analyzed is fed to a set of two filters – namely, a low-pass filter H₀(Z) and a high-pass filter H₁(Z) whose outputs are decimated by factor two. They are called analysis filters. The reconstruction is performed from two sequences in which every other sample is null, filtered by a low-pass filter G₀(Z) for one and a high-pass filter G₁(Z) for the other, which are called synthesis filters.

As explained in Section 10.1, due to undersampling at analysis filter output, the image bands which are added to the useful signal are expressed by H₀(−Z)X(−Z) and H₁(−Z)X(−Z). These undesired signals disappear at the output of the synthesis filters if the following condition is met:

(11.1)

Then, it is sufficient to have the same filters in both subsets and the relations:

(11.2)

The reconstruction condition takes the form:

(11.3)

K is the delay incurred ...