Orthonormal Wavelet Representation

The key property of the o.n. wave packets is that all phase space cell (i.e., wave packets) have the same shape

$$\Delta t = \frac{2\pi}{\varepsilon}$$

$$\Delta\omega = \varepsilon\,.$$

Suppose, however, we must represent a signal which looks like the one in Figure 2.15.

Figure 2.15: A function whose large scale and small scale structures are of equal importance.

In other words, upon closer examination, the signal on a small scale is similar to the signal on the larger scale. In that case the large scale structure is represented ``most economically'' by a sequence of wide low frequency wave packets. The qualifier ``most economically'' means representing the signal with the fewest number of non-zero wave packet coefficients. The existence of o.n. wave packets which are wide and narrow is the important new feature.

Let us apply the general idea of constructing o.n. wave packets in Section (2.4.1) to obtain o.n. wave packets with Fourier domain windows of variable width. In fact, we shall construct o.n. wave packets with adjacent, but non-overlapping, windows in the Fourier domain, with each window extending exactly over one octave. Thus each positive frequency window is twice as wide as its neighbor on the left.

In the time domain, all o.n. wave packets have the same half width, namely $\frac{2\pi}{\varepsilon}$ . They are different in that they are related to one another by discrete equal shifts $\frac{2\pi}{\varepsilon}$ in time and also by equal shifts $\varepsilon$ in frequency. If a signal changes ``slowly'' over time, i.e., does not change appreciably over a time interval less than the inter-wave packet spacing $\frac{2\pi}{\varepsilon}$ , then the signal can be represented quite efficiently by a finite wave packet sum. If, however, the signal changes ``abruptly'', i.e., it changes appreciably over a time interval small compared to the width, and hence the spacing, of the wave packets, then the wave packets representation becomes less efficient. The wave packet sum must contain many high frequency packets that reinforce each other on one side where the abrupt change occurs and cancel each other on the other side of that change.

What is needed is an o.n. set of variable width wave packets. In effect, instead of having a uniform sampling rate, the sampling rate should be variable to accomodate abrupt changes in the signal. Orthonormal wavelets fullfill this requirement.

Lecture 17