First consider the simpler problem in which the wavelet need not be causal.
We can easily find a symmetric wavelet with any spectrum
(which by definition is an energy or power).
We simply take the square root of the spectrum--this is the
**amplitude spectrum**.
We then inverse transform the amplitude spectrum
to the time domain,
and we have a symmetric wavelet with the desired spectrum.

The **prediction-error filter** discussed in chapter
is theoretically
obtainable by spectral factorization of an inverse spectrum.
The **Kolmogoroff** method of spectral factorization,
which we will be looking at here,
is much faster than the time-domain, least-squares methods
considered in
chapter and the least-squares methods given in FGDP.
Its speed motivates its widespread practical use.

Figure 8

Some simple examples of spectral factorization are given
in Figure 8.
For all but the fourth signal,
the spectrum of the minimum-phase wavelet
clearly matches that of the input.
Wavelets are shifted to *t*=0 and turned backwards.
In the fourth case, the waveshape changes into a big pulse at zero lag.
As the **Robinson** theorem introduced
on page suggests,
minimum-phase wavelets tend to decay rapidly after a strong onset.
I imagined that hand-drawn wavelets with a strong onset
would rarely turn out to be perfectly minimum-phase,
but when I tried it, I was surprised
at how easy it seemed to be to draw a minimum-phase wavelet.
This is shown on the bottom of Figure 8.

To begin understanding spectral factorization, notice that the polar form of any complex number puts the phase into the exponential, i.e., .So we look first into the behavior of exponentials and logarithms of Fourier transforms.

- The exponential of a causal is causal.
- Finding a causal wavelet from a prescribed spectrum
- Why the causal wavelet is minimum-phase
- Pathological examples
- Relation of amplitude to phase

10/21/1998