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Next we see why the causal wavelet *B*(*Z*), which we have made
from the prescribed spectrum,
turns out to be minimum-phase.
First return to the original definition of minimum-phase:
a causal wavelet is minimum-phase if and only if
its inverse is causal.
We have our wavelet in the form *B*(*Z*)= *e*^{C(Z)}.
Consider another wavelet *A*(*Z*) = *e*^{-C(Z)},
constructed analogously.
By the same reasoning, *a*_{t} is also causal.
Since *A*(*Z*)*B*(*Z*)=1, we have found a causal, inverse wavelet.
Thus the *b*_{t} wavelet is
**minimum-phase**.
Since the **phase** is a Fourier series,
it must be periodic; that is, it cannot increase
indefinitely with as it does for the nonminimum-phase wavelet
(see Figure 19).

** Next:** Pathological examples
** Up:** SPECTRAL FACTORIZATION
** Previous:** Finding a causal wavelet
Stanford Exploration Project

10/21/1998