Why the causal wavelet is minimum-phase

 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 $\omega$ as it does for the nonminimum-phase wavelet (see Figure 19).