Minimum Phase Means Fastest Decay

The previous example is an instance of the following general result:

$\parbox{0.8\textwidth}{% Among all causal signals $h_i(n)$\ having identical magnitude spectra, the minimum-phase signal $h_{\hbox{\tiny mp}}(n)$\ has the \emph{fastest decay} in the sense that \begin{displaymath} \sum_{n=0}^K \left\vert h_{\hbox{\tiny mp}}(n)\right\vert^2 \geq \sum_{n=0}^K \left\vert h_i(n)\right\vert^2, \qquad K=0,1,2,\ldots\,. \end{displaymath}}$

That is, the signal energy in the first $K+1$ samples of the minimum-phase case is at least as large as any other causal signal having the same magnitude spectrum. (See [60] for a proof outline.) Thus, minimum-phase signals are maximally concentrated toward time 0 when compared against all causal signals having the same magnitude spectrum. As a result of this property, minimum-phase signals are sometimes called minimum-delay signals.