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Minimum Phase Means Fastest Decay

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

$\textstyle \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.


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``Introduction to Digital Filters with Audio Applications'', by Julius O. Smith III, (September 2007 Edition).
Copyright © 2014-03-23 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
CCRMA