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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 i...
...K \left\vert h_i(n)\right\vert^2,
\qquad K=0,1,2,\ldots\,.
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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Minimum-Phase/Allpass Decomposition
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Maximum Phase Filters