Editing Minimum phase (section)

=== Relationship of magnitude response to phase response ===
{{See also|Kramers–Kronig relations#Magnitude (gain)–phase relation}}
A minimum-phase system, whether discrete-time or continuous-time, has an additional useful property that the [[natural logarithm]] of the magnitude of the frequency response (the "gain" measured in [[neper]]s, which is proportional to [[Decibel|dB]]) is related to the phase angle of the frequency response (measured in [[radian]]s) by the [[Hilbert transform]]. That is, in the continuous-time case, let
<math display="block">
 H(j\omega)\ \stackrel{\text{def}}{=}\ H(s)\Big|_{s=j\omega}
</math>
be the complex frequency response of system {{math|''H''(''s'')}}. Then, only for a minimum-phase system, the phase response of {{math|''H''(''s'')}} is related to the gain by
<math display="block">
 \arg[H(j\omega)] = -\mathcal{H}\big\{\log\big(|H(j\omega)|\big)\big\},
</math>
where <math>\mathcal{H}</math> denotes the Hilbert transform, and, inversely,
<math display="block">
 \log\big(|H(j\omega)|\big) = \log\big(|H(j\infty)|\big) + \mathcal{H}\big\{\arg[H(j\omega)]\big\}.
</math>

Stated more compactly, let
<math display="block">
 H(j\omega) = |H(j\omega)| e^{j\arg[H(j\omega)]}\ \stackrel{\text{def}}{=}\ e^{\alpha(\omega)} e^{j\phi(\omega)} = e^{\alpha(\omega) + j\phi(\omega)},
</math>
where <math>\alpha(\omega)</math> and <math>\phi(\omega)</math> are real functions of a real variable. Then
<math display="block">
 \phi(\omega) = -\mathcal{H}\{\alpha(\omega)\}
</math>
and
<math display="block">
 \alpha(\omega) = \alpha(\infty) + \mathcal{H}\{\phi(\omega)\}.
</math>

The Hilbert transform operator is defined to be
<math display="block">
 \mathcal{H}\{x(t)\}\ \stackrel{\text{def}}{=}\ \hat{x}(t) = \frac{1}{\pi} \int_{-\infty}^\infty \frac{x(\tau)}{t - \tau} \,d\tau.
</math>

An equivalent corresponding relationship is also true for discrete-time minimum-phase systems.