Editing Minimum phase (section)

== Minimum phase as minimum group delay ==

For all [[causal]] and [[BIBO stability|stable]] systems that have the same [[frequency response|magnitude response]], the minimum phase system has the minimum [[group delay]]. The following proof illustrates this idea of minimum [[group delay]].

Suppose we consider one [[Zero (complex analysis)|zero]] <math>a</math> of the [[transfer function]] <math>H(z)</math>. Let's place this [[Zero (complex analysis)|zero]] <math>a</math> inside the [[unit circle]] (<math>\left| a \right| < 1</math>) and see how the [[group delay]] is affected.
<math display="block">a = \left| a \right| e^{i \theta_a} \, \text{ where } \, \theta_a = \operatorname{Arg}(a)</math>

Since the [[Zero (complex analysis)|zero]] <math>a</math> contributes the factor <math>1 - a z^{-1}</math> to the [[transfer function]], the phase contributed by this term is the following.
<math display="block">\begin{align}
\phi_a \left(\omega \right) &= \operatorname{Arg} \left(1 - a e^{-i \omega} \right)\\
&= \operatorname{Arg} \left(1 - \left| a \right| e^{i \theta_a} e^{-i \omega} \right)\\
&= \operatorname{Arg} \left(1 - \left| a \right| e^{-i (\omega - \theta_a)} \right)\\
&= \operatorname{Arg} \left( \left\{ 1 - \left| a \right| \cos( \omega - \theta_a ) \right\} + i \left\{ \left| a \right| \sin( \omega - \theta_a ) \right\}\right)\\
&= \operatorname{Arg} \left( \left\{ \left| a \right|^{-1} - \cos( \omega - \theta_a ) \right\} + i \left\{ \sin( \omega - \theta_a ) \right\} \right)
\end{align}</math>

<math>\phi_a (\omega)</math> contributes the following to the [[group delay]].

<math display="block">\begin{align}
-\frac{d \phi_a (\omega)}{d \omega} &= \frac{
\sin^2( \omega - \theta_a ) + \cos^2( \omega - \theta_a ) - \left| a \right|^{-1} \cos( \omega - \theta_a )
}{
\sin^2( \omega - \theta_a ) + \cos^2( \omega - \theta_a ) + \left| a \right|^{-2} - 2 \left| a \right|^{-1} \cos( \omega - \theta_a )
} \\
&= \frac{
\left| a \right| - \cos( \omega - \theta_a )
}{
\left| a \right| + \left| a \right|^{-1} - 2 \cos( \omega - \theta_a )
}
\end{align}
</math>

The denominator and <math>\theta_a</math> are invariant to reflecting the [[Zero (complex analysis)|zero]] <math>a</math> outside of the [[unit circle]], i.e., replacing <math>a</math> with <math>(a^{-1})^{*}</math>. However, by reflecting <math>a</math> outside of the unit circle, we increase the magnitude of <math>\left| a \right|</math> in the numerator. Thus, having <math>a</math> inside the [[unit circle]] minimizes the [[group delay]] contributed by the factor <math>1 - a z^{-1}</math>. We can extend this result to the general case of more than one [[Zero (complex analysis)|zero]] since the phase of the multiplicative factors of the form <math>1 - a_i z^{-1}</math> is additive. I.e., for a [[transfer function]] with <math>N</math> [[Zero (complex analysis)|zero]]s,
<math display="block">\operatorname{Arg}\left( \prod_{i = 1}^N  \left( 1 - a_i z^{-1} \right) \right) = \sum_{i = 1}^N \operatorname{Arg}\left( 1 - a_i z^{-1} \right) </math>

So, a minimum phase system with all [[Zero (complex analysis)|zero]]s inside the [[unit circle]] minimizes the [[group delay]] since the [[group delay]] of each individual [[Zero (complex analysis)|zero]] is minimized.

[[File:Minimum and maximum phase responses.gif|frame|center|Illustration of the calculus above. Top and bottom are filters with same gain response (on the left : the [[Nyquist diagram]]s, on the right : phase responses), but the filter on the top with <math>a = 0.8 < 1</math> has the smallest amplitude in phase response.]]