Minimum Phase - Minimum Phase As Minimum Group Delay

Minimum Phase As Minimum Group Delay

For all causal and stable systems that have the same 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 of the transfer function . Let's place this zero inside the unit circle and see how the group delay is affected.

Since the zero contributes the factor to the transfer function, the phase contributed by this term is the following.

contributes the following to the group delay.

-\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{d \phi_a (\omega)}{d \omega} = \frac{ \left| a \right| - \cos( \omega - \theta_a ) }{ \left| a \right| + \left| a \right|^{-1} - 2 \cos( \omega - \theta_a ) }

The denominator and are invariant to reflecting the zero outside of the unit circle, i.e., replacing with . However, by reflecting outside of the unit circle, we increase the magnitude of in the numerator. Thus, having inside the unit circle minimizes the group delay contributed by the factor . We can extend this result to the general case of more than one zero since the phase of the multiplicative factors of the form is additive. I.e., for a transfer function with zeros,

So, a minimum phase system with all zeros inside the unit circle minimizes the group delay since the group delay of each individual zero is minimized.

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