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.

Read more about this topic:  Minimum Phase

Famous quotes containing the words minimum, phase, group and/or delay:

    After decades of unappreciated drudgery, American women just don’t do housework any more—that is, beyond the minimum that is required in order to clear a path from the bedroom to the front door so they can get off to work in the mourning.
    Barbara Ehrenreich (20th century)

    The problem of the twentieth century is the problem of the color-line—the relation of the darker to the lighter races of men in Asia and Africa, in America and the islands of the sea. It was a phase of this problem that caused the Civil War.
    —W.E.B. (William Edward Burghardt)

    A little group of willful men, representing no opinion but their own, have rendered the great government of the United States helpless and contemptible.
    Woodrow Wilson (1856–1924)

    Face troubles from their birth, for ‘tis too late to cure
    When long delay has given the evil strength.
    Haste then; postpone not to the coming hour: tomorrow
    He’ll be less ready who’s not ready now.
    Ovid (Publius Ovidius Naso)