For real , a plot of real and imaginary parts of *Z* is the circle
.A smaller circle is .9*Z*.
A right-shifted circle is 1+.9*Z*.
Let *Z _{0}* be a complex number, such as

(34) | ||

(35) | ||

(36) |

Figure 18

Real and imaginary parts of *B* are plotted in Figure 18.
Arrows are at frequency intervals of .Observe that for the sequence of arrows
has a sequence of angles that ranges over ,whereas for the sequence of arrows
has a sequence of angles between .Now let us replot equation (37)
in a more conventional way, with as the horizontal axis.
Whereas the **phase** is the angle of an arrow in Figure 18,
in Figure 19 it is the arctangent of .
Notice how different is the phase curve in
Figure 19 for than for .

Real and imaginary parts of *B*
are *periodic* functions of the frequency ,since .We might be tempted to conclude that the phase
would be periodic too.
Figure 19 shows, however, that for a nonminimum-phase filter,
as ranges from to ,the phase increases by (because the circular path in Figure 18 surrounds the origin).
To make Figure 19 I used the Fortran arctangent function
that takes two arguments, *x*, and *y*.
It returns an angle between and .As I was plotting the nonminimum phase,
the phase suddenly jumped discontinuously
from a value near to , and I needed to add to keep the curve continuous.
This is called ``**phase** **unwinding**.''

phase
Left shows real and imaginary parts and phase angle
of equation ((37)),
for . Right, for . Left is minimum-phase and right is nonminimum-phase.
Figure 19 |

You would use phase unwinding
if you ever had to solve the following problem:
given an **earthquake** at location (*x*,*y*), did it occur
in country X?
You would circumnavigate the country--compare
the circle in Figure 18--and see if the phase angle
from the earthquake to the country's boundary accumulated to (yes)
or to (no).

The word ``minimum" is used in ``minimum phase"
because delaying a filter can always add more phase.
For example,
multiplying any polynomial by *Z* delays it and adds to its phase.

For the minimum-phase filter, the group delay applied to Figure 19 is a periodic function of .For the nonminimum-phase filter, group delay happens to be a monotonically increasing function of .Since it is not an all-pass filter, the monotonicity is accidental.

Because **group delay** is the Fourier dual
to **instantaneous frequency** ,we can now go back to Figure 5 and explain
the discontinuous behavior of instantaneous frequency
where the signal amplitude is near zero.

10/21/1998