     Next: Phase of a rational Up: PHASE OF A MINIMUM-PHASE Previous: PHASE OF A MINIMUM-PHASE

## Phase of a single root

For real , a plot of real and imaginary parts of Z is the circle .A smaller circle is .9Z. A right-shifted circle is 1+.9Z. Let Z0 be a complex number, such as x0+iy0, or ,where and are fixed constants. Consider the complex Z plane for the two-term filter (34) (35) (36) origin
Figure 18
Left, complex B plane for . Right, for .

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 Figure 19 Left shows real and imaginary parts and phase angle of equation ((37)), for . Right, for . Left is minimum-phase and right is nonminimum-phase.      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.     Next: Phase of a rational Up: PHASE OF A MINIMUM-PHASE Previous: PHASE OF A MINIMUM-PHASE
Stanford Exploration Project
10/21/1998