## Graphical Phase Response Calculation

The phase response is almost as easy to evaluate graphically as is the amplitude response:

If is real, then is either 0 or . Terms of the form can be interpreted as a vector drawn from the point to the point in the complex plane. The angle of is the angle of the constructed vector (where a vector pointing horizontally to the right has an angle of 0). Therefore, the phase response at frequency Hz is again obtained by drawing lines from all the poles and zeros to the point , as shown in Fig.8.4. The angles of the lines from the zeros are added, and the angles of the lines from the poles are subtracted. Thus, at the frequency the phase response of the two-pole two-zero filter in the figure is .

Note that an additional phase of radians appears when the number of poles is not equal to the number of zeros. This factor comes from writing the transfer function as

and may be thought of as arising from additional zeros at when , or poles at when . Strictly speaking, every digital filter has an equal number of poles and zeros when those at and are counted. It is customary, however, when discussing the number of poles and zeros a filter has, to neglect these, since they correspond to pure delay and do not affect the amplitude response. Figure 8.5 gives the phase response for this two-pole two-zero example.

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