# Z Transform Transfer Functions 1 Pole Frequency Response.

Cuthbert Nyack
The transfer function of a system with a single pole at p1 and n zeros at the origin is shown below.
The frequency response is obtained by substituting ejwT = ej2pw/ws for z.
The applet below shows the frequency response around the unit circle. n is the number of zeros of G(z). -ve n means poles at the origin. Realizable transfer functions should have more poles than zeros.
The gray curve above the magenta lines show the response between 0 and ½ ws while the gray curve above the green lines show the response between ½ ws and ws. Orange lines show the phase variation. White disc shows the unit circle, black line is the real z axis and cyan line is the imaginary z axis. Pole location is shown by the "x".
When the q parameter is zero the view is along the imaginary axis from the top.
-90° corresponds to a view along the real z axis from the left -¥ and
+90° represents a view along the real z axis from the right +¥.

Note that having a pole near the unit circle +1 or -1 produces a peak in the response at 0 or ws/2.

Changing Fn to 1 shows the unfolded spectrum from 0 to 2ws. Both n and p1 can be varied to illustrate their effect on the frequency response. Zeros at the origin do not affect the magnitude of the frequency response but they affect the real and imaginary parts and the phase. When there are more zeros than poles, then the phase is +ve for part of the frequency range between 0 and ws/2.