# Z Transform
Transfer Functions 1 Pole Frequency Response.

Cuthbert Nyack

The transfer function of a system with a single pole at p_{1}
and n zeros at the origin is shown below.
The frequency response is obtained by substituting
e^{jwT}
= e^{j2pw/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 w_{s} while the
gray curve above the green lines
show the response between ½ w_{s}
and w_{s}. 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 w_{s}/2.

Changing Fn to 1 shows the unfolded spectrum from 0 to
2w_{s}. 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 w_{s}/2.

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© 2000 Cuthbert A. Nyack.