# Z Transform
Transfer Functions 1P 1Z Frequency Response

Cuthbert Nyack

The Z transform of a system with a single pole at p_{1} and
a single zero at z_{1} along the real Z axis is given by:-
The frequency response is obtained by substituting
e^{jwT}
= e^{j2pw/ws}
for z.

The applet below shows the frequency response of the system which is
also the Z transform along the unit circle in the Z plane.
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 orange line is the imaginary z axis. Pole is
shown by the "x" and the zero by an "o". When the "Theta" 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 produces a peak in the
response while having a zero on the unit circle produces a
zero in the response.

Setting Fn = 1 shows the unfolded spectrum between 0 and
2w_{s}. Numerical values can be seen by changing wL.

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Copyright 2008 © Cuthbert A. Nyack.