Z Transform Transfer Functions 1 Pole Impulse, Step and Ramp Response.

Cuthbert Nyack

The Z transform of a single pole system is shown below.

And the corresponding difference equation is:-
y(k) = x(k + n - 1) + p₁y(k - 1)
If n = 2, (more zeros than poles) then y(0) = x(1) + 0 for a sequence which begins at zero ie the output produced by x(1) appears before x(1).

The variation of the impulse(cyan) response with the pole location is illustrated by the following applet with Fn = 0. The unit circle is shown in magenta, the real Z axis is in black, the imaginary Z axis is in pink and the pole location is shown as a green "x". When the pole is between 0 and 0.99 response(red) is an exponential decay. The response is a constant(step) when the pole is at 1 and increases exponentially when the pole is at a location greater than 1. When the pole is between 0 and -0.99, the response alternates and decays.

Changing n shows the effect of changing the number of zeros or poles (n < 0) of the function at the origin of the Z plane. An additional zero advances the output, moves it to the left. An additional pole delays the output, moves it to the right. If n > 1, then the output begins before the input. This is an unrealizable system.

Fn = 1 shows the step response.
Fn = 2 shows the ramp response.
Fn = 3 shows numerical values of the impulse response.
Fn = 4 shows numerical values of the the step response.
Fn = 5 shows numerical values of the the ramp response.

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