# Z Transform Transfer Functions 2 Poles Impulse, Step and Ramp Response

Cuthbert Nyack
The Z transform of a 2 complex conjugate pole system is shown below:-
n zeros at the origin are included in the transfer function. and its difference equation is given by:-
y(k) = x(k + n - 2) + 2pcos(fT) y(k - 1) - p2y(k - 2)

The impulse response of a 2 pole system is shown in red below and its dependence on the pole locations can be seen by changing p and f.
The unit circle is shown in magenta, the imaginary Z axis is in pink and the pole locations are shown as green "x"s. When the poles are off the real +ve axis but inside the unit circle, the impulse response has the characteristic of a damped sinusoid as for a continuous system.
eg (0.96, 20.0, NA, NA, NA, NA, NA, 2, 0, 1.0) 18 (360/20) samples per period.

eg (0.96, 90.0, NA, NA, NA, NA, NA, 2, 0, 2.0)
The response is still a damped sinusoid but at 0.25 ws ie 4 samples per period, 2 at the peaks and 2 at the zeros of the sinusoid.

eg (0.96, 180.0, NA, NA, NA, NA, NA, 2, 0, 0.3)
The response is a damped sinusoid at 0.5 ws ie 2 samples per period at the peaks of the sinusoid.

The step and ramp response can be seen by changing Fn.