Z Filter 3 Poles Impulse, Step and Ramp Response

Cuthbert Nyack

The Z transform of a 3 pole system is shown below :-

and its difference equation is:- y(k) = x(k + n - 3) + (2r cos(fT) + p)y(k - 1) - (p2r cos(fT) + r²)y(k - 2) + pr²y(k - 3)

The impulse response is shown in red in the applet below and its dependence on the pole locations. The unit circle is shown in magenta, the imaginary Z axis is in pink and the pole locations are shown as green "x"s.

Parameters (0.98, 20.0, 0.950, NA, NA, NA, NA, 3, 1, 0.05) show one example of the step response of a 3rd order system when all poles contribute to the response.

Parameters (0.96, 20.0, 0.07, NA, NA, NA, NA, 3, 1, 0.65) show one example of the step response of a 3rd order system when the complex conjugate poles dominate the response.

Parameters (0.25, 20.0, 0.9, NA, NA, NA, NA, 3, 1, 0.6) show one example of the step response of a 3rd order system when the real pole dominate the response.

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