Poles and Zeros of Time-Domain Response Functions

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This Demonstration displays the response of a system with domain representation to an impulse, step or sinusoidal input. is a transfer function generated by multiplying the poles and zeros together. Poles are at locations marked with a red X and have the form . Zeros are at locations marked with a blue O and have the form . You can drag the poles and zeros, but because the generating differential equation is assumed to have real coefficients, all complex poles and zeros occur as complex conjugates.


The poles and zeros of a system enable reconstruction of the input/output differential equation.


Contributed by: Aaron Becker  (November 2016)
(The University of Houston)
Open content licensed under CC BY-NC-SA



The poles and zeros are movable locators that are disabled when inside the gray rectangle. Poles are roots of the denominator of , while zeros are roots of the numerator.

Poles in the right-hand plane of the domain (with positive real components) represent unstable modes with time-domain responses that either increase to or decrease to as time increases. This region of the domain is colored red. A single right-hand plane pole dominates the system response and makes the system unstable. If the poles are all in the blue region, the response is damped and it overshoots the final value by no more than 15%.

Transfer function zeros cannot change the stability of the system but can alter the response. Each pole generates a response in the time domain. Poles further to the right influence the time-domain response more than poles to the left, because the time-domain responses of poles with large negative components decay to zero quickly. An oscillatory response is due to complex pole pairs.


[1] F. T. Ulaby and A. E. Yagle, Engineering Signals and Systems in Continuous and Discrete Time, 2nd ed., Allendale, NJ: National Technology and Science Press, 2016.

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