In this Demonstration,

.

Using the impulse invariance method,

is directly generated from

using a mapping that depends on the sampling period and the locations of the poles of

. Because the input is an impulse, the system transfer function

is the same as the Laplace transform of the response

.

The method starts by expressing the Laplace transfer function

in partial-fraction form

, where the

poles are located at the points

. Then the discrete system can be written as

, where

is the sampling period for the analog system. This formula applies when the poles

are all distinct. In the case of a pole of order two, which pertains to the damping ratio

,

.

Plots of the magnitude of the frequency response are generated for both

and

to compare the effect of changing the system parameters, including the sampling period

. Aliasing effects (which occur in the impulse invariance method) can be observed by making

larger and comparing the frequency response shapes of the analog and discrete systems. In these plots, the frequency axis (

axis) is a linear scale (not the more usual logarithmic scale) to better illustrate the method.

Locations of the poles of

and

show that stable poles in the left

-half-plane are mapped to stable poles inside the unit circle in

-space.

This Demonstration can also be used to analyze the impulse response of a second-order system as the system's natural frequency and damping ratio are varied.

You can change the system damping ratio

, the natural frequency

, and the sampling period

to observe how the poles of

and

change. The analytic forms of

and

are displayed at the top-center of the graphic with the numerical values of the poles. The system response

is plotted, with the option to scale the

axis and the

axis manually.

Reference: A. V. Oppenheim and R. W. Schafer,

*Digital Signal Processing*, Upper Saddle River, NJ: Prentice Hall, 1975 pp. 201–203.