This Demonstration illustrates the impulse invariance method used to convert an analog to a discrete system representation. The analog system consisting of the Laplace transfer function is converted to the discrete system , the Z transfer function. This analog system is the response of a standard second-order system (with damping and stiffness) to a given impulse with zero initial conditions. The functions and are displayed with their pole locations.
In this Demonstration, stands for .
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 T. 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.
References
A. V. Oppenheim and R. W. Schafer, Digital Signal Processing, Upper Saddle River, NJ: Prentice Hall, 1975 pp. 201-203.