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Analog-to-discrete system conversion using impulse invariance

Nasser M. Abbasi

May 3, 2010 compiled on — Wednesday July 06, 2016 at 08:25 AM
Applied to second order system, sampling, poles, zeros.

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 H (s)  is converted to the discrete system H (z)  , 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 H(s)  and H (z)  are displayed with their pole locations.

In this Demonstration, j  stands for √---
 − 1  .

Using the impulse invariance method, H (z)  is directly generated from H (s)  using a mapping that depends on the sampling period and the locations of the poles of H (s)  . Because the input is an impulse, the system transfer function H (s)  is the same as the Laplace transform of the response y(t)  .

The method starts by expressing the Laplace transfer function H(s)  in partial-fraction form

        N∑    Ai
H (s) =    s-−-p-
        i=1      i

Where the N  poles are located at the points pi  . Then the discrete system can be written as

H (z) =    ------T-Ai------
        i=11 − z−1 exp (piT )

where T  is the sampling period for the analog system.

This formula applies when the poles p
 i  are all distinct. In the case of a pole of order two, which pertains to the damping ratio ζ = 1

       ∑N   T zexp (T pi)
H (z) =    -------------2-
       i=1 (exp(piT )− z)

Plots of the magnitude of the frequency response are generated for both H(s)  and H (z)  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 T  larger and comparing the frequency response shapes of the analog and discrete systems.

In these plots, the frequency axis (x  axis) is a linear scale (not the more usual logarithmic scale) to better illustrate the method.

Locations of the poles of H (s)  and H (z)  show that stable poles in the left s  -half-plane are mapped to stable poles inside the unit circle in z  -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 ζ  , the natural frequency ωn  , and the sampling period T  to observe how the poles of H (s)  and H (z)  change. The analytic forms of H (s)  and H (z)  are displayed at the top-center of the graphic with the numerical values of the poles. The system response y(t)  is plotted, with the option to scale the x  axis and the y  axis manually.


A. V. Oppenheim and R. W. Schafer, Digital Signal Processing, Upper Saddle River, NJ: Prentice Hall, 1975 pp. 201-203.