Analog-to-discrete system conversion using impulse invariance

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Analog-to-discrete system conversion using impulse invariance

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

∑N 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.

References

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