Plotting responses to z-plane transfer functions

In summary, you should inverse Laplace to get to the time domain and use a stem plot to plot the time response.
  • #1
Jimbo
10
0
Hi

I am looking for any tutorials in how to plot a time response plot from a z-plane transfer function

For example, if you have G = 4 / z + 2

I understand that plotting this on the z-plane would result in a point at -2 (although I am not sure about how the numerator of 4 might affect this) but I am unsure about the steps to be taken to convert this to a time response plot?

Any help is much appreciated!
 
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  • #2
If you're talking about what I think you're talking about (control systems... my experience uses 's' instead of 'z'), then to plot the time domain, you need to take the inverse Laplace to get to the time domain.

In this case, you're looking at e^-2t for a time domain response to a step input.
 
  • #3
You'd only use the inverse Laplace tranform to find the time response if this were a continuous-time tranfer function. Since we're talking about the z-plane, I'll assume you're talking about a discrete-time system. You should specify the form of the input function, since the output function response will depend on the input convolved with the transfer function; since you didn't say, I'll assume you're looking for the unit impulse response (that's what the tranfer function is).

Anyhow, let's see if I can do this without messing up:

[tex] Y(z) = G(z)X(z) = \frac 4 {z+2} = 4 \frac{z^{-1}}{1+2z^{-1}} = 4z^{-1}\frac 1 {1+2z^{-1}} [/tex]

(X(z) = 1)

which is the same as
[tex] Y(z) = 4\frac{1}{1+2z^{-1}} [/tex] time shifted one sample to the right.

so
[tex] y[n] = 4(-2)^{n-1}u[n-1] [/tex]

where u is the unit step function.

The standard way to plot this would be to use a stem plot
(so it kind of looks like a bunch of lollipops).
y[0] = 0
y[1] = 4
y[2] = -8
y[3] = 16
and so on...
 
  • #4
If you want the inverse Z-transform, then it involves a contour integration. You should have a table in your textbook that would include Z-transforms of the basic functions. If you want the general rule:

x(n) = (1/2πi) integral of { X(z) zn-1 dz }

where x(n) is the discrete time signal, X(z) is the Z-transform of it, and the integration is around a closed contour (but I can't remember what the rules are for the contour; it probably has to contain all of the poles and the origin). In your case, X(z) is G and x(n) is the response to a Kronecker Delta input of δ0n.
 
  • #5
Great!

Thanks for all the help!

Just what I needed!
 

1. What is a z-plane transfer function?

A z-plane transfer function is a mathematical representation of the relationship between the input and output signals of a system in the z-domain. It is commonly used in control systems and signal processing applications to analyze and design systems.

2. How do you plot a response to a z-plane transfer function?

To plot a response to a z-plane transfer function, you first need to determine the transfer function itself. This can be done by taking the Laplace transform of the system's differential equations. Once you have the transfer function, you can use software tools such as MATLAB or Python to plot the response using either the impulse or step function.

3. What is the significance of the poles and zeros in a z-plane transfer function?

The poles and zeros in a z-plane transfer function represent the frequencies at which the system's response is either amplified or attenuated. The location of these poles and zeros can provide insight into the stability and performance of the system.

4. How do you interpret the shape of a response plot in the z-plane?

The shape of a response plot in the z-plane can provide information about the system's behavior. A flat response indicates a steady-state response, while a peaked response indicates an overshoot or ringing behavior. A gradual decrease in amplitude indicates a decay in the response.

5. Can a z-plane transfer function be used to analyze digital systems?

Yes, z-plane transfer functions are commonly used to analyze and design digital systems. Since digital systems operate in discrete time, the z-plane provides a more suitable domain for analysis and design compared to the Laplace domain which is used for continuous-time systems.

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