Sketching Frequency domain repsonses

In summary, the problem involves sketching the z-plane pole zero diagram and finding the time and frequency domain responses for the given function G(z). The important equations are G(z) = Y(z)/X(z), zeros when Y(z) = 0, and poles when X(z) = 0. The magnitude and phase frequency responses can be calculated by considering the distances from zeros and poles at different points around the unit circle. The calculation should start from 1 on the real axis and move anti-clockwise around the unit circle.
  • #1
ineedmunchies
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Homework Statement


Sketch the z-plane pole zero diagram for:
[tex]G(z) = \frac{z^{2} + z + 1}{z^{3}}[/tex]

Also sketch the time and frequency domain repsonses, the latter in amplitude and phase.

Homework Equations



[tex] G(z) = \frac{Y(z)}{X(z)}[/tex]

Zeros when [tex]Y(z) = 0[/tex];

Poles when [tex]X(z) = 0[/tex]

For frequency reponse:
[tex] |G(\omega)| = \frac{\prod Distance from Zeros}{\prod Distance From Poles}[/tex]

[tex] \angle G(\omega) = \sum Angles from Zeros - \sum Angles from Poles[/tex]


The Attempt at a Solution



I have found that there is a pole at zero and two zeros at -0.5 on the real axis.

However I cannot figure out how to calculate or even sketch from inspection the magnitude and phase frequency response.

I have looked at a few different sources and have got somewhat confused at to what point this calculation should be made. Say for example I wanted to figure out the magnitude response at [tex]\omega = 0[/tex], [tex]\pi[/tex] and [tex]2\pi[/tex]. Would I calculate these distances to a point on the unit circle that corresponds to this value of [tex]\omega[/tex] or to a point on the real axis, or another point altogether.

I'm confused! Any help at all would be appreciated!
 
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  • #2
Ok so I figured this out straight away after posting. I meant to say that I found two zeros at -1. I then realized that I need to calculate for points around the unit circle, starting from 1 on the real axis and moving anti clockwise around the unit circle. Thanks for anyone that took time to read this.
 

1. What is a frequency domain response?

A frequency domain response is a representation of a system's output as a function of frequency. It shows how the system responds to different frequencies of input signals and is useful for analyzing the behavior of systems in various applications, such as signal processing and control systems.

2. How is a frequency domain response different from a time domain response?

A time domain response shows the behavior of a system over time, while a frequency domain response shows the behavior of a system at different frequencies. Time domain responses are typically represented by waveforms, while frequency domain responses are represented by plots of magnitude and phase as a function of frequency.

3. How is a frequency domain response related to the system's transfer function?

The frequency domain response is derived from the system's transfer function, which is a mathematical representation of the system's response to a given input. The transfer function can be used to calculate the frequency domain response, allowing for analysis and design of the system's behavior.

4. What is the purpose of sketching frequency domain responses?

Sketching frequency domain responses allows for visual representation and analysis of a system's behavior at different frequencies. This can help in understanding the system's stability, bandwidth, and frequency response characteristics, and can aid in designing and optimizing the system for desired performance.

5. What are some common techniques for sketching frequency domain responses?

Some common techniques for sketching frequency domain responses include using Bode plots, Nyquist plots, and Nichols plots. These plots display the magnitude and phase response of the system as a function of frequency, allowing for easy visualization and analysis of the system's behavior.

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