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=3-D Visualization of Z-Transform=
 
=3-D Visualization of Z-Transform=
  
== '''1. Objective:''' ==
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== '''1. Objective''' ==
  
 
This is the page that help students to visualize the Z-Transform in 3-D domain. The relationship between Z transform and Fourier Transform will also be illustrated in 3-D form.
 
This is the page that help students to visualize the Z-Transform in 3-D domain. The relationship between Z transform and Fourier Transform will also be illustrated in 3-D form.
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Conventionally, when we draw the 2-D Z transform plot, we assign x-axis as the Real axis and y-axis as the Complex axis. Values on the complex plane will be assigned as numbers or "x"s(when we encounter a pole). Like the figure below:
 
Conventionally, when we draw the 2-D Z transform plot, we assign x-axis as the Real axis and y-axis as the Complex axis. Values on the complex plane will be assigned as numbers or "x"s(when we encounter a pole). Like the figure below:
  
[[File:4381.png|thumb|Figure 1]]
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[[File:4381.png]]
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                      Figure1: 2D Z-Transform representation
  
 
However, if we made this plot in 3D, the entire transfer function will be clearer and more straightforward. Any point that has no value(zeros) will obtain a height of zero and all poles will expand to infinity.  
 
However, if we made this plot in 3D, the entire transfer function will be clearer and more straightforward. Any point that has no value(zeros) will obtain a height of zero and all poles will expand to infinity.  
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Take the high pass filter y[n] = 1/2*(x[n]-x[n-1]) as an example.
 
Take the high pass filter y[n] = 1/2*(x[n]-x[n-1]) as an example.
  
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[[File:4382.png]]
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                      Figure2: Fourier Transform of a high pass filter
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      [[File:4383.png]]
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                  Figure3: 3D Z-Transform representation of the high pass filter
  
 
== 3. Obtain Fourier Transform from the Z Transform plot ==
 
== 3. Obtain Fourier Transform from the Z Transform plot ==
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 +
Suppose we have a 3D Z-Transform plot like the following:
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[[File:4384.png]]
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                  Figure4: 3D Z-Transform representation of something
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 +
When we want to obtain the Fourier Transform from this plot, we just need to have a cylinder with radius 1, and "cut" the 3D image vertically:
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[[File:4385.png]]
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                  Figure5: 3D graph is cut by a cylinder
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Then, take out the extra part, Leave the cylinder and the edges cut from the Z-Transform:
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[[File:4386.png]]
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                  Figure6: Leave the cut part only
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Finally, expand the cylinder onto a plane, now you get the Fourier Transform!
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[[File:4387.png]]
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                  Figure6: Fourier Transform from -pi to pi
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 +
Note: This process could also explain why DTFT always have a period of 2pi. When we are drawing graphs outside [-pi, pi], we are actually trace the edge again and again around the circle in Figure6.

Latest revision as of 21:40, 30 November 2017


3-D Visualization of Z-Transform

1. Objective

This is the page that help students to visualize the Z-Transform in 3-D domain. The relationship between Z transform and Fourier Transform will also be illustrated in 3-D form.

2. Draw the Z Transform plot in 3D format

Conventionally, when we draw the 2-D Z transform plot, we assign x-axis as the Real axis and y-axis as the Complex axis. Values on the complex plane will be assigned as numbers or "x"s(when we encounter a pole). Like the figure below:

4381.png

                      Figure1: 2D Z-Transform representation

However, if we made this plot in 3D, the entire transfer function will be clearer and more straightforward. Any point that has no value(zeros) will obtain a height of zero and all poles will expand to infinity.

Take the high pass filter y[n] = 1/2*(x[n]-x[n-1]) as an example.

4382.png

                      Figure2: Fourier Transform of a high pass filter
     4383.png
                 Figure3: 3D Z-Transform representation of the high pass filter

3. Obtain Fourier Transform from the Z Transform plot

Suppose we have a 3D Z-Transform plot like the following:

4384.png
                 Figure4: 3D Z-Transform representation of something

When we want to obtain the Fourier Transform from this plot, we just need to have a cylinder with radius 1, and "cut" the 3D image vertically:

4385.png

                 Figure5: 3D graph is cut by a cylinder

Then, take out the extra part, Leave the cylinder and the edges cut from the Z-Transform:

4386.png

                 Figure6: Leave the cut part only

Finally, expand the cylinder onto a plane, now you get the Fourier Transform!

4387.png

                 Figure6: Fourier Transform from -pi to pi

Note: This process could also explain why DTFT always have a period of 2pi. When we are drawing graphs outside [-pi, pi], we are actually trace the edge again and again around the circle in Figure6.

Alumni Liaison

Correspondence Chess Grandmaster and Purdue Alumni

Prof. Dan Fleetwood