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=3D Visualization of Z-Transform=
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=3-D Visualization of Z-Transform=
  
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== '''1. Objective''' ==
  
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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.
  
Put your content here . . .
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== 2. Draw the Z Transform plot in 3D format ==
  
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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:
  
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[[File:4381.png]]
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                      Figure1: 2D Z-Transform representation
  
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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.
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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
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== 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.

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Prof. Math. Ohio State and Associate Dean
Outstanding Alumnus Purdue Math 2008

Jeff McNeal