Inverse Z transform - contour integration

In summary, the conversation discusses finding the inverse z transform of a function, specifically $$X(z)=\frac{1}{2-3z}$$, using the definition formula $$x(n)=\frac{1}{2\pi j}\oint_{C}^{ } X(z)z^{n-1}dz$$ and another method. However, it is mentioned that this method may not be practical without prior knowledge of complex variables and contour integration. It is also noted that there is a similar technique for finding inverse Laplace transforms, but it is not commonly used.
  • #1
etf
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Hi!
Here is my task:
Find inverse z transform of $$X(z)=\frac{1}{2-3z}$$, if $$|z|>\frac{2}{3}$$ using definition formula.
I found that $$x(n)$$ is $$\frac{1}{3}(\frac{2}{3})^{n-1}u(n-1)$$ (using other method). But how can I find it using definition formula, $$x(n)=\frac{1}{2\pi j}\oint_{C}^{ } X(z)z^{n-1}dz$$ ?
Thanks in advance
 
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  • #2
Forget it, unless and until you have taken a course in functions of a complex variable with contour integration.
Sample: http://ocw.mit.edu/resources/res-6-008-digital-signal-processing-spring-2011/study-materials/MITRES_6_008S11_sol06.pdf
 
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  • #3
There is a similar integral technique for finding inverse Laplace transforms:

http://en.wikipedia.org/wiki/Inverse_Laplace_transform

No one uses it.

It's more practical to get your transform (Laplace or Z) into a form which can be looked up in a table.
 
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FAQ: Inverse Z transform - contour integration

1. What is the purpose of the inverse Z-transform?

The inverse Z-transform is used to find the original time-domain sequence from its corresponding Z-transform. This allows us to analyze and understand the behavior of discrete-time systems in the time domain.

2. How is the inverse Z-transform related to contour integration?

The inverse Z-transform can be calculated using the method of contour integration, which involves integrating along a closed contour in the complex plane. This allows us to efficiently compute the inverse Z-transform for complex functions.

3. What is the difference between the inverse Z-transform and the inverse Laplace transform?

The inverse Z-transform is used for discrete-time signals, while the inverse Laplace transform is used for continuous-time signals. Additionally, the Z-transform is defined for both finite and infinite sequences, whereas the Laplace transform is only defined for causal signals.

4. What are the properties of the inverse Z-transform?

The inverse Z-transform has several important properties, including linearity, time shifting, and convolution. These properties allow us to simplify and analyze complex systems using the inverse Z-transform.

5. How is the inverse Z-transform used in practical applications?

The inverse Z-transform is used in a variety of practical applications, such as digital signal processing, control systems, and telecommunications. It allows us to analyze and design systems that operate in discrete time, which is commonly used in modern technology.

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