Inverse Z Transform
Overview
- The purpose of this page is to...
- I. Define the Z Transform and Inverse Z Transform
- II. Provide Example Problems of the Inverse Z Transform
I. Definitions
- Z Transform
$ X(z)=\mathcal{L}(x[n])=\sum_{n=-\infty}^{\infty}x[n]z^{-n} $
- Inverse Z Transform
$ x[n]=\mathcal{L}^{-1}(X(z))=\frac{1}{2\pi j}\oint_{c}X(z)z^{n-1}dz $
II. Example Problems of the Inverse Z Transform
- We will find the Inverse Z transform of various signals by manipulation and then using direct Inversion
Ex. 1 Find the Inverse Z transform of the following signal
$ X(z)=\frac{1}{1-z}, \text{ ROC } |z|<1 $
- note: It is important to realize that we are not going to try to use the direct formula for an inverse Z transform, Instead our approach will be to manipulate the signal so that we can directly compare it with the Z transform equation and by inspection obtain the Inverse Z transform.
First we need to manipulate the given ROC equation to be in the following form, with 'A' being some expression that contains z
In this case this is already satisfied with
The reason for this is so that we can use the formula for an infinite geometric sum and so that we are operating in the correct ROC (this will change your answer) Using a infinite Geometric sum we can obtain following...
$ X(z) = \sum_{n=0}^{\infty} 1(z^{n}) = \sum_{n=-\infty}^{\infty} z^{n} u[n] $
$ \text{let } k = -n \text{ then, } X(z) = \sum_{k=-\infty}^{\infty} z^{-k} u[-k]= \sum_{k=-\infty}^{\infty}u[-k] z^{-k} $
By comparison with the Z Transform definition, we can determine $ x[n] $
$ x[n] = u[-n] $
