Inverse Z Transform
Overview
- The purpose of this page is to...
- 1. Define the Z Transform and Inverse Z Transform
- 2. Provide Example Problems of the Inverse Z Transform
1. 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 $
1. 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
