If we want to analyze a system, which is already represented in the frequency domain, a discrete time signal then we go for inverse Z transformation.

Mathematically it can be represented as :

x(n) = z

^{-1 }X(Z)
x(n) = Z-1 X(Z)

Here x(n) is the signal in time domain and X(Z) is the signal in the frequency domain to be represented.

And here also define if we want to represent the above equation in integral format then we can write it as

x(n) = (1/2∏j) ∫ X(z) Z-1 dz

Here the integral is over close path C. This is within the region of conversions (ROC) of the x(z) and it does contain the origin. Now here this article also gives the information about how to find inverse Z transform.

**Method to find Inverse Z - transform :**

We follow the following four-way to determine the inverse Z transformation system.

- Long division method
- Partial fraction expansion method
- Contour or residue integral method