The z-transform of a sequence is defined as

There is a close relationship between the z-transform and the Fourier transform of a discrete time signal, which is defined as

Summary: A brief definition of the z-transform, explaining its relationship with the Fourier transform and its region of convergence, ROC.

The z-transform of a sequence is defined as

There is a close relationship between the z-transform and the Fourier transform of a discrete time signal, which is defined as

In order to get further insight into the relationship between the Fourier Transform and the Z-Transform it is useful to look at the complex plane or z-plane. Take a look at the complex plane:

Z-Plane |
---|

The Z-plane is a complex plane with an imaginary and real axis
referring to the complex-valued variable

The region of convergence, known as the ROC, is
important to understand because it defines the region where
the z-transform exists. The ROC for a given

For

For