Each of the transforms will be accompanied by an image of the solution and a video of how the computation is performed.

In the discrete time domain, a signal is usually defined as a sequence of real or complex numbers which is then converted to the z-domain by the process of z-transform.

The z-transform is a very useful and important technique, used in areas of signal processing, system design and analysis and control theory.

The first part is the formula as shown above and the second part is to define a region of convergence for the z-transform.

Both parts are needed for a complete z-transform as a z-transform without a ROC would not be of much help in signal processing.

Even each of their respective formulas are also quite similar, which is often overlooked.

So before going any further, lets look at the formulas for both the DTFT and the z-transform for a signal xn.

Thus the final DTFT formula can be written as:,mathcalX(ejomega)sumn-inftyinftyxne-jomega n, Which if we choose the variable omega becomes:,mathcalX(omega)sumn-inftyinftyxne-jomega n, Taking r1 for z which gave us zejomega means that we are dealing with a circle which has a radius 1 and is centered around the origin on the z-plane.

Thus we can conclude that the z-transform of the signal can exist anywhere on the z-plane but the DTFT of the signal can only exist on the unit circle.

One is the mathematical formula and another is a region definition which is know as the region of convergence.

This region of convergence needs to be defined for all z-transforms, for values of z as it will define the points on the z-plane where the z-transform converges and in other words the z-transform exists.

Again in simpler terms, we know that the z-transform is a summation, so for the summation to exist, it needs to converge at some point.

If the summation converges then the z-transform fall within the region of convergence and the z-transform exists otherwise the summation diverges and we say that the z-transform does not exist.

It should be remembered always that for a z-transform, the region of convergence cannot contain any poles.

For each of these three types of signals we have three different types of region of convergence.

This means that the signal can have many other poles but the one that is farthest away from the origin forms a circle around the origin, beyond which the z-transform is convergent.

However since we are assuming this plot to be of a right sided signal, the region of convergence is formed by the region beyond a circle with radius 3 and centered at the origin.

So for example xn could be xn(2n)u-n-1 which is a left sided signal which starts at n-1 and goes on to negative infinity.

This means that the signal can have many other poles on the z-plane but the one that is closest to the origin forms a circle around the origin, inside which the z-transform is convergent.

Since this is a left sided signal, the region of convergence is within the circle with radius 1, centered at the origin and going through the pole at z1.

The region of convergence for two sided signals usually forms the shape of a doughnut.

There can be many poles for the signal but the region of convergence is alway bounded by a doughnut shaped ring which goes through two poles and the ring does not contain any poles inside.

This doughnut shaped ring can be formed by any two poles however with different poles we will get different criterias for the region of convergence for the signal.

However for this particular system of signals, the region of convergence is at z1 and z.

Z Transform Calculator