Why z transform is used

Stack Overflow for Teams — Collaborate and share knowledge with a private group. Create a free Team What is Teams? Learn more. What is the significance of Z-transform? Ask Question. Asked 6 years, 7 months ago. Active 6 years, 7 months ago. Viewed 12k times. It has nowhere mentioned why we are even using z-transform my apologies if this question is too basic or if it shouldn't belong here. Improve this question. We will use a different technique that is a bit more complicated mathematically but will, in the long run, yield some physical insights not afforded by the simpler technique.

To understand the sampling process in this paradigm, first consider a signal that is a train of evenly spaced impulse functions. There is a double-sided Z Transform that takes the limit of the summation from negative to positive infinity much like the double-sided Laplace Transform.

We will not consider the double sided transform here. We can rewrite the sampled signal as. Generally when we consider a sequence we will implicitly assume a sampling interval, T, and simply use x[k].

Since we now have a time domain signal, we wish to see what kind of analysis we can do in a transformed domain. Let's start by taking the Laplace Transform of the sampled signal:.

Since x[k] is a constant, we can because of Linearity bring the Laplace Transform inside the summation. This obviously looks different than the Laplace Transforms we have seen in the past.

For one thing, there is an infinite sum. For another, we are used to seeing functions in the Laplace domain that are the ratios of polynomial in the variable 's,' not exponentials in 's. This may seem a complicated way to define sequences, but it turns out that many sequences of interest to us impulse, unit step, In discrete time systems the unit impulse is defined somewhat differently than in continuous time systems.

Note that this is the same as the Laplace Transform of a unit impulse in continuous time. The fact that the Z Transform of an impulse is unity will yield many of the same properties, and allow for many of the same analysis techniques i.

This is called the region of convergence ROC. This method is generally preferred for more complicated functions. Simpler and more contrived functions are usually found easily enough by using tables. We have to worry about the region of convergence, and stability issues, and so forth. However, in the end it is worthwhile because it proves extremely useful in analyzing digital filters with feedback.

For example, consider the system illustrated below. Similarly, given a rational function, it is easy to realize this function in a simple hardware architecture. The z-transform proves a useful, more general form of the Discrete Time Fourier Transform. More on the region of convergence will be discussed below. Although the z-transform achieved by directly applying this formula, the inverse z-transform requires some mathematical manipulations that is related to the power series and geometric series.

More on this will be discussed in the next sections. There is a very close relation between DTFT and z-transform. Even each of their respective formulas are also quite similar, which is often overlooked. In other words it is a unit circle 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.

As we have seen earlier that the z-transform comes in two parts. 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.

In general we have three types of signals which are: right sided, left sided and two sided. For each of these three types of signals we have three different types of region of convergence. If we consider a signal x[n], then the signal is right sided if x[n] is greater than zero for positive values of n as defined by the unit step function.