Introduction to Z-Transform

From this lecture, we are going to start a new chapter, and the chapter is Z-Transform.

And we have already completed continuous time Fourier transform and the Laplace transform.

And we know we use CTFT to convert continuous time domain signals to their corresponding frequency domain representations, and we use CTFT to analyze the continuous time signals.

On the other hand, Laplace transform is the general case of CTFT, and we use Laplace transform to analyze continuous time systems.

We can comment about the stability, causality, etc. of a continuous time system using the Laplace transform.

Now we will talk about the existence of CTFT, CTFT will exist for energy and power signals only, whereas Laplace transform will exist for energy signals, power signals and also for neither energy nor power signals, but Laplace transform will exist for any NP signals up to certain extent only.

Now if you remember the previous chapter, we had discussion on discrete time signals and systems, and like CTFT, we have DTFT, that is discrete time Fourier transform.

And we use DTFT to convert discrete time domain signals to their corresponding frequency domain representations.

And we use it to analyze the discrete time signals, so DTFT is equivalent to CTFT, and Z-transform is equivalent to Laplace transform.

And Z-transform is the general case of DTFT, and we use Z-transform to analyze the discrete time systems.

We can comment about the stability, causality, etc. of a discrete time system using the Z-transform. Now let's talk about the existence.

Like CTFT, DTFT will exist for energy and power signals only, whereas like Laplace transform, Z-transform will exist for energy signals, power signals and also for neither energy nor power signals, but it will exist for any NP signals up to certain extent only. Now out of DTFT and Z-transform, we will first have discussion on Z-transform, and once we are done with Z-transform, we will move on to discrete time Fourier transform.

Now let's begin our discussion on Z-transform, and I will first give you the transform, let's say there is a discrete time signal X of N, and we want to calculate its corresponding Z-transform, so we will perform the Z-transform on this signal.

And let's say its Z-transform is uppercase X inside the bracket Z, so when we perform the Z-transform of this signal, we have a new signal, which is X Z. Now when we perform the inverse Z-transform on this signal, we will have the original discrete time signal X N.

And therefore, we call X N and X Z as the Z-transform pairs. Now to calculate X Z, we will perform the summation from N equal to minus infinity to plus infinity, the discrete time signal whose Z-transform we are calculating multiplied to Z power minus N.

Now the summation part is clear, we are performing the summation from N equal to minus infinity to infinity.

And in this part, we have the discrete time signal whose Z-transform we are calculating, and then we have Z power minus N, this N is an integer, it is an integer, and Z is a complex variable, it is a complex variable, and Z is equal to R multiplied to E power J omega.

This is the polar form of the complex variable Z. Now this R is the magnitude of the complex variable Z.

So R is equal to mod Z, and this omega is the complex argument.

From this polar form, we will have a polar plot like this, you can see that we have a circle because when we change omega slowly, slowly, R will remain same, and therefore we have a circle like this. Now we will talk about bidirectional Z-transform and the unidirectional Z-transform. This Z-transform I have written is known as bidirectional Z-transform.

We are calling it bidirectional Z-transform because you can see that we are performing the summation from N equal to minus infinity to plus infinity. Now if we perform the summation from N equal to 0 to plus infinity.

Then we call the Z-transform unidirectional Z-transform, so this one here is known as unidirectional Z-transform. Now to implement what we have discussed till now, let us solve one example.

In this example, the discrete time signal X of N is equal to A power N multiplied to U N, and we are required to calculate the corresponding Z-transform along with the region of convergence.

So let's move on to the solution of this example problem. Discrete time signal X N is equal to A power N U N, therefore the corresponding Z-transform according to this is equal to summation N equal to minus infinity to plus infinity X N, and X N is equal to this, so we have A power N multiplied to U N multiplied to Z power minus N.

And because of U N, we will perform the summation from N equal to 0 to infinity.

And we have summation of A power N multiplied to Z power minus N. Let us write A power N multiplied to Z power minus N as A multiplied to Z power minus 1 power N.

Now when N is equal to 0, we will have 1 because A multiplied to Z power minus 1 power 0 is equal to 1, so this is our first term, and to get the second term, we will put N equal to 1, so we have A multiplied to Z power minus 1.

Similarly, the third term will be A multiplied to Z power minus 1 square all the way to the term which is obtained when N is equal to infinity.

Here you can see that we are having the sum of infinite GP, the common ratio we are getting is equal to A multiplied to Z power minus 1.

And we know the sum of infinite GP is equal to the first term divided by 1 minus common ratio, and we can use this formula when the mod of common ratio is less than 1.

So let's use it to solve our summation, we are having 1 as the first term, so we will write 1 in the numerator divided by 1 minus common ratio, common ratio is A multiplied to Z power minus 1.

So this is what we have as the Z-transform, but we should also be vigilant about this condition here, that is mod of the common ratio A multiplied to Z power minus 1 should be less than 1.

From here we can say that mod Z is greater than A, and this is our region of convergence, mod Z is equal to R, the magnitude of the complex variable, so ROC is equal to R greater than A.

And hence, the Z-transform we have calculated will exist only when R is greater than A. Now we will quickly plot the ROC.

This is our Z-plane, and we will first draw a circle having the radius equal to A.

According to the condition we are having here, R should be greater than A.

This means the region of convergence is all the region outside this circle.

And we know the significance of region of convergence, the Z-transform will exist only within the region of convergence, outside region of convergence, Z-transform will not exist. Therefore, in this particular case, the Z-transform will exist outside this circle, inside the circle, the Z-transform will not exist, so it is important to mention the region of convergence along with the expression of.