 Now, you see let us look at the properties of z transform as we did for the discrete time Fourier transform. Let us first look at the linearity or otherwise. In fact, it is almost trivial. It is almost trivial to show that the z transform is linear or it is a linear operator. What I mean is, if you think of the z transform as an operator which takes you from the set of sequences and you know the z transform is denoted this way a script z often. It is also sometimes denoted like this script z. Both of these are used as symbols for saying take the z transform. If h n leads you to h z with an r o c r h, so you know you must always write it like this. An expression h z with an r o c or region of convergence script r h. Script r h is a region between two concentric circles. Is that right? So, if h n and in fact, now we use the same shorthand notation if h 1 and 2 n respectively have the z transforms h 1 2 with regions of convergence r h 1 2, then alpha times h 1 plus beta times h 2 would have the z transform or you can write z transform alpha capital H 1 plus beta capital H 2. Now comes the question of region of convergence. What do you think we can say about the region of convergence? The region of convergence is at least the intersection of r h 1 and r h 2. It could be bigger. R o c at least r h 1 intersection r h 2 could be bigger. This is often the case with the regions of convergence. The region of convergence is of course, at least the intersection. It could expand beyond the intersection and the expansion takes place because the trouble created by one sequence can be outdone by the other. Let us take an example. You see, let us take the sequence h 1 n is delta n plus half raised to the power of n u n and let h 2 n be minus half raised to the power of n u n. It is very easy to see that of course, h 1 n plus h 2 n has the region of convergence is of course, the z transform is essentially the z transform of delta n and that is one for all z and the region of convergence is all z. On the other hand, the region of convergence of either h 1 or h 2 is mod z greater than half. So, here the intersection is mod z greater than half, but the region of convergence has expanded beyond the intersection because the trouble created by one sequence has been outdone by the other. Is that right? So, much so for the linearity of the z transform. Now, we look at some other properties. What happens when we delay or shift that is very easy. So, if x n has the z transform of a variety, let us use x n. We will use one of them in future. The z transform x z with the region of convergence r x. What is the z transform of x n minus d? d is an integer. It is very easy to answer this question. We only need to write down the summation. The summation n going from minus to plus infinity x n minus d z raise the power minus n. You see, it can be evaluated by putting n minus d equal to m. Now, when n runs over all the integers, so does m for a fixed integer d. Therefore, this can be rewritten as summation m going from minus to plus infinity x m z raise the power minus m plus d. Therefore, that is z raise the power minus d x z. The only difficulty is what happens to the region of convergence. Now, the region of convergence is going to be affected by the factor z raise the power minus d. So, the only effect that the factor z raise the power minus d can have is to affect what happens at the boundaries. So, essentially it is almost r h as the region of convergence possibly except boundaries. The region of convergence will almost be the same. The boundaries have to be carefully seen. Let me give you an example. Again, a very simple example. Take delta n minus 2. Delta n minus 2 had the z transform z to the power minus 2. We have seen that before. The region of convergence is the entire z plane excluding z equal to 0. Advance this by 2. So, take n plus 2 in place of n there and you get delta n. Now, the boundary also gets included. So, because of the multiplication by z raise the power minus d, you need to worry about what is happening at the boundaries. Otherwise, the rest of the region of convergence is unaffected. The interior is unaffected. The boundaries could be affected. So, much so for delaying or shifting. Now, we see a very interesting variant. We will see one variant of a property today and then we will look at more properties in the next lecture. The important variant we will see is that of differentiation and here we differentiate the z transform. So, here we go the other way. Let x z be the z transform of x z with the region of convergence r x. We ask the question what is it that has the z transform d x z d z or can we do something to figure out what happens to x n when we take the derivative with respect to z if it exists. Let us write down the expression. So, d x z d z is d z times summation n going from minus to plus infinity x n z raise the power minus n. Now, let us assume that this derivative exists. So, if it exists then we can evaluate the derivative term by term. If it exists then it is analytic. So, we see normally we expect now we are going to consider this as a subtle point. Functions in the complex plane can be analytic or not analytic in the region of. Analytic means they have an infinite number of continuous derivatives. Now, we will initially deal only with z transforms which are in fact, all through this course we will deal largely with z transform which are analytic in the region of convergence. So, in the region of convergence they have an infinite number of derivatives. Of course, one can always conceive of functions that do not have this property, but we shall not do it in this course. It is more of mathematical interest than practical. So, anyway we can then take the derivative term by term and that gives us minus n, see it becomes summation n going from minus to plus infinity minus n times x n. Remember x n is a constant with respect to z minus n minus 1. Therefore, if you take minus z times d x z d z that is summation n going from minus to plus infinity n x n times z raise the power minus n. This is very interesting. So, what we are saying is that the z transform of n times x n is essentially the derivative of the original z transform of x n multiplied by minus z. Now, we have assumed the z transform to be analytic in its region of convergence. So, all over the region of convergence interior to the region of convergence the derivative is valid. So, the region of convergence continues to exist at the derivative. I mean the derivative can spread all over the region of convergence. So, the region of convergence R h is definitely included here. Again, because of a multiplication by minus z you need to worry about the boundary. So, region of convergence is at least R h if not more at least the interior. So, what we have said effectively is n times x n has the z transform minus z d x z d z and the region of convergence is definitely the interior of R h if not more and the more essentially refers to the boundaries. So, this brings us to a very interesting point. In fact, we intentionally took this property before others, because this property can also be taken to the discrete time Fourier transform. If x n has a DTFT, we can ask what is the discrete time Fourier transform of n times x n and in a way it can be answered by this property. If you can evaluate the z transform on the unit circle, unit circle means where R equal to 1. Then you can evaluate essentially you are saying the z transform evaluated on the unit circle. The unit circle is a circle with the radius of 1 or R equal to 1. So, if you can evaluate the z transform on the unit circle, then you can find out what is the z transform or what is the sequence whose z transform is the derivative multiplied by minus z and that gives you the z transform of n times x n. When you multiply a sequence by the time index, you multiply a sequence by n. The effect in the z domain is to take the derivative with respect to z and then multiply by minus z. These operations cannot be interchanged. You must first take the derivative with respect to z and then multiply by minus z and in particular if you do this on the unit circle, you get what happens to the discrete time Fourier transform. That is an interesting property. Now we have to answer several other questions about the z transform. The first question is what happens when you convolve two sequences, the million dollar question when you talk about linear shift invariant systems. So, when I want to see what happens to unstable systems when I give them inputs, I need to know what happens to z transform under convolution. I also need to answer similar questions about the z transform with respect to inversion. Can I invert the z transform? Well, let me mention that it is most common to invert the z transform by experience. That means we associate inverses for certain typical forms and we use those inverses to calculate the inverse of a more complicated z transform. But anyway, in the next lecture, let us see a little more about some of these questions. Thank you.