 A warm welcome to the 7th session of the 4th module in signals and systems. We have commenced looking at the properties of the Laplace transform and the Z transform. We saw the most important property namely linearity in the previous session and we understood the important variation that you need to worry about when you look at the Laplace and the Z transforms namely the region of convergence. You see linearity was a property even in the context of the Fourier transform it was a property in the context of the discrete time Fourier transform. But we did not have to worry about regions of convergence there that is something which you need to bother about and bother about carefully when you talk about the Laplace transform and the Z transform. In this session we shall look at some more properties of the Laplace transform and the Z transform. Now note that we are taking these transforms together this is I am again emphasizing somewhat different from how many textbooks treat it because they tend to separate the two transforms and treat them differently. I personally prefer to treat these two transforms together I think there is a lot of similarity and there are also important dissimilarities or differences which should be emphasized. Let us proceed to the second property. In this second property we are going to look at what happens when you shift on the independent variable. So for example if you are told that a continuous independent variable signal x of t has the Laplace transform and this is the symbol that I am going to use to denote the fact that capital X of s is the Laplace transform of small x of t with the region of convergence r subscript subscript s x. Now what we are going to ask is what is the Laplace transform of x of t minus t0 for some constant t0 and in parallel we are going to also ask if the Z transform of x of n the sequence x of n is capital X of z with the region of convergence script r and subscript capital X what is the Z transform of x of n minus n0 and what is its region of convergence for some integer constant. These questions are not very difficult to answer let us proceed to answer them. Before that let us freeze this notation we will use script L to denote Laplace transform of so you know basically when you put script L as you did here on the previous statement that we made here we are saying that the Laplace transform of x of t is capital X of s with the region of convergence script r subscript capital X and when we write like this as we have written here we are asking what is the Laplace transform of x of t minus t0 for some constant t0 going further script Z as we have written it here would mean Z transform of and therefore you should read this here as the Z transform of x of n is capital X of z with the region of convergence script r subscript capital X and you are asking here what is the Z transform of x of n minus n0. So this is a standard notation that we are going to freeze later we shall also be using the following notation that I write we are going to use script L inverse to denote the inverse Laplace transform and script Z inverse to denote the inverse Z transform of course the preposition of is required. So for example you could say that 1 by s minus 2 with the region of convergence real part of s greater than 2 has the inverse Laplace transform given by e raised to the power 2tut and similarly 1 by 1 minus 2 Z inverse with the region of convergence of mod Z greater than 2 has the inverse Z transform given by 2 raised to the power of n u n. So this is the notation that we are going to freeze and let us use this in future. Now coming back to the property let us investigate the Laplace transform of x of t minus t0 very easy put t minus t0 is another variable let us call it lambda where upon t is lambda plus t0 and of course d t is d lambda also when t runs from minus to plus infinity then lambda also runs from minus to plus infinity there is just a shift after all. So with this substitution what do we have we would be replacing this by lambda here and this by lambda plus t0 and this by d lambda where upon you get so x of lambda e raised to the power minus s into lambda plus t0 d lambda which we can now expand. Now it is very easy to see that what we have here is essentially the Laplace transform of x of t and of course the region of convergence is implied this factor will only be a multiplying factor and as long as it does not tend to infinity there is no problem. So it is not going to affect very seriously. So in fact we can straight away now write down the Laplace transform of x of t minus t0 x of t minus t0 has the Laplace transform essentially the Laplace transform of x t namely capital X of s multiplied by e raised to the power minus s t0 and the region of convergence is essentially the same except possibly the extreme contours as we call them. You know this is the word of caution that we have to give when we shift then everything else remains the same in the region of convergence there is no essential difference but you have to worry about the extremities. Do the extreme contours continue to be included or do they have to be excluded or do they have to continue to be excluded or do they get included you know. So it can happen both ways something that was not initially included could get included something that was included had to get excluded both things can happen. You see let us take an example let us take the Laplace transform of the unit impulse in continuous time very easy all that the unit impulse does is to pick it picks the value of e raised to the power minus s t at t equal to 0 and that is of course equal to 1 and this is unconditional that means we do not have to worry what is the real part of s and so on. So the region of convergence is the entire s plane everywhere in contrast let us look at what happens when you shift the impulse. So let us take delta t minus 1 its Laplace transform is going to be e raised to the power minus s into 1 times 1 and the only problem here is when real part of s starts tending to infinity you know when real part of s starts tending to minus infinity the infinite contours are the problem for real part of s tending to minus infinity this diverges. So that part has to be excluded in other words the region of convergence now becomes the entire s plane excluding real part of s tending to minus infinity that is important. So you know it is very interesting you have this whole s plane but you have to exclude one extreme contours. Now the same thing has to be seen if I shift the impulse backwards so let us do that now I take delta t plus 1 shifted backwards and calculate the Laplace transform. So it is going to be e raised to the power s into 1 and this is going to get multiplied by 1 and here you are going to have trouble in case real part of s tends to plus infinity. So the region of convergence is the entire s plane except the extreme contour real part of s tends to plus infinity that is what I meant when I said essentially the same region of convergence but you do have to worry about the extremities. Now let us look at a similar case for the z transform so what happens when I shift now remember in the case of a sequence you can only shift by integer steps there is no real meaningful interpretation in general for shifting a sequence by non-integer steps you have to ascribe a meaning to it which we have not yet done. So you can only shift by integer let us come back to it. So suppose a sequence x of n has the z transform capital X of z with the region of convergence script R subscript capital X what is the z transform of x of n minus n 0 well simple make the summation and use the same trick put n minus n 0 is another integer variable m of course n 0 is integer. So n is m plus n 0 and when n runs from minus to plus infinity so does m run from minus to plus infinity and therefore we can make the replacement again here you can replace this by m you can replace this by m plus n 0 and you can essentially use the same summation on m thereby what we get is summation m going from minus to plus infinity x of m z to the power minus m plus n 0 which gives us z to the power minus n 0 summation same thing x m z to the power minus m which is essentially z to the power minus n 0 times the z transform of x and the region of convergence is essentially the same except for the extreme contours. Now what are the extreme contours here what is this mean the extreme contours are z in magnitude tending to infinity and z in magnitude tending to 0. So it does not have to do with the real part or imaginary part has to do with the magnitude remember for the z transform essentially your regions of convergence were between circles located with their center at the origin and different radii. So here it is the radius of the circle center at the origin that is of importance 0 radius and the infinite radius that is the radius tending to infinity infinity in the z plane is a concept not a point there are many ways of putting an infinity in the complex plane one is to take a vertical contour as we did the other is to take the circular contour in fact virtually any contour can start growing and becoming infinite. So infinity is a concept there are many different ways of arriving at infinity in the complex plane. So here again we need to worry about the extreme contours we shall see more of this in the next session. Thank you.