 A warm welcome to the 8th session of the fourth module in signals and systems, we have been looking at the properties of the Laplace transform and the Z transform. We had looked at the property of shifting, what happens when you shift and we had been shifting a discrete sequence, let us complete that discussion. So, there again let us take the unit discrete impulse. What is the Z transform of the unit discrete impulse? Delta of n, very simple, this simply picks up the value of the rest of this exponential at n equal to 0 and therefore, it simply leaves you with 1, Z to the power minus 0. The region of convergence is the entire Z plane. Now, what happens when you shift backward and shift forward? Let us see. So, let us take delta n plus 1 and query its Z transform. It is very easy to see that this would be essentially this summation with only one term really non-zero and that is Z and here, all the Z plane can be considered except mod Z tending to infinity. So, this poses a problem at mod Z tending to infinity. So, the region of convergence is the entire Z plane except mod Z tending to infinity. So, you have to include all of the Z plane, but exclude that extreme contour, funny situation. Now, let us see what happens if you shift forward. Shifted forward by one step, again simple summation which gives you Z to the power minus 1 and here, we have a problem when mod Z tends to 0 because the inverse of 0 is not defined and therefore, the region of convergence is the entire Z plane except mod Z tending to 0 of that matter Z tending to 0. It is the beauty is that when you are talking about moduli, then mod Z tending to 0 is the same as Z tending to 0. There is no problem, but mod Z tending to infinity is there are many ways of defining it. Of course, here you would think of it as a circle with a growing radius. Anyway, so the extreme contours are a cause of worry when we shift. Now, let us come to a very important, well you know in a sense dual property of the Z transform or the Laplace transform. What happens when you multiply a signal by another signal of exponential nature or essentially a modulator kind of a signal. So, let us see, we call it the modulation property. So, in other words the question that we are asking is if I have X t with the Laplace transform of capital X of S and script R subscript capital X as its region of convergence. What is the Laplace transform of e raised to power a t times X t? Of course, it depends on a, I agree. A is a constant, could be a complex constant in general, very simple. Let us see. Now, you can see that when we make this expansion everything else looks very similar except that we are evaluating the Laplace transform of X t at a different value of S. We are evaluating it at S minus a and therefore, S minus a needs to belong to the region of convergence not S. Now, let us take a very simple example. Let us take e raised to power 2 t u t, our favorite signal hall this while. Let us look at its Laplace transform. We know what it is. It is 1 by S minus 2 with the region of convergence real part of S greater than 2. This is X of t. Let us multiply it by e raised to power 3 t and query its Laplace transform. What is e raised to power 3 t X t? This is after all essentially e raised to power 5 t u t. So, simple and therefore, its Laplace transform is going to be 1 by S minus 5 with the real part of S greater than 5, not very difficult. Essentially, that can be obtained by putting S minus 3 in place of S and S minus 3 in place of S and S minus 3 replacing S here as well. So, real part of S minus 3 is greater than 2 or real part of S is greater than 5. Simple enough. So, what happens here? Here there can be a substantial change in the region of convergence. In fact, the region of convergence can shift backward or forward depending on the exponential by which you have multiplied. Now, this whole concept of multiplying by an exponential is important. You see multiplication by just a pure exponential with 0 real part, a rotating complex number whose magnitude remains constant has a physical significance. It is a part of multiplication by a sine wave and that is for example, important in what is called amplitude modulation. How do you analyze what happens in amplitude modulation in a communication system? That is why this word modulation has been used here. Modulation essentially means multiplying by a fixed signal. So, a chosen signal if you might want to call it that. So, let us look at what happens when you modulate a sequence now. So, suppose you had x of n with a z transform capital x of z and region of convergence script R subscript big x. What is the z transform of alpha to the power of n x z? We need to query an answer. Let us do that. Simple enough. So, you can see essentially what we have done after all this algebra is to obtain the z transform of x of n with z replaced z replaced by z into alpha inverse. Of course, alpha is a complex constant in general and there again now when you talk about the region of convergence it is z alpha inverse which must lie in the region of convergence not z. Let us take an example to illustrate. Let us consider x n are favorite to raise the power of n u n. We know it is z transform 1 by 1 minus 2 z inverse with mod z greater than 2. Now, let us consider one third to the power of n times x n and query its z transform. So, one third to the power of n x n is essentially 2 by 3 to the power of n u n and this z transform is very easy to write. It is 1 by 1 minus 2 by 3 z inverse and this z transform is with mod z greater than 2 by 3. We could also obtain that directly. We could replace z by 3 times z essentially and then 3 times z needs to have a magnitude greater than 2 where upon mod z needs to be greater than 2 by 3 as expected. It tallies and of course these 2 also tallies. When I take 1 minus 2 z inverse with z replaced by 3 z it gives me this 1 minus 2 by 3 z inverse simple enough. Essentially, what modulation does is to make a replacement of the variable in the s domain or in the z domain in the Laplace transform or in the z transform. There is a replacement of the variable and when you make a replacement of the variable there is also replacement of the variable in the region of convergence. So, in terms of the original variable s or z the region of convergence can change. Now, we are going to ask a very important property of Laplace and z transforms to be elaborated. Let me explain what we are going to now discuss. We are going to ask what happens when we convolve. That is one of the difficult operations in the natural domain. So, let x of t have the Laplace transform. First let us take the continuous independent variable. Let x of t have the Laplace transform capital X of s with the region of convergence script R subscript capital X and similarly h t. Let y of t be the convolution of the 2. We ask what is its Laplace transform and what is the region of convergence. And we ask a similar question for the z transform. But first let us answer the question for the Laplace transform. A very important question here. Consider the Laplace transform of y t and expand y t. Of course, the d tau needs to come into this integral actually. But let me take all the integral together that is easier to do. And now let us make a replacement of variable. So, let us have tau going as it is and t minus tau being replaced. If lambda is t minus tau for fixed tau for t going from minus to plus infinity implies lambda also goes from minus to plus infinity. And therefore, I can rewrite this. Further t is lambda plus tau. Let me make these replacements here. Let us expand this now and notice that we can now separate the integrals. The separation gives us and clearly you can see that this is the Laplace transform of small x and this is the Laplace transform of small h with the appropriate regions of convergence. Now, you see the difficult question here is what happens to the region of convergence in tau tau. That is when you try and consider the convergence of this product. The first thing that we have to note is that we have a very significant result here. Look at the expression. Essentially, it is a product of the Laplace transform. A very significant result. In fact, this result is so important that we are going to discuss it in much greater detail in the next session. So, stay and listen to the next session very carefully that you understand this important property. Thank you.