 A warm welcome to the ninth session of the third module in signals and systems. We are now proceeding in a direction which tells us the same truth approached by two different paths. In one path, we asked all the sinusoids which have the same samples at the same points and we also asked what happens if you bring them all together. So, you have the original sine wave and you have all these other sine waves which are in some sense imposters, if you would like to call them that, they are deceptive because they seem to have the same samples at the same points and if the original true sine wave comes together with all the imposters, what does it give you? That is the question we asked and we had answered that to some extent by seeing that it gives you the product of the original sine wave and an expression which involved essentially a constant plus a combination of sinusoids. So, now let us put that down first. What we said was if we take the original sine wave plus the imposters, all the imposters have the same samples at the same points. The original sinusoid as you recall was essentially a0 cos 2 pi by t0 t plus pi by 4. When you add these, you get the original sinusoid multiplied by a combination of a constant, let us say some c0 plus summation k going from 1 to infinity as n grows larger. Some constant, in fact, they had the same constant all over. So, this is interesting. We came to this conclusion in the seventh session. Now, let us also ask what happens when we go the other way? This was one path. The other path was multiply the original wave form by very narrow pulses and make the pulses narrower and narrower, but in session 8, we tried to do this and we said if we just make the pulses narrower, they ultimately the pulses vanish and nothing remains. So, if you make the pulse narrower, you must compensate by increasing the height proportional. In other words, you must move towards impulses. So, there we are. What we said was the alternate path is multiply the original sinusoid by a train of very narrow pulses. So, remember these were going towards impulses which were uniformly spaced. We are talking about this situation. So, we discussed this in session 8 and what did we see in session 8? We saw that essentially this means that you are going to multiply the original sinusoid by the 4-year expansion. So, you multiply the original sinusoid by the 4-year expansion of this periodic wave form. The wave form is periodic with period T s. You can see that. We worked out the 4-year expansion of this periodic wave form in session 8 and we started taking limits and now what we shall do is to write the 4-year expansion directly recognizing that we are talking about impulses. So, let us now write the 4-year expansion directly. So, let us isolate one period around 0 and of course, you have this very narrow pulse which has gone to an impulse. Let us recognize it as an impulse and let us operate it like an impulse and I do agree that when writing the 4-year expansion, we are in some sense now writing a 4-year expansion involving generalized functions. They are not functions in the true sense, but let us do that because it does not really do any harm per se. So, how would we write the 4-year? Let us write this time for variety. Let us write the complex 4-year series. We know how to do that. That is easier to do also here. So, complex 4-year expansion or complex 4-year series. So, those should be of the form. Summation, let us say L going from minus to plus infinity, some gamma L times e raised to the power J 2 pi by Ts times LT and we need to find the coefficients gamma L and finding gamma L is very easy. All that we need to do is to recognize that gamma L is obtained. These are all orthogonal. So, you take the original wave form, take one period of the original wave form as we had it. Take this one period. Let us call it say Pt standing for pulses going towards impulses and let us write down gamma L in terms of Pt. That is easy. Multiply Pt by the complex conjugate of the orthogonal function and now remember Pt has only one impulse. This impulse simply lifts the value of the rest of the integrand and what is the value of the rest of the integrand? This is the rest of the integrand. So, its value is simply 1 and therefore, gamma L becomes 1 by Ts into 1. So, simple. And therefore, the complex Fourier expansion becomes, well summation gamma L, but you do not need to, all the gamma L's are the same. You do not need to write separate gamma L's. So, 1 by Ts as simple as that. 1 by Ts, summation L going from minus to plus infinity, e raised to the power j 2 pi by Ts times LT. So, simple. And now we can combine positive and negative L and rewrite this with L equal to 0 separately. That is 1 by Ts for L equal to 0 plus summation L going from 1 to infinity and write the positive and negative. And now this is a very simple expression that you have here. This is simply 2 times cos 2 pi by Ts times LT. Very interesting. That is exactly what we had. Do you notice now? That is exactly what we had when we brought all those imposters together with the original way. So, now we have a correspondence. We have taken two different paths and we have a conclusion. In this path, I notice that I am multiplying the original sinusoid with the train of uniform impulses. And if I bring all the imposters together, algebraically I am doing the same thing. So, it convinces me that essentially bringing all these imposters together, bringing all the sine waves with the same samples or the same points together is equivalent to multiplying the original sinusoid with a uniform train of impulses with impulses located at a sampling instance. It is very interesting. Let us look at this expression again carefully. So, let us mark it prominently. This is the same as what we got in session 7. Recognize it. And what conclusion we draw here? Let us write that down also. It is a very important conclusion. Bringing all these imposters' sinusoids together with the original one amounts to multiplying the original sinusoid with a uniform train of impulses located at the sampling instance. In fact, what it also says directly or indirectly is these imposter's sinusoids. What we conclude here is that these sinusoids have a constructive interference at the points of sampling and destructive elsewhere. An important conclusion. We have more to do with this in the next session. Thank you.