 Welcome to the 34th session in the first module of the course signals and systems. In the previous session, we had looked at the operation of convolution. We just about gave it a name at the end of the session. And now we will use that name again and again. We would not just keep saying operation, operation between input and impulse response. The operation is convolution. We are going to convolve the input and the impulse response to obtain the output. And we are going to do it in this session by using what we call the train platform analogy. Let me now build that train platform analogy of convolution. So, let us take the same sequence that we were dealing with. Xn is respectively minus 195p at minus 1012 and hn is 1 minus 12 at 012. And now we put down the platform and the train. So, we have the platform. The platform has markers indexed by k. So, k equal. Let us put only a few of them. Let us put that I said starting from minus 4. You know, you must have seen railway platform. They have markers. And now we put the train. The train is hn minus k. And people on the platform, which is x of k. Now this train moves. That is the whole beauty. This is the location of the engine. And the train moves with n or as per n. So, now we can take specific locations. We could take the location. Now you know, you can now look at this carefully and you can find out from which location we need to begin. We do not need to take all the infinite locations for n. Let us look at the situation. So clearly, you see at n equal to minus 2, there is nothing to be done. Because when n comes here, let me mark it for you. When n is here, in fact, when n is before this, there is no handshake at all. You see, you have to visualize a situation where these are people inside the bogies. So, see these are essentially bogies with people inside. So, we call these people inside the bogies or coaches. So, at the 0th bogie or you know, you could call it the nth point, you have the person sitting, number 1. At bogie number n minus 1, you have minus 1. Of course, when I say n minus 1, I am talking about the position of the bogie right now. But actually, it is bogie number 1. And bogie number 2, I have the person 2 sitting inside. So, now, you know, you can think of it like this. At multiplication, x of k times h of n minus k is like a handshake between the person on the platform and the person in the corresponding bogie. The engine moves one step at a time. So, you can visualize just one person sitting in each coach, which essentially is the value of the sequence at that point. And the person sitting inside the train shakes hands with the person sitting outside on the platform. At the same corresponding location. And you have handshakes all over the train. And the combined effect of the handshake is what you record as the value of the convolution at that point n. Let us write this down explicitly. So, what we are saying essentially is this expression y of n is equal to summation k going from minus infinity to plus infinity x k h n minus k can be interpreted as follows. The person on the platform at location k shakes hands. The person in the bogie or coach at same location. And y of n is the combined effect of all these handshakes. So, now if you went back to the particular sequence that we have, you notice that till the engine reaches minus 1, there is nothing to be very recorded as a combined handshake. Let us look at that. We will see that if we look at the platform and the train here. So, when n is here, up to this point in n, for n less than equal to minus 2, no combined effect. Now, when n becomes minus 1 that is here, the combined effect begins. And what is the combined effect? When n reaches minus 1, it is essentially minus 1 multiplied by 1. So, now let us start finding the combined effect step by step. Let us write down the train and the platform once again and write down the combined effect everywhere. So, we write down the platform once again and the people on the platform and we write down the train on the side. And we now write down y of n. So, as you see it begins only with n equal to 0. So, I will write in a different color. So, n equal to minus 1. Now, you need to take this with this 1 into minus 1. So, you get minus 1. Now, n equal to 0. When n equal to 0, 1 is multiplied by 9, 9 and minus 1 is multiplied by minus 1 that is 1. So, you get 10, n equal to 1. 1 is multiplied by 5 that is 5 minus 1 by 9. So, minus 9, 5 and minus 9, minus 4 and 2 into minus 1 that is minus 2. So, minus 9 plus 5 that is minus 4 and minus 2 minus 6. Now, n equal to 2 you have 1 getting multiplied by 3 that is 3 plus minus 1 getting multiplied by 5. So, 3 plus minus 5 that is minus 2 and finally, 2 getting multiplied by 9, 18. So, 18 minus 2 that makes 16 for you and then n equal to 3, n equal to 3 gives you 1 multiplied by well at 3 there is nothing here. So, 1 does not get multiplied well it gets multiplied by 0. So, there is nothing left there. So, minus 1 gets multiplied by 3 that is minus 3 for you and 2 gets multiplied by 5. So, 10 minus 3 that is 7 and finally, at n equal to 4 you have the last point the last person in the bogey that is 2 gets multiplied by 3 and that gives you 6 and again y of n is equal to 0 for all n greater than equal to 5 and of course, y of n is equal to 0 for all n less than equal to minus 2. It is only in the region between minus 1 and 4 that you have an output. You see it is only in that region that there can be a handshake. Otherwise, for each person inside the train there is nobody outside on the platform and for each person on the platform there is nobody inside the train beyond 4 here and before minus 2 there. I hope all of you understood the train platform analogy. I recommend that you practice it for a few examples. Let me give you an exercise to enable you to practice let me put down exercise practice exercise. Use the train platform analogy and convolve. So, I have given you this exercise here. Now, you know notice I have made the exercise a little more troublesome. Here I have made a slightly longer sequence of people sitting on the platform and the train also has samples on you know both the train and the platform have samples on both sides of 0. Not very difficult, but you will understand the train platform analogy when you work out this exercise properly. Now, when doing this exercise I recommend that you also answer the following question. We can now going to put down as the second slightly probing question. Let me put it down. So, question to probe, probe your understanding. Again use the train platform analogy xn is known to be non-zero only between n equal to n1 and n equal to n2 where minus infinity is strictly less than n1 which is less than equal to n2 and n2 is strictly less than plus infinity. Similarly, hn between n3 and n4 in what region is yn non-zero? I am asking you in other word if I have two finite length sequences and I know the extents in which these sequences lie. The leftmost point and the rightmost point of each sequence. Can I say something about the leftmost point and the rightmost point of the convolution of these sequences? What can I say is what I want you to think probe and answer. We shall see more about convolution in the next session. Thank you.