 A warm welcome to this discussion where as the instructor I would like to respond to some of the discussion that has been taking place on the forum which I am very happy to see actually. I am first very keen to convey to all of you that having this discussion on the forum is really the purpose of a course like this because it allows you to learn things from one another in addition to what you learn from all of us I mean me as the instructor and all of the teaching associates that are associated with this course. I would actually urge a little more response also from the persons other than those making the first post. So, you will notice that my teaching associates are answering your questions quite keenly but at the same time they are waiting for a while because they would like to hear responses from many of you before coming to a point where they respond on their own. So, you know it is very important I mean I do not feel comfortable when I see just one person having posted a discussion item and no one else from among the students has responded. It is alright you know the teaching associates respond at times but we have responded wherever we think it is immediately important but at times we have been waiting to actually encourage discussion amongst all of you before we respond immediately. So, please in future it is very important that when one person makes a post we would be very happy if other people also responded to that post and since there are thousands of people really involved in this whole course do not feel awkward about responding it does not harm if you say something which is incorrect because if you write something and somebody else responds to it both of you learn in the process the discussion is after all not an evaluation at all. So, it is very important that you participate this is the first message that I wanted to convey to you after looking at the discussion forum. Now, let me go through some of the questions of the discussion forum. So, you will notice that one of the questions that has come repeatedly in this discussion forum is this whole idea of the shifting property or you know the idea of selection that is taking place in the. So, let me now draw essentially the unit impulse located at a point T naught again and there was you know there was a very interesting discussion which has now been pinned up about this whole business of the unit impulse and so on. So, you have this very narrow pulse with the width of capital delta and a height of 1 by delta centered at T 0 on the time axis and you have a continuous function which I am sketching in red and calling x of t. So, this is the unit impulse well it is not really unit impulse it is a unit pulse I should call it and you can see that if I call this unit pulse well delta sub capital delta of t well since it is shifted so you know delta sub capital delta of t would look like this it would be the same thing, but located at 0 and therefore, it goes from minus capital delta by 2 to capital delta by 2. Let us first sketch small delta sub capital delta t which is a unit pulse around 0 going from minus capital delta by 2 to plus capital delta by 2 and with a height of 1 by delta. Why is it called a unit pulse unit because the unit area is involved here there is a unit area 1 by delta into delta which is 1. Now, what we have here is the unit pulse at t 0 so to speak and that is written as delta capital delta t minus t 0. Now, you will agree with me that the way I have drawn delta capital delta t here delta capital delta t is equal to delta capital delta minus t and therefore, you could switch and write delta sub capital delta t 0 minus t here both of them have an interpretation. Now, what you also understand and this is what a lot of people I think have been discussing on the forum is that if I multiply if I fix the t 0 with fixed t 0 the product x t times delta sub capital delta t minus t 0 dt integrated over all t essentially gives you this you know the value here you are extracting this part of the function so to speak and you are multiplying it by 1 by delta and so on. So, you know you are essentially looking at this area multiplied by 1 by delta but remember you know when this becomes smaller and smaller this area also becomes smaller and smaller there is a confusion about that this green area would tend to 0 if you did not have this height scaling up to 1 by delta. So, ultimately what is going to happen is that essentially when you would make this capital delta go to 0 you would have an area of 1 by delta times the value of x at the point t 0 times delta. So, you know delta goes to 0 but 1 by delta tends to infinity. So, it lifts the value of x at the point t 0 and this is meaningful if x is continuous at the point t 0 if it is not continuous then one cannot apply this principle at these isolated points of discontinuity that should be kept in mind. Now, all this is alright I think you know and in fact you also understood that you could switch. So, you could write delta sub capital delta t minus I mean t 0 minus t and then you know you also had a situation where you could interpret the integral the other way. So, you could write x t delta t 0 minus t dt integrated over all t is essentially x at t 0 and here what I was emphasizing is that there is an important different interpretation that you need to give it that is that now this holds for all t 0 where x is continuous. Now, how do you visualize this you know you should visualize it like this think of an impulse a very narrow pulse with an capturing an area of 1 located at t 0 and visualize t 0 moving. So, what you are saying is every time you take a particular position of this impulse multiply the function x and integrate out you pick up a particular value of x and as you keep doing this for different t 0 you can reconstruct x at every point while both these equations are really the same equation per say when you write it like this when you write it in this form the second interpretation becomes important namely that you can actually move t 0 around the reason why I have written it like this is that later we are going to actually look at this expression you know here you will recognize this as a convolution between x t and delta t notionally you know it is a convolution between x t and delta t you will understand this later even if you do not understand now and that is very important you know because that will also lead us to a very basic principle that governs linear shift invariant systems. So, you know it is the same equation written in different ways can help you do different things now that is all fine I just clarified what all this means, but you know I do realize that understanding this in continuous time is a little more difficult than understanding it in discrete time. So, what I would like to do before I you know conclude this discussion session is to actually talk about the discrete time version of this which will make it much easier for you to understand. So, let us now take the discrete version of this. So, firstly what do you mean by a discrete sequence or a discrete signal a discrete signal I am kind of anticipating I am going a little ahead perhaps a discrete signal is a mapping from the integers to the complex numbers. And we denote it by writing script z mapped to script c. So, this is you know you can look at the analog you know just as a continuous signal maps the real access to the set of complex numbers. So, in a discrete signal we often use the independent variable n and we talk about x of n square bracket you know the square bracket is often used to denote the discrete nature of the signal and we call it a sequence. Now, let me take a very simple sequence let me take a sequence x of n which is 3 respectively 3, 4 minus 2 and 6 at the points n equal to 1, n equal to 0, n equal to minus 1 and n equal to minus 2. Now, let us define a very simple sequence which we call the impulse sequence. Now, this is very easy to define this does not require any peculiarities like infinite values and so on. The impulse sequence is denoted by delta n and that is equal to 1 for n equal to 0 and 0 else very simple. So, you know in discrete time it is very easy to define the impulse sequence. Now, you can very easily see that x of n can be written as 3 times delta n minus 1 plus 4 times delta n minus 2 times delta n plus 1 plus 6 delta n plus 2. Now, like in this or bring out the analogy see the similarity with integral minus infinity to plus infinity you can take k going from minus 2 to plus 1. So, you know you could in fact put down that here you know you could write k equal to 1 here k equal to 0 here and so on and you could say that x of n is a summation on x of k times delta n minus k summed over the integers k that is what you are saying here. Now, this will put things in perspective it is easier I think to understand this by starting with the discrete time interpretation which is very clean and neat and then moving to continuous time. Now, I would encourage you I am not giving the full explanation I have just sort of triggered a discussion from this discussion of mine or this response of mine and I would like some of you to talk about this actually I have written this answer also partly on one of the discussions, but I would like many of you to take this discussion further and try and link the two ideas together bring out their commonalities and differences. There is a lot of other discussion that is taken place one observation that I would like to make is that you know you need to also develop the skill of deciding what kind of tools to understand what concept sometimes some things are better understood graphically some things are better understood by writing down the algebra. One must learn to use both tools both strategies depending on the context. We will talk more as we go along, but I would once again encourage you to participate actively in the discussion. The other message that I had for all of you is please do not get scared from slightly difficult questions some of the quizzes. I have also conveyed this to all of you on the email we are trying to make some of the questions challenging because we want to challenge you we want to challenge you to think deeply, but we do not want to make the course so difficult that you cannot grasp it. So, please give us your feedback I have asked for feedback my teaching associates are keenly listening to your feedback and we will moderate and modulate the questions as we see your feedback our aim is to give you as good the material that we can give as we can, but at the same time we do not want to make it too difficult to grasp. So, you know we do need to challenge you a little bit and of course, you know you can always I think you have a chance of dropping some of the quizzes and still being considered for you know for the total and so on. Getting an A grade of course should be difficult, but we think those who are diligent with the course those are put in the required effort should have no difficulty in completing the course successfully that is what we feel and we will definitely moderate the remaining quizzes in accordance with that and we would all the time want you to engage with us to make this happen well we will meet again wish you all the best again. Thank you.