 A warm welcome to this session of response where we respond to your questions and comments. Well, you must have noticed that I sent all of you an email a day or so ago and one of the things that I emphasized in this email was that you are nearing the end of the first half of the first course, so the first module, so to speak, where you were essentially looking at systems in their natural domain. By natural domain, we mean the domain in which the systems are described in which they occur. So, normally one could talk of time domain systems, it could be space domain in the case of two dimensional images or a combined space and time domain where you have two dimensions of space and one dimension of time, if it is a video sequence, natural domain, the domain in which the signal naturally occurs. Of course, we have been confining ourselves to one-dimensional systems, though some of you have given examples of multi-dimensional systems on occasion. Now, a few things, of course, we have had a summary of it in the very last recording of this first module, we have a summary of what we have done, but I still wanted to emphasize a few things. The first thing I want to emphasize to all of you is the operation of convolution. Convolution, in fact, as the names give the word convolved or convoluted, actually the word convoluted in common parlance, in common speech, means something that is highly twisted and mixed. So, when we say a convoluted argument, as I think I have also said somewhere else, we actually mean an argument which is very difficult to understand, highly twisted, too many ramifications, too many clauses and bylaws. So, convolution, the name itself in some sense could be termed unfortunate because it appears to be complicated. Actually, it is not so complicated at all. Convolution is a consequence of the fact that you have a contribution at every point in time by every point of the input. That is what convolution in some sense says. Let us look at it carefully. So, let us look at convolution. You have studied convolution in great depth in the last few sessions. Let us look at the continuous independent variable convolution first, which says that if I convolve x and h to get y, y at t is an integral. It is an integral of x tau multiplied by h t minus tau integrated over all tau and this integral runs over all tau, of course. Now, what are we really saying? We are saying the following. Let us visualize the situation properly. We are saying first look at, now you know there are two axes here. There is an axis t and there is an axis tau. Let us take the axis t. We are trying to calculate the output at a particular t. So, let us mark this t. Let us put it equal to t 0. So, in particular if I have y at t 0, I will make those corrections here in red. So, y at t 0 is everything else very similar, except that you have h of t 0 minus tau. Now, let us take any particular tau. So, what are we saying? For example, for if tau has this value, we are saying that x took a certain value there, which is x of tau. And because there was an impulse located at this tau, the impulse response which emanated from this particular tau contributes to every t, in particular to t equal to t 0. So, you know you had h of t 0 minus tau. So, you must sort of understand this correctly. Let me draw it in a separate page. At every point tau, I will take a few points. Let us call them tau 1, tau 2, tau 3. There is a particular value of x. So, I will mark these values. This is x of tau 1, I am marking them in green, x of tau 2 and x of tau 3. Now, you already understood the shifting property or the idea of constructing an x from many impulses. What you are saying in effect is that at tau 1, there is an impulse with a strength of x of tau 1, similarly at tau 2, similarly at tau 3. And each of these impulse response with an impulse response. So, for example, let me assume that my impulse response has disappearance. This is what h of t looks like. Now, I am going to put red impulse responses at every one of these tau's, appropriately scaled by the strength of x at that point. So, let us say this is 1 here and so on. So, at every one of them, there is an impulse response appropriately shifted and scaled by the strength of x. Now, let us take just these three. So, I will extend these. So, what I will do is let me just keep drawing it in continuity like this. And you can visualize that each of these red responses here would keep going down. See, it would continue as they are here. And now, take any particular point here. Let me mark it here. Let us take t equal to t 0 here. You will see there is a contribution from each of these. For each of these tau's, there is a contribution to that particular point and that is what that integral is capturing. That is what this integral seeks to capture. You are saying that to calculate the output at a particular t, you need to see the contribution of every point of the input. And the contribution of a given point of the input is weighted by the extent to which the impulse response reaches the current point from that point. You know, the impulse response changes. If there were an impulse at a particular point, it would cause a consequence in principle all over time. And depending on where is that point t equal to t 0 is located with respect to the impulse which is causing that part of the effect, you need to put a weight. How much is that weight? X of tau multiplied by H of t minus tau. The same interpretation. In fact, I would encourage you to give the same interpretation in discrete time as in fact, you might well guess. In discrete time, it might be easier for you to write the interpretation. So, I strongly encourage you on the discussion forum. You should put down a similar interpretation for discrete time. Now, this is a point wise interpretation of convolution. The other is an operational introduction or an operational interpretation which we have given. I have talked about the train and platform. So, you have, you know, the train moving against the platform. And in fact, that is the best way to understand the operational behavior of convolution. So, convolution is not complicated at all. Convolution is not convoluted at all. It is very simple. In fact, the best part of it is you will see when you go to module 2 that you can actually make this whole business of convolution much easier if you go into an alternate domain. In fact, in module 4, you will see you can do it very generally. But at least in module 2, we will be able to do it for a wide class of signals. So, now you should in that sense look ahead to module 2 and you should be very keen to see how a lot of things can actually become much more easy and much easier to appreciate, to relate to real life when you look at signals in an alternate domain. You are going to think of signals as a combination of sinusoids. And actually that is what we often want to do when you want to describe systems in terms of their behavior. So, do keep your attention as alert as it is now for module 1. In module 2, module 2 is going to bring a lot of new material to you, take you to a different domain of representation. And you must be able to correlate what you have learned in module 1 in the so-called natural domain to what you are learning in module 2 in the so-called frequency domain. And you must also understand the context in which the operations in module 2 would be applicable. They are not applicable to all signals, but a very wide class of signals. In fact, most practical signals. Good. Now, a few remarks about, so the first thing I want to draw your attention to is that, you know, you will notice that on the discussion forum I have made an invitation to all of you. This time I have put a post which is supposed to excited discussion. I have asked you to identify 32 types of systems. And I have also explained to you how to number these types. So, you know, you have these five properties, additivity, homogeneity, shift invariance, causality, stability. And these are all independent properties. None of them depend on the other. You can have any combination of them, any subset of them, including the null subset. And in fact, if you have the null subset, that means none of the properties were true, then you would call it type 0. So, for each property, you can assign 0 if the property is not obeyed and 1 if it is obeyed. And therefore, you could get all the numbers starting from 0 to 31 represented in binary. So, that means each type of system has a unique number here. And I am very happy to see that one gentleman, I assume, has already answered with one example of a very simple type 31 system, which is an identity system. And he is quite clearly and correctly shown that it satisfies all the five properties. He is also correctly noted the system is memory less. Now, how do I pronounce that? Texan Jimbo. So, Texan Jimbo also like to now introduce an example of the same type of system, maybe which has all these five properties, type 31. But where there is memory, that is not too difficult to do. So, I encourage Texan Jimbo to put up that kind of an example. In fact, I am looking forward to many of you putting up examples of many other types. There are 32. So, at least there should be 32 responses, if not more to this and that should be many more. In fact, very interesting to see your responses. I do notice that several of you have found some questions in the current quiz a little difficult. Do not get too flustered, as I said. Our purpose is to challenge you at times. And if you put a little bit of thought and a little bit of time into the questions, my teaching associates are quite convinced that you should be able to do them without too much of a difficulty. And I strongly encourage that you do that because then it will really help you go deep into the ideas that you are learning in the respective weeks. So, do not get frightened if the quizzes are difficult. I have already told you that our aim is not to make life difficult for you, but to make the whole study of system very enjoyable and to allow you to probe very deep to see the depth and the beauty of the field. Now, a few remarks about, you see, I did not notice that some people had some difficulty in understanding stability. And I believe that, Sushrutan, I have recorded a discussion where you try to explain to you how you get that particular input which shows you the necessity of that condition of stability, the absolute summability or absolute integrity. I know that is the difficult part. Showing the sufficiency is not so difficult, but showing the necessity is a little difficult and you have to choose a strategic input there. You explain how that strategic input was arrived at. And then, of course, I have seen a lot of other questions which you have asked and I am very happy to see my teaching associates have answered them very enthusiastically. Please keep posting your questions. There are some questions which I would like to personally answer. Many of the questions have been answered by my teaching associates and rest assured that I am looking at every one of your posts, even though it is not physically possible for me to answer each one of them. I am very happy to see the posts that you have, whether they are observations on the quizzes or their doubts that you are asking, whether they are additions or thoughts on what you have learned, they are all very valuable. And we think discussion is the best process in this whole course in terms of learning. So, please do continue. Now, remember we are going into module 2 and you must pay careful attention to how a different domain of representation makes life easier in many ways. Thank you.