 So, warm welcome to this response video and I would like to begin by looking at the screen of discussion here and expressing my pleasure on a post that has just come, so if you notice this D. Kohat, D. K. O. H. U. T. 1 has asked why I have not responded for the last few weeks and I am very glad that this student, this participant has enjoyed the course and he liked the course and he has asked me what I think of the course. Well, I am actually very happy with the way participants are responding in this course and I am particularly happy that some of you have liked this and in fact inspired by your post I am responding now formally. There are several different issues that I want to talk about in this response. One is that in module 2 in general I did notice a few posts which brought out some of the technical difficulties that people were facing, but one of the reasons why I did not create a response video was that I did not see enough discussion on the technical issues. So, I am wondering what the reason was? Was it that the concepts were rather difficult? Was it that some ideas were not getting across? If so, we would appreciate it if you put up on the discussion forum the ideas that are difficult to grasp because I do appreciate this module is a little hard in comparison to the previous one. I mean this requires a considerable use of calculus again and again and again and in fact in a way this also requires appreciation of a generalization of geometry to multiple or why I mean to infinite dimensions. So, you have to now visualize some kind of geometry in an infinite dimensional space and then talk about perpendicularity of functions seen as vectors and this generalized sense. So, there must be some concept which posed a challenge, but we would appreciate it if you put up on the discussion forum the topics that you found tricky or difficult to appreciate and of course, you will notice that my teaching associates have been answering your queries with great enthusiasm and dedication and I am sure they will continue to do that. So, please do not restrain yourself from expressing your doubts, comments, queries on the discussion forum. I have also emphasized this in the email that I have sent all of you very recently. Now, I would like to take up a few of the questions that were raised on the forum. In fact, I would not go through the post, I will just quickly come to the questions themselves. So, one of the questions that seemed to confuse a lot of people was the whole idea of the frequency response or that integral which created the eigenvalue becoming independent of the variable of integration. Let me go through it in two or three different ways. So, let us again review what happens when you pass the rotating phasor, the complex exponential e raised to the power j omega t into a linear shift invariant system. Let the impulse response of the system be h of t. So, it is easy to see the output will be obtained by convolution. So, you have minus 2 plus infinity integrated h tau e raised to the power j omega t minus tau d tau and we can expand this. So, you would have an e raised to the power j omega t coming out clean and then we have minus 2 plus infinity h tau e raised to the power minus j omega tau d tau. Now, here this quantity that I am emphasizing here in red is an integral over tau. The moment we integrate over tau, the resultant becomes independent of tau that is universal for integration. The variable upon which you integrate in the integration is removed in the answer of the integral so to speak. So, here for example, you have integrated the product h of tau e raised to the power minus j omega tau with respect to tau and therefore, this integral over tau becomes independent of tau. Now, what that means is it is a quantity which depends only upon the other unknowns in the integral. So, here in this integral it is this omega which remains as an unknown and therefore, it is a function only of this unknown now. So, that is why we wrote it as capital H as a function of capital omega multiplied of course, by what is left e raised to the power j omega t that is carried from here. Now, what is important here see it does not matter what you call the variable of integration you can call it tau you can call it t. So, for example, if you were to take the Fourier transform of a signal let us write that down now you have a signal x t and we take its Fourier transform the Fourier transform is obtained by again by integrating x t you can call it if you like into e raised to the power minus j omega t dt. Now, this is the same as x tau multiplied by e raised to the power minus j omega tau d tau the variable of integration is inconsequential it does not matter what name you give that variable the integral becomes independent of that variable. Now, one important thing that I would like to emphasize I am going to preempt a little bit in the last week of this course you are going to deal with the same Fourier transform. So, you will see the material as you go along you will come to it and there I want to reemphasize this issue of the factor of 2 pi. So, at places you may have to make this correction please do. In fact, I wanted at this point to appreciate a couple of you I think one of them is Eugene who pointed out that there was a mistake in the constants at one point and of course, my teaching associates very quickly identified that mistake and had preemptively corrected that mistake in the notes. So, in fact that goes to the credit of my teaching associates, but I want to mention that I am very glad that some of you are looking carefully at the constants dotting your eyes and crossing your t's as they say working the problems out or working the solutions out with complete clarity and with every detail intact. And in that sense there is one detail to which I wish to draw your attention I might just have goofed up in a couple of places be vigilant on that count. So, I will just tell you what I am talking about you see when you invert the Fourier transform. So, if x t has the Fourier transform and here if x t has the Fourier transform capital X of omega and here we are talking about angular frequency omega capital omega. Then we can reconstruct x t from capital X of omega with the geometrical intuition that we employed. So, essentially we are saying multiply each component you know you can think of capital X of omega as a component along capital omega, multiply the component by the vector along that component so to speak and then put all such components together this integral essentially means put all such components together the only catch is that you need a factor of 2 pi when you put it in the angular frequency sense. And if I take capital omega equal to 2 pi times the cycles per second frequency here I can make a change in this integral. I can now write this down let me write on the next page x of t is 1 by 2 pi integral over all omega from minus to plus infinity. So, here the 2 pi can be removed by cancellation and now you can think of x of omega also as a function of f rather than of omega. So, what I am trying to emphasize is in an integral you could also make a change of the variable of integration not just by nomenclature but also by relationship. And in that case you need to worry about constants if it is such a straightforward change if there are of course there are more complicated changes and more things have to be done. And this might occur currently in many situations in the Fourier transform. So, beware of this issue of the factor of 2 pi at places you know in the discussion we might have omitted the 2 pi but that is just to bring the main point out and to focus on the main relationship. But when you are actually as they say dotting your eyes and crossing your t's you should be careful to put the 2 pi where it is needed and not put the 2 pi where it is not supposed to be there. That is just a point of warning I wanted to give you as you enter the last week of this module. Now a couple of other things one of the important things that you saw in this module was that you could look at signals and systems in an entirely different domain. But you also realize that this is true only for a limited class of signals all signals do not have a Fourier transform we have been giving hints to that here and there at times. In fact some useful signals like the unit step strictly speaking does not have a Fourier transform you can of course bring in a Fourier transform to unit step also by generalization using generalized functions and so on. In fact now I am going to tell you something interesting which I have not quite discussed elaborately in the videos. However one of my teaching associates has given you a question which will indicate this idea. Periodic signals strictly speaking do not have a Fourier transform in the sense of functions because periodic signals are not square integrable or absolutely integrable. There is a problem there you can of course see you notice we have done something very clever we have as far as periodic signals go we have confined ourselves to a period and we have thought of that periodic signal as a sum of discrete sinusoids over that period and then of course that has gone on to every period. So, the periodic signal is that decomposed into what we called a Fourier series. But then if a signal has finite energy you could think of this complex phase are running all over the time axis and then you get a Fourier transform. So, the Fourier transform of a periodic signal we have not probably talked about in explicit detail at all and now I would like to see a little bit about that. You see how would you get a Fourier series starting from some quantity as a function of all omega. For example, suppose I just show you how I get a few terms. Now, let me assume there is a periodic signal with period t and let us assume that it has only two harmonics the fundamental. So, on the omega axis this is the only frequencies present complex phase of frequencies present 2 pi by t minus 2 pi by t because these together will make a sinusoid of frequency angular frequency 2 pi by t and then twice that. So, let us assume there is expansion looks like this it is a 0 plus a 1 plus a 2 and so on. Now, of course, you can break this into its exponential components. Let me do that. Now, the question is what should I put on this omega axis. So, that when I calculate the inverse Fourier transform I get this expression here. I mean let us phrase the question what should I put on the omega axis and the answer is very simple. What are you going to do when you invert? You are going to calculate integral minus infinity to plus infinity whatever something as a function of omega let us call it capital P of omega here this is the so quote unquote Fourier transform of that periodic signal e raise the power j omega t d omega and then divide by 2 pi. And as you notice when you go back to the expression that we have specific values of omega coming out look at this here and this here and these and in fact here also you have omega equal to 0. So, what it really means is we need to put impulses at omega equal to 0 omega equal to plus minus 2 pi by t and omega equal to plus minus 2 into 2 pi by t. And the strengths of those impulses should push out those coefficients you see what I mean the strengths of the impulses should push out these coefficients this one and this one and then this one and then this one. In fact now remember this is not the only part of the coefficient this together with this that is the coefficient here becomes a 1 by 2 e raise the power j phi 1 no that is the coefficient and the same is true of the others here it is a 1 by 2 e raise the power minus j phi 1 and here it is a 2 by 2 e raise the power j phi 2 and similarly here those are the coefficients that come out. So, when you give strengths to the impulses you must put these numbers with impulses, but remember there could be a factor of 2 pi. So, now my exercise for you which I am going to give you exercise put down appropriate strengths for these impulses to get the required expansion. So, essentially the Fourier transform of a periodic signal has impulses located at all multiples of the fundamental and of course also at 0 that is also multiple after all this should be useful also in a subsequent module in the next course. So, it is worth knowing about this so do think more about it. Once again let me conclude this response period by urging you again to ask questions to post your doubts and your queries on the forum. My teaching associates are more than eagerly waiting to answer them and we do hope that you will participate with as much enthusiasm as my teaching associates are putting into answering your questions. Thank you so much.