 A warm welcome to the 27th session in the second module of the course Signals and Systems. In this module, we will continue from what we did in the previous two modules where we had worked very hard to generalize the Fourier series decomposition to the context where the signals are not periodic. The way we looked at it was that if the signal is not periodic, it amounts to saying the signal is periodic with a period of infinity. In other words, we make the fundamental frequency go towards 0, smaller and smaller. I mean, you know, it is not quite meaningful to think of the fundamental frequency as being equal to 0, that is exactly what the period going to infinity would imply. But what we must try and visualize is that we bring the points on the frequency axis closer and closer and closer together until we get the whole continuum. So, we get the whole continuum of the frequency axis which we have to populate with appropriate values. And those values are really obtained by using the ideas of vectors generalized to the context of functions. So, let me review a few key ideas which we have built in the last two sessions from which we will go ahead in this session to take an example and to build a few properties. So, we said the Fourier transform of the function h t is integral from minus to plus infinity h t raised to the power minus j omega t with respect to t and this is denoted by capital h of omega. So, we use capital letters to denote the corresponding Fourier transform and this is essentially an inner product of h t and the rotating complex phase a e raised to the power j omega t. This makes a lot of sense. What we are saying is that the inner product or dot product of the function with a rotating complex number rotating with an angular velocity of omega, now remember omega could take positive and negative real values and positive and negative real values really correspond to anticlockwise and clockwise rotation respectively. So, there is nothing very special about positive or negative values they are equally real or unreal as you might think. So, this idea of projection you can think of it as projecting the function onto a rotating phasor rotating with an angular velocity of omega. And what we worked hard in the last session to show was that if I took two such phasors rotating with angular velocities of omega 1 and omega 2, these two phasors are orthogonal in the limit and that orthogonality was in a slightly more generalized sense. What we showed in the previous session was that if I took the inner product of these two rotating phasors e raised to the power j omega 1 t and e raised to the power j omega 2 t, it would look something like this. In fact, this would essentially be an impulse or you could write it as delta omega 1 minus omega 2 times some constant let us call it kappa 1, kappa 1 a constant. Let us not bother too much about the constant. But the idea is that you know you have an impulse. Now you know you have to take recourse to generalized functions here because if you look at this integral carefully, this integral is divergent in the conventional form. You know if you look at this integral diverges in the sense of conventional function and that is why an impulse is required. Now what it means in loose language is that if I take two rotating phasors rotating with different angular velocities, even if that angular velocity is slightly different, they are in a certain notion orthogonal, perpendicular. Of course, I told you this is very loose language, not very precise. What one should say to be precise is that when I take the inner product of one rotating phasor with another, it really goes towards a generalized function located at omega 1 equal to omega 2 in the omega 1 omega 2 plane. But if one wants to be informal and explain this idea informally, one can say with some correctness that when we take the Fourier transform, we are projecting this function ht onto a set of orthogonal vectors, a mutually perpendicular. And once we recognize that and once we also accept here, I have not proved, but we will just accept that this constant k 1 which I wrote here, you know in this discussion here, this constant k 1 here is independent of the particular values of omega 1 and omega 2. So, with that background, we can now go on to reconstruct, reconstruct ht from h omega essentially using what we call the inverse Fourier transform. And the principle in the inverse Fourier transform, multiply each component by the corresponding orthogonal vector and integrate over all such vectors. And we express this mathematically as, so you notice that this is the component, this is the so called orthogonal vector and this is the combination or the integral. And this constant we recognized was to normalize the vector to make it a unit vector, makes a lot of sense. Multiply the component by the unit vector in the direction of that component, integrate over all such components and you hope to get back the original function ht. Now, there is a subtle point here. We have decomposed along several components for omega going from minus infinity to plus infinity. Do these components contain all that there is in the original ht? That is a subtle question. And frankly, we do not have the wherewithal in this course to answer that. So, those of you who are interested in answering this question should take up a book on Fourier analysis. Perhaps refer to text on functional analysis or real analysis or transform domain analysis as written by mathematicians. Here in this course, we are not too worried about that issue. We will just take it that this set is complete in the sense that if we take all these components for omega going from minus to plus infinity, it contains all that we need to reconstruct. We have not lost anything in the process. So, this decomposition and reconstruction is invertible. Now, you know to be very precise, it is not true for all functions ht. In fact, we had talked about certain conditions for which a Fourier series decomposition exists, namely the Dirichlet conditions. So, let us review the Dirichlet conditions. Let us review them in a few words. A finite number of maxima and minima in a finite interval. A finite number of discontinuities in a finite interval and bounded variable. These are the three things we need. In a finite interval, there must be only a finite number of maxima and minima, not like sine of 1 by x, if you remember that example. In a finite interval, there must be only a finite number of discontinuities, not like that step function which steps after every half of the remaining interval and steps by only half of what remains. That was an example. You could construct many others like that. But such functions are not amenable to Fourier analysis. And finally, bounded variation. You do not want the function to go to infinity loosely, indeed, at any point here. Now, of course, you know, there are generalizations or there are what we call boundary cases or boundary ways to understand the Fourier transform even when this bounded variation condition is not obeyed. We have the periphery of the Fourier transform if you might understand. But for the moment, let us understand these Dirichlet conditions as being adequate to ensure that there is the possibility of a Fourier transform. So, the moment you see Dirichlet conditions being violated, you can sense there is trouble. Fourier transform might not exist. So, it is not as if the Fourier transform is guaranteed for all functions. There is a class of functions for which you have a Fourier transform when you can invert the Fourier transform. In fact, there is also some amount of ambiguity in inversion, particularly at the points of discontinuity. We shall take that up in a discussion. But for the moment, let us assume everything is hunky dory. In particular, if you have no discontinuities at all, things are very good. Point wise, if the Dirichlet conditions are satisfied, there are no discontinuities. Point wise, you can reconstruct the function from its Fourier transform. There is no trouble. Now, there are also other issues here. For example, you take the simplest of the functions, a sine wave. It obeys all these. Now, it has a Fourier transform, but not a Fourier transform in the conventional sense of function. So, the Fourier transform of a sine wave is not a conventional function. It is a generalized function. So, the Dirichlet conditions are necessary, but they do not guarantee a Fourier transform in the sense of conventional functions. The Fourier transform might require generalized functions to come in. We shall now illustrate these ideas with a few examples in the next session. Thank you.