 A warm welcome to the 21st session of the 2nd module in the core signals and systems. We were talking about the Dirichlet conditions in the last session, essentially conditions that forbid Fourier decomposition that means if the Dirichlet conditions are violated that periodic waveform cannot be decomposed into a Fourier series. I would like just to mention informally these conditions, I am not going too much into formal proofs or formal statements, I just want to mention the situations in which Fourier decomposition is forbidden, some situations in which Fourier decomposition is forbidden. Now I had pointed to one situation the last time, when you have an infinite number of maxima and minima in the period, there is one more situation which forbids Fourier decomposition which I would now like to explain. So let us take one more forbidden situation, the Fourier decomposition, I would best explain it with an example, let us visualize a waveform x of t as follows, it is periodic with period capital T and let us look at it in the period from v0 to t. Let us divide this period into 2, so you have t by 2 here, let us assume that the waveform is to reach its maximum value of 1 ultimately, so the waveform is going to rise, let us say from a very small value here to an ultimate value, an asymptotic value, let us say this value is 0.1, the asymptotic value is 1 at t. Now at t by 2, it overcomes half the gap, so 1 minus 0.1 that is 0.9 divide by 2, so 0.45, so it rises by 0.45 and then this continues until you reach one half of the remaining end. Now again you divide this interval into 2, you get the middle of this interval and once again you cause the waveform to rise by now, so it has come here to 1 minus 0.1 is 0.9 divide by 2 that is 0.45, so 0.45 plus 0.1 is 0.55, so now again it would rise here by 1 minus 0.55 by 2 and this continues until this point if you have identified the middle. And now you can see how I am constructing this phase point, I am taking the midpoint of the remaining interval, I am causing the waveform to rise by one half of what is left up to the asymptotic value of 1 and keeping that persistent up to that midpoint. So each time I am creating a discontinuity at the midpoint and that discontinuity takes you only half the way to the asymptotic value. So how many discontinuities are we going to thus create in the interval from 0 to t and infinite number of them because whatever interval is left, I am going to put a discontinuity in the middle of that interval no matter how small. And that discontinuity is going to take you only half the way to the asymptotic point and therefore it is like steps that rise infinitely slowly but asymptotically towards the asymptotic value. The problem here is an infinite number of discontinuities, let me make that very clear. This has an infinite number of discontinuities in the period and this is also not amenable to Fourier handling. So you know now there are two situations which you see, there are also others as I said I am not formally discussing the traditional condition, I am just making you aware that there are some peculiar signals which you can think of conceptually or theoretically which do not have a Fourier series decomposition. But of course we would seldom encounter such signals in practice. So we are quite on safe grounds to say that most practical periodic signals that we deal with are amenable to Fourier decomposition and we shall use that principle in most practical situations. Now I must say a little bit about Dirichlet actually was again a mathematician and he was trying to explain where there were problems with the Fourier series decomposition. In fact there is something peculiar about a Fourier series at a point of discontinuity. Before I start explaining what it is I would like you to discover what it is. So I am going to give you an exercise. I recommend you work that exercise out, see something for yourself and then we will discuss it. So let me give you a periodic wave. I give you a square wave which occupies only half the period which is 1 between 0 and t by 2 and 0 between t by 2 and t. This is x of t in the interval 0 to t and x of t equal to x of t plus capital T for all t. This is the periodic waveform that you are going to deal with. So a decompose this into its Fourier components. I would recommend you use what is called the sinusoidal decomposition. So construct it of the form k equal to 1 to infinity ak cos 2 pi by tk t plus pi k sum from 1 to infinity plus a0. Part p, finally plot the partial sum. For example, you could take n equal to let us say 2 first then 4, then you could take 10, you could take 12, then you could take 15, perhaps you can go on like that. Plot for a fine grid. So for example, you could take t equal to let us say you know you could take 2048 points you know t divided by 2048 multiplied by n for n going from 0 to 2047. So you will have 2048 points in that fine interval. So essentially we are trying to take as many points as we can. I mean 2048 is just an indicative number but you could as well take large number of points and plot this partial sum. So you know you are plotting it finely because you want to get a feel for the continuous function and you are plotting it as a function of how many terms of the Fourier series you have included. And observe what happens as you increase n. Do you go closer and closer to the periodic waveform that you aspire to reach? Are there problems at some points? Especially look at the point of discontinuity, let me mark that down for you in the exercise. So specially look at the point of discontinuity, look at and around the point of discontinuity and which is that point of discontinuity t equal to capital T by 2. What happens around that time? Do you see something peculiar? So I am going to leave this for you to do, let us give you some time to do it and then we will probably talk about it in one of the subsequent discussions. But you know this is what you are going to see when you do this exercise is what you might call a problem with the Fourier series, a problem in the sense that at the points of discontinuity the Fourier series has some issues which need to be dealt with carefully and that is even for very reasonable waveforms like this square wave. Let us go back to the Fourier series of the output that we had and we will make one observation about it. So if you remember the Fourier series decomposition of the output in the RC circuit look like this. Summation T going from minus to plus infinity Ck 1 plus j and you notice that these where the Fourier series coefficients of the output. Now let us compare Ck and C minus k. So for k we will have Ck 1 plus j 2 pi by t times k Cr and for minus k we will have C minus k. Now it is very easy to see that the complex conjugate of this is equal to this that is because denominators are complex conjugates and C minus k is equal to Ck complex conjugate. So that complex conjugate property of the coefficients provided you are dealing with the real input x t is preserved at the output. I want to emphasize this because we have started with a complex exponential decomposition and if you want to convert it into a sinusoidal decomposition we can. We will see more about Fourier analysis in the next session. Thank you.