 So, warm welcome to the 18th session of the second module in the course signals and systems. So, we have been decomposing periodic waveforms into their components periodic sinusoidal components. We would like now look at a variation of the same decomposition this time using not sinusoids, but complex exponentials not a very difficult thing because if you have decomposed into sinusoids you could as well change that into a decomposition in terms of complex exponential functions. However, we might want to a priori decompose into complex exponential components and that is what we would like to see today. So, let us assume that we wish to decompose xt periodic with period t, which means that x of t plus capital T is equal to xt for all t. I am repeatedly reminding you of the meaning of this symbol after a while I will stop doing that. So, you have to understand that this symbol means for all anyway what we wish to do is to write xt in the following form, we wish to write xt is summation k going from minus to plus infinity C k e raised to the power j 2 pi by t k t. So, essentially complex exponentials rotating at angular frequencies of 2 pi by t times k k over all the integers. Now, what we are really saying here is that instead of thinking of xt as a combination of sinusoids as we have been doing all this while, think of it as a linear combination of all the possible rotating complex exponentials that you can have which can contribute to this waveform and which are those are all the rotating complex exponentials which have periods equal to t. Of course, when you say periods equal to t, the period can also be a sum multiple of t. So, if the period is t by 2, the period is also t. I mean if something is periodic with period t by 2, it is also periodic with period t or if it is periodic with period t by 3, it is also periodic with period t. So, a sum multiple is also acceptable this should be understood. So, all harmonics so to speak. Moreover, we need to take both the anticlockwise rotating complex number and the clockwise rotating complex number. In fact, if you look at it, when we talked about the sinusoid at every frequency at every angular frequency, we had two orthogonal components a sine component and a cosine component. Now, here the two orthogonal components are not going to be sine and cosine, but one complex exponential rotating in the anticlockwise direction and the other complex exponential rotating in the clockwise direction. So, the first thing we need to do is to show the orthogonality of these rotating complex exponentials over the period t and we will now do that. So, what we want to do is to prove, we want to prove that these two rotating complex exponentials e raised to the power j 2 pi by t k t and e raised to the power j 2 pi by t l t are orthogonal for integers k not equal to l over a period t of course, very easy just take the dot product or they are in a product to use formal language complex conjugate the second argument and integrate with respect to t for an interval of t. Now, what this means is any contiguous interval of t, you know you can take all these possible examples for a contiguous interval it could be for example, 0 to t or it could be minus t by 2 to plus t by 2 or whatever any contiguous interval of t that is what is meant here. Now, we need to evaluate this integral. So, the integral going back to the previous one we are trying to evaluate this integral here and that is very easy to do it is simply integral over any let us take 0 to t does not matter e raised to the power j 2 pi by t k minus l into t d t and remember k is not equal to l this is a very easy integral to evaluate it is simply e raised to the power j 2 pi by t k minus l times t divided by the argument except for t. So, j 2 pi by t k minus l evaluated from 0 to t. Now, this is equal to e raised to the power j 2 pi into k minus l of course, t and t cancel minus when you substitute t equal to 0 here you would get 1. So, minus 1 divided by j 2 pi by t into k minus l remember k is not equal to l. So, here this is of course, equal to 1 e raised to the power j 2 pi into k minus l is equal to 1 when k minus l is not 0. So, it is 1 minus 1 and therefore, this is equal to 0 when k is not equal to l. So, it is very clear that these two rotating complex exponentials are orthogonal they are perpendicular in the sense of the inner product that we are familiar with. So, our orthogonality is established now we know once you have orthogonal dimensions along which you want to project you must find the unit vectors now let us take the inner product of any one of these with itself that is very easy to do let us do it next. So, if you took the inner product of any of these orthogonal components with itself it should essentially give you e raised to the power j 2 pi by t times k t into e raised to the power j 2 pi by t times k t complex conjugate d t which is very easy to evaluate it simply you know this product is going to be the modulus squared and you know what the modulus is the integral is over any contiguous interval of t. So, it is just 1 d t that is equal to capital T. So, now we know what to do you see here again we can use the same strategy the inner product of one of those so called vectors with itself is t. So, that is the square of the magnitude. So, we should be dividing by the square root of t if you want to make it a unit vector, but instead of trying to divide by the square root of t on both sides on the side of decomposition and on the side of reconstruction we might as well divide by t on one of the two sides. So, let us understand what we are trying to say we are saying instead of dividing by square root of t on both decomposition and reconstruction side we simply write C k that is the decomposition side is the dot product of x t with e raised to the power j 2 pi by t k t over any contiguous interval of t divided by t. So, here what we have done is to absorb the square root of t which you would have put on the decomposition side and on the reconstruction side together on the decomposition side. So, as to avoid the square root business you have square root of t the whole squared on the decomposition side that makes matters easy to deal with. So, now we have a very clear beautiful simple and elegant relationship let us write that down in total x t a periodic wave form with period t the complex exponential decomposition and we call this a Fourier series decomposition complex exponential Fourier series decomposition x t is the sum of C k e raised to the power j 2 pi by t k t essentially all the rotating complex exponentials with angular frequencies equal to 2 pi by t times k for all integer k and C k here is obtained according to what we just wrote down it is 1 by t integral over any contiguous interval of t x t e raised to the power j 2 pi by t k t d t complex conjugate but complex conjugation is equivalent to putting a minus sign. So, here we have the decomposition expression and reconstruction expression and notice that the 1 by t here takes care of the 1 by square root of t that you need on the reconstruction and the 1 by square root of t that you need on the side of decomposition simple beautiful elegant. In fact, now I shall show you in a minute that we can get the same expression if you know the sinusoidal decomposition getting this is very easy let us write that down in a minute if we know the sinusoidal decomposition. So, if we know x t is of the form summation k going from 0 to infinity a k cos 2 pi by t times k t plus phi k then we can write down this is also equal to summation k going from 1 to infinity let us take the 0 separately a k by 2 e raised to the power j 2 pi by t times k t into e raised to the power j phi k plus the conjugate of this and a 0 will be taken separately constant is 0 frequency these are all the positive case and these are the negative case and of course, you also have the c case. So, for example, these are the c case and correspondingly you could get the c case for k negative just by comparing terms. We shall see more about these decompositions in the next session. Thank you.