 A warm welcome to the 22nd session in the second module of the course Signals and Systems. We had seen a few points about the Fourier series decomposition in the previous session. In this session, we are going to do a generalization, then I take the following situation. I have a general linear shift invariant system, let us call it script S, to which I give a periodic input xt with period T. So of course it is very clear that we can get a Fourier series representation of xt and the certain circumstances. So assume that xt is Fourier decomposable, let the impulse response of the system Vh of T. We want to find the output now in general and we want to make certain observations about it. So let us use the linearity of this. Let us take any one of those components first, the kth component to be specific and give it to the linear shift invariant system S with impulse response xt. Well the output is very easy to calculate. In fact, you can calculate it by the convolution integral. It is integral from minus to plus infinity. You could keep the xt as it is. So make that h lambda or h tau as you desire and write ck e raised to the power j 2 pi by T times k into T minus tau d tau. That becomes essentially an integral with respect to tau of the product of four quantities h tau, ck e raised to the power j 2 pi by T k T and e raised to the power minus j 2 pi by T k tau. Now notice that this quantity is independent of tau and that can be pulled outside the input. Let us indeed do that. So this would become ck e raised to the power j 2 pi by T times k T multiplied by a quantity. Essentially this is a complex constant depending on k. So it is very interesting. When you give any one of these components that linear shift invariant system, what comes out is the very same component multiplied by a complex constant. What is the physical interpretation of that? The magnitude of this complex constant is essentially the magnitude change that the original rotating complex number and which undergoes. And the angle of that complex constant is the change of angle which the original rotating complex number undergoes. Starting angle. Let us write that down explicitly, the expression that we have been writing. So you know you can write this. I am writing it very clearly in a different color. Mode ck and we will give this complex constant a name. We call it capital H evaluated at 2 pi by T k where we could use a different symbol for this whole argument. We can call this whole argument omega. We call this whole argument here equal to capital omega. And therefore this whole argument here is capital omega. And therefore what we are saying is that this is essentially integral from minus to plus infinity h tau e raised to the power minus j omega tau d tau with omega at a specific point. So now coming back to this expression here you have mod ck e raised to the power j angle of ck times e raised to the power j 2 pi by T times kT times modulus h 2 pi by T times kT e raised to the power j angle the same thing. So you notice that these angles add this angle and this angle angles add and magnitude multiply. There is a magnitude here and there is a magnitude here. Magnitudes multiply. Let us write that down all together. So it is mod ck times mod h 2 pi by T times k. This is the composite magnitude into e raised to the power j angle of ck plus angle of h evaluated at 2 pi by T times k. This is the composite angle multiplied by a complex rotating number e raised to the power j 2 pi by T times kT. This is the composite overall output for the kth component. And of course you can sum overall k that is because the system is linear. So if I give a linear combination of several inputs the corresponding output is the same linear combination of the corresponding output for each individual one. So now we have a general way to deal with these periodic inputs which are Fourier decomposable when you know the impulse response of the system. Let us write that expression down together just to be clear. You write it there. You have this system S we have given xT equal to summation k going from minus to plus infinity ck e raised to the power j 2 pi by T times kT and what we get is in fact the Fourier series representation of the output and that is summation k going from minus to plus infinity ck h evaluated at 2 pi by T times k multiplied by e raised to the power j 2 pi by T times kT where capital H evaluated at any omega is essentially integral. Now you know there is this quantity capital H with an argument coming in here. What is this quantity and can we use that quantity to go beyond only periodic functions? We need to reflect on that quantity. You see let us look at that quantity more fully. We have been talking about inner products before and let us try and interpret this quantity in terms of inner products. Let us go back to this point. We will give it a name. We call it the Fourier transform as opposed to Fourier series by the way. We call it the Fourier transform of team pulse Ht and we understand this Fourier transform a little better. What exactly is it? So we have capital H of omega is minus to plus infinity integrated h tau e raised to the power minus j omega tau d tau which we can also write as minus to plus infinity h tau multiplied by the complex conjugate of e raised to the power j omega tau integrated over all tau. Now you recall our understanding of inner products. When we were taking the inner product of two functions, we did that all the while when we were talking about Fourier series decomposition. Remember that we said that essentially we need to think of finding components in particular directions and we use the idea of an inner product or a dot product to find those components. How did we take a component? We essentially took the inner product of the function for which a component had to be found, inner product of that function with a unit vector in the direction of that component. The moment do not worry about unit vector, worry about a vector in that direction. So if you take the inner product of the function you are trying to decompose with a vector or a function in the direction in which you want to find a component, you are doing well. Now look at this expression from that perspective. Essentially this is like trying to find a component. You see that you are taking an inner product of the impulse response with this rotating complex number. The only point to be noted is that the impulse response is not periodic. Therefore, you could also say it is periodic with an infinite period. Now note that that is a new development. If a function is not periodic, let us make a note of that. If a function is not periodic, you could think of it as a function periodic with an infinite period. That means only after infinite time does it repeat itself. Of course, it never repeats itself. So you have taken the entire real axis as your period and you have taken a dot product or inner product of the impulse response with this rotating complex number. Makes sense, does not it? If I were to take this dot product, I should actually be calculating what contribution that impulse response has to that particular rotating complex number and that is exactly what we are using. We will see more about this in the coming sessions. Thank you.