 A warm welcome to the 31st session in the second module of the course signals and systems. You will recall that in the previous sessions, we have been looking at various properties of the Fourier transform. In this particular session, we shall look at a very important property of the Fourier transform. In fact, the property that makes for a change of paradigm, a change in the way we deal with signals and systems by using the Fourier transform and that property is called the convolution property. So, let us study the convolution property of the Fourier transform. Now, let us formulate the question first. We realize that convolution is an operation in itself. That means we can talk about convolving a signal x1 t with a signal x2 t in general. Suppose x1 t has the Fourier transform given by capital X1 of omega and similarly, x2 t has the Fourier transform given by capital X2 of omega. Now, in future, I am going to use a symbol to denote the operation of Fourier transformation, which I am going to write like this script F. This essentially implies Fourier transformation. So, the question is suppose x1 t has a Fourier transform of capital X1 omega and x2 t has a Fourier transform of capital X2 omega, of course, meaning that both of them do have Fourier transform in the first place. Then what is the Fourier transform of their convolution? In other words, when we convolve two signals, what happens to the Fourier transform? What happens in the Fourier domain as it were? That is the question that we would like to now answer. Now, towards that objective, let us write down the Fourier transform of the convolution in the first place. So, let us write down the convolution and take its Fourier transform. Let us do that. So, first, let us write down x1 t convolved with x2 t. It is essentially the integral overall tau of x1 tau into x2 t minus tau with respect to tau. And now we take its Fourier transform, which gives us an integral over all t, the convolution evaluated at t multiplied by e raised to the power minus j omega t dt and we now need to evaluate this double integral. Now, of course, we have the strategy of evaluating double integrals. What we do is, we first bring those two integrals together, we look at the combined element of integration on both the variables and then we try and make a transformation to another set of variables, which gives us some new insights. Let us do the same thing in this particular integral. So, we go back and we show the double integral together in the following way. And now we make the following transformation of variable. We put alpha equal to tau and beta equal to t minus tau. So we have this transformation, alpha is 1 times tau, beta is minus 1 times tau plus 1 times t. So, now we look at essentially a transformation of the element of integration, tau is of course equal to alpha and t is equal to tau plus beta, which is equal to alpha plus and d tau dt is essentially the modulus of the determinant of this transformation times d alpha d beta. And as you can see, the modulus of this determinant, the modulus of the determinant of this transformation is equal to 1 and therefore, this is essentially 1, which means that d tau dt is the same as d alpha d beta. And of course, we can also see that when tau and t run independently from minus infinity to plus infinity, alpha and beta also run independently from minus to plus infinity. So, the limits of integration remain the same minus infinity to plus infinity for both of them alpha and beta and the element of integration also remains the same because the Jacobian is essentially 1 and therefore, we could now write down the modified integral. So, x 1 t convolved with x 2 t Fourier transformation is essentially minus infinity to plus infinity on both the variables x 1 alpha x 2 beta e raise the power minus j omega alpha plus beta d alpha d beta. Now, notice that we can take the beta integrals all together, so let us understand which are the beta integrals here. This of course, can be split as you see. These are essentially the terms contributing to the beta integral. So, let us aggregate them. We keep the alpha integral outside, beta integral inside and encapsulate this and there we are. Now, this quantity here is very familiar, this one. What is it? You will notice that essentially it is the Fourier transform of x 2. We have agreed that the Fourier transform of x 1 and Fourier transform of x 2 converge. So, look at this expression here. This expression here is essentially the Fourier transform of x 2 which we have called x 2 capital omega and that is independent of alpha. So, I can bring it outside the integral. So, now we have capital x 2 of omega integral from minus to plus infinity x 1 alpha e raise the power minus j omega alpha d alpha and now this is as familiar to us as the previous expression. So, this is essentially the Fourier transform of x 1 which we have called capital x 1 of omega and therefore, we have a very interesting conclusion. I am now going to highlight x 1 convolved with x 2 has a Fourier transform given by the product of the Fourier transforms of individually x 1 and x 2. In other words, what we are saying is convolution in time or in the natural domain moves to multiplication in the Fourier domain. This is very interesting. Multiplication is an operation which is understood more easily. Convolution as the name suggests is a slightly difficult operation. It requires twisting and mixing. If you twist and mix two signals, the good thing about multiplication is it is a point wise operation. Point by point you operate on the two Fourier transforms. Different omegas do not interact. In contrast, convolution is the other extreme. To calculate any output point, you in principle require the whole of the two signals to interact. The whole of the first signal interacts with the whole of the second signal in the natural domain. In the Fourier domain, it is only that particular omega which interact. There is a decoupling of different frequencies. Let us write this down formula. We should take notes. We shall try and explain why these are very attractive. In the Fourier domain, convolution translates to multiplication. So, x 1 t convolved with x 2 t has a Fourier transform given by capital X 1 omega into capital X 2 omega. To calculate a convolution, convolution at t requires the whole of the signal. What I mean by that is you require to involve the signal all over the t axis for x 1 and for x 2 for both of them. You require the whole of the signal to interact. In fact, for every point, you need the whole signal to interact with the whole signal. Whole of x 1 to interact with the whole of x 2. In contrast, here it is a point wise interaction or memory less. Frequencies are decoupled. That is a very important idea here. Now, we need to interpret this decoupling of frequencies a little bit. What exactly does it mean? You see, let us again take the context of the linear shift invariance system to interpret this decoupling. So, we have a linear shift invariance system with impulse response h t. You give it an input x t and calculate the output y t. And of course, you know that y t is equal to x t convolved with h t. Suppose all of them have a Fourier transform. So, x t has the Fourier transform capital X omega and so to h t capital H omega and y t capital Y of omega. Then of course, we know from the convolution theorem that Y of omega is x omega h omega. Incidentally, the convolution theorem is what we have just proved a few minutes ago that when you convolve two signals in the natural domain, then they are multiplied in the Fourier domain. Now, what are we saying here? When we say capital Y of omega is equal to capital X of omega times capital H of omega, what are we saying? We are saying the output angular frequency omega essentially requires only the input angular frequency omega and the impulse response contribution to the frequency omega. We will see more about this in the next session because this requires a little bit of explanation. So, let us meet again in the next session. Thank you.