 A warm welcome to the 32nd session of the second module of the core signals and systems. We will continue in this session to talk about some of the properties of the Fourier transform and specifically we shall see a little more about the convolution property that we built in the previous session. Let us recall what the convolution property said. The convolution property said, if I convolve two signals in time, their Fourier transforms are multiplied. We call this the convolution theorem. Let us write it down to be complete in the beginning of this session. So, the convolution theorem of the Fourier transform. If two signals x1 and x2 have a Fourier transform and a convolve and if the convolution also has a Fourier transform, then this Fourier transform is the product of x1 and x2. Now, we were trying to interpret this convolution theorem in the context of a linear shift invariant system. So, when we applied it to the context of a linear shift invariant system noting that the output is the convolution of the input and the impulse response, we saw that after all y of omega is equal to x omega times h omega and we were trying to give this an interpretation. We said that there is a point wise decoupling, there is a memorylessness in the Fourier domain. What it means is that the output at angular frequency omega has only to do with the input at angular frequency omega and none other and the impulse response at angular frequency omega and none other. Now, what is the meaning of this? We need to spend a little time and understand. You see, where does this come from? Now, this comes from the fact that if you look at capital X omega, it tells you what is the component of the input at the angular frequency omega. So, you could in principle visualize capital X omega times e raised to the power j omega t going into the LSI system. That is I mean notional, but let us write it like that. So, what we are saying is notional. We have capital X omega times e raised to the power j omega t going into this LSI system with impulse response h t and we know that what would come out. Remember, this is a complex number. We know that what will come out is capital X omega, you know, when e raised to the power j omega t goes in, what comes out is capital H omega e raised to the power j omega t. This is what will come out and this would happen at every individual omega. So, omega by omega there is a decoupling that is because essentially e raised to the power j omega t when it goes into the LSI system comes out unchanged in form, but multiplied by a constant. It gets multiplied by the projection of the impulse response along e raised to the power j omega t. That is peculiar to e raised to the power j omega t. It is not true for arbitrary signals. When an arbitrary signal goes into an LSI system, it does not come out of the same form. But here the form is preserved e raised to the power j omega t goes in and comes out as e raised to the power of j omega t in form, but multiplied by a complex constant. So, essentially what we are saying here is that if you look at this quantity, this is essentially the component of e raised to the power of j omega t in y t which is actually that y of omega. So, we have y of omega is equal to X omega times H omega that is why this decoupling. So, you know it is like saying that if I have an object or an operator acting upon a force and I know what that operator does. So, you know for example, I know that if I can decompose the force into its X, Y and Z components and I know what happens when there is a force in the X direction, I know what happens through that operator or object when there is a force in the Y direction and I know what happens when there is a force in the Z direction. And to get the overall effect, I can sum the effects of the force in the X direction, in the Y direction and in the Z direction. What would I do? I would decompose the force into its X component, Y component and Z component, use the behavior of that object or that operator in the X direction to obtain the consequence of the force in the X direction similarly for the Y and the Z directions and then add them. That is what we are saying here. So, you resolve the input into its components along the different omegas, for every omega there is a component. And you know what the system does to that component in a decoupled way, it multiplies it by H of omega and you bring together all these components to form Y. In a way, you know it is a very beautiful idea, it is an idea of being able to decouple the action of the linear shifting variance system when you think of the input as comprising of different complex exponentials rotating at different angular frequencies, both positive and negative. Of course, this is from a systemic perspective, this is the interpretation, but we can also make an interpretation from a signal perspective. When we multiply to Fourier transforms, I am going the other way now. When we multiply to Fourier transforms, their underlying signals are convolved. So, in fact, now I can start interpreting things in the frequency domain. Now, you know, I want to make a kind of statement here about what a transform really means. So, let us look at that in some depth. What is a transform more generally? A transform is a change of paradigm. Paradigm by the way means world view. What do we mean by this change of world view of change of paradigm? You see, the whole context, the system and the signals, all of them are transformed. So, x t is transformed to capital X of omega, h t to capital H of omega and y t to capital Y of omega and this operation of passing the input through the LSI system is transformed to multiplication. So, you know, if you recall, we thought of a signal as a mapping from the real axis in general. Here, we are talking about continuous independent variable signals. So, it is a mapping from the real axis to the set of complex numbers. That is a signal, a continuous independent variable signal. What is a system? It is a mapping from signals to signals. Let us write this down. We should now get this whole thing clear because you must see the transform in a slightly more philosophical, more generalized sense. So, a signal is a mapping from the reals to complex numbers. So, essentially the natural domain to the set of complex numbers. Now, of course, this could be time or it could be space or whatever you like. A system is a mapping from signals to signals and a transform is a mapping of the whole context altogether signals and systems to what we might call transform signals and systems. So, it is a mapping of the whole world view, so to speak, or the whole context, the signal system, all of them get mapped. That is why I am saying it is a change of paradigm, a change of world view, a change in the way we look at the situation. And in fact, if you are talking about convolution, hopefully multiplication is an easier operation. In general, it is easier to multiply two Fourier transforms rather than convolve the underlying signals. In some cases, the convolution may be easy to do. In many cases, it is not so easy. Of course, we need to go back, that is where the catch lies, whether it is a good thing to convolve directly or whether it is a good thing to first go to the Fourier domain that is convert each signal into the Fourier transform, multiply the Fourier transforms and then of course, if you want to go back to the original, if you want to find out, for example, suppose you have convolved xt and ht to get yt directly in the natural domain, you have done the operation of convolution once, right. What would you do if you wanted to go to the Fourier domain? Let us see. Let us put down the steps. So if I wanted to use the Fourier transform for convolution, how would I do it? I would Fourier transform x1t, similarly x2t, I would take a Fourier transform. I would multiply the Fourier transforms. So I would have capital X1 omega, capital X2 omega and I would take the inverse Fourier transform. I will denote the inverse Fourier transform by script f inverse and I would get x1t convolved with x2t. Now here, you are doing two Fourier transform operations and one inverse Fourier transform operation. Naturally, this is benefit if the Fourier transform operations are easy to do. Otherwise, it is not straightforward that it is always better to go through the Fourier domain. Better or worse depends on what kind of signals we are dealing with. Now I will give you an example where I will show you. It is easier for us to convolve in time, but if you wish to find the Fourier transform of the convolution, it is easier to work in the Fourier domain. So I will put down the example. I will allow you to think a little bit about it and we will come back to it in the next session. So the example that we are going to take is as follows. X1t is equal to x2t and it is equal to a rectangular pulse. Essentially, let us put a rectangular pulse. Let us make it simple. Let it be a rectangular pulse going from minus t to plus t symmetric and of height a. Now the question is, obtain the Fourier transforms of x1 and x2 and then the Fourier transform of x1t convolved with x2. We shall look at this in the next session to make a few points. Thank you.