 A warm welcome to the 35th session of the second module in the course signals and systems. We been looking at duality in the previous session and duality was a very interesting property where you derived one principle in detail and you got one principle free. So, in fact I would like to make this remark little more explicitly here. So, what is duality? Learn one and get one free is duality. This is the beauty of electrical engineering. In fact, I wanted to mention to you that this is only one instance of duality in the domain of electrical engineering. We encounter duality in other contexts in electrical engineering as well. For example, we encounter it in graph theory, in network theory, we also encounter it in Boolean algebra. So, duality is a very interesting general philosophy where you have an interchangeable situation. You have situation A being analogous to situation B and with a little bit of change situation B can be modified to become analogous to a version of situation A and that makes your learning easy. We have seen to some extent how it does it in the previous session in the context of the Fourier transform, but now we will look at it in more detail in the same context of the Fourier transform. We will see a much more serious consequence. Last time of course, we saw a consequence in terms of multiplication convolution parallel. Now, we will see a consequence again multiplication convolution, but taken the other way. So, what we had derived was that if I have x t, x 1 t convolved with x 2 t, x 1 t of the Fourier transform capital X 1 omega and x 2 t of the Fourier transform capital X 2 omega, then x 1 t convolved with x 2 t has the Fourier transform capital X 1 omega into capital X 2 omega provided all these Fourier transforms exist. Now, this is just highly potent as a situation to apply duality. So, suppose I reverse, I apply duality on each of them individually. So, I take as a consequence of duality capital X 1 of t having the Fourier transform 2 pi small x 1 minus omega and capital X 2 t the Fourier transform 2 pi small x 2 minus omega. Then it should be true that capital X 1 t times capital X 2 t, we can ask what is its Fourier transform. Now, you know from the last point here, we already know the answer, it has to do with x 1 convolved with x 2. So, let us write that down. So, take the last expression that you had here used duality on this and I will write it on the next page. Capital X 1 t capital X 2 should have the Fourier transform small x 1 convolved with small x 2 evaluated at minus omega and of course, multiplied by 2 pi. Now, we need to figure out what is small x 1 convolved with small x 2 evaluated minus omega in terms of x 1 and x 2, each evaluated again at minus omega. So, let us now write down this expression. You want to find out x 1 convolved with x 2 evaluated at minus omega. So, in fact, we will come to a general question here. Let me take x 1 t convolved with x 2 t first and replace t by omega as we should be doing here. So, that is x 1 lambda times x 2 capital omega minus lambda d lambda and suppose you were to convolve x 1 minus omega with x 2 minus omega, what would it give you? It would be essentially x 1 minus lambda well x 2 now you need to evaluate at omega. So, x 2 omega you know you have. So, let us put. So, you know if we get confused here let us put x 2 minus omega equal to x 3 omega. So, we should write x 3 here first let us do that. So, instead of x 2 let us write x 3 and then we will write x 2. So, x 3 omega minus lambda d lambda and now we will substitute pi x 2. So, x 3 omega minus lambda is x 2 lambda minus omega. Now, let us do the trick of replacing minus omega by some other variable. So, let us write this down clearly first x 1 minus omega convolved with x 2 minus omega is essentially integral as shown here. So, x 1 minus omega I am repeating here x 1 minus omega convolved with x 2 minus omega is essentially an integral x 1 minus lambda x 2 lambda minus omega d lambda put minus lambda is some alpha. So, of course, as usual minus d lambda is d alpha when lambda goes from minus to plus infinity alpha goes from plus to minus infinity. But then absorbing the minus sign in minus d lambda we have x 1 minus omega convolved with x 2 minus omega is essentially integral minus 2 plus infinity x 1 alpha x 2 minus omega minus alpha d alpha which is essentially x 1 convolved with x 2 evaluated at minus 2. So, as you see if I have x 1 convolved with x 2 and now if I replace that with x 1 with its independent variable reflected. So, omega replaced by minus omega similarly for x 2 then in the convolution the same thing happens. You could think of it in time for example, what we are saying is if I have x 1 t convolved with x 2 t and if I replace t by minus t in each of x 1 and x 2 what does replacement of t by minus t mean it means reflection about the point t equal to 0. So, you have t equal to 0 you have reflected mirrored around that. It is as if you have reversed the order of time you are going from time t equal to plus infinity towards t equal to minus infinity. If you do it for x 1 and if you also do it for x 2 you are also doing it for the convolution that is what we have written here. So, essentially what you have is that x 1 of minus omega convolved with x 2 of minus omega is indeed x 1 convolved with x 2 which would have normally evaluated with again its independent variable time reversed. Now, we have a very interesting conclusion that we can draw. Let us draw that conclusion. So, what we have seen so far is that capital X 1 of t has the Fourier transform 2 pi small x 1 minus omega and capital X 2 of t has the Fourier transform 2 pi small x 2 evaluated minus omega. Now, what would be the Fourier transform of capital X 1 t into capital X 2 t? If we go by duality we have shown it to be 2 pi x 1 convolved with x 2 evaluated at minus omega, but x 1 convolved with x 2 evaluated at minus omega is what we have just shown to be x 1 at minus omega convolved with x 2 at minus omega. So, you could multiply both sides by 2 pi. So, I have 2 pi times x 1 convolved with x 2 evaluated at minus omega is now you know you will wonder why I am doing this, but you will understand in a minute I will say 1 by 2 pi times 2 pi x 1 convolved with x 1 at minus omega of course convolved with 2 pi x 2 at minus omega. Now, here I am using the property that if I multiply any of the functions by a constant the convolution also gets multiplied by the same constant. So, if I look at this expression I am multiplying x 1 by 2 pi and x 2 by 2 pi and therefore, I have also divided by 2 pi. So, I have only 1 2 pi left which I have on the other side. So, 1 2 pi here there are 2 2 pi is here, but I have cancelled one of them by division I am doing that intentionally because now you can identify something very interesting. This is essentially this I am showing them in different colors and this is essentially this and what we are saying here is that we are convolving this with this the red with the green and then dividing by 2 pi. So, the conclusion we have drawn is that when I multiply 2 functions their Fourier transforms are convolved and then divided by 2 pi if I am talking about the angular frequency domain. So, if I have expressed the Fourier transform in the angular frequency domain I have a very simple and a beautiful expression for the Fourier transform of the product of two functions. And that is the convolution of the corresponding Fourier transforms divided by 2 pi when expressed in terms of angular frequency. Let me this is a profound conclusion. Let me again emphasize this based on what we have just written down. So, let us look at this again what I am saying is here I have x 1 t I am underlying the relevant things x 1 t has the Fourier transform 2 pi small x 1 minus omega capital X 2 t has the Fourier transform 2 pi small x 2 minus omega. And we have seen that capital X 1 t into capital X 2 t essentially has the Fourier transform 1 by 2 pi the Fourier transform of capital X 1 t convolved with the Fourier transform of capital X 2 t. So, multiplication of time functions results in convolution of their Fourier transforms that is what we are saying. Let us now make the statement formal theorem if say y 1 t has the Fourier transform capital Y 1 omega and y 2 t has the Fourier transform capital Y 2 omega then y 1 t into y 2 t has the Fourier transform 1 by 2 pi y 1 convolved with y 2 evaluated at which can be written as follows minus infinity to plus infinity integrated what you see in the integrand and the element of integration capital Y 1 lambda capital Y 2 omega minus lambda d lambda integrated over all lambda. Now, provided this is of course provided all the Fourier transforms exist all these transforms exist we have to keep putting that condition because we are not always confident that those Fourier transform would exist. But if they do essentially we have said that multiplication in the time domain translate to convolution in the Fourier domain a beautiful second consequence of duality. We will see more in the next session. Thank you.