 A warm welcome to the 37th session of the second module of the core signals and systems. The duality property or the duality concept has been giving us very important conclusions over the last few sessions. In fact, let us recall a very, very important conclusion that we drew from just the session gone by, namely an equivalence between inner products, let us write that down explicitly. So we saw the inner product in time, which is minus to plus infinity y 1 t, y 2 bar t dt is equivalent to an inner product in frequency. And what is that inner product in frequency? A product of the corresponding Fourier transform, the second one complex conjugated and integrated overall frequency. I know this has a very important place in the whole subject of signals and systems. This is what is called the passable theorem. Passable is the name of a researcher or scientist who came up with this idea. It is write down passable theorem and its glory. Of course, this is provided the Fourier transforms exist. Now, you know, we need to reflect on this a little bit. If come up with this conclusion with some work, I mean, we have to first go through a multiplication theorem or first go through a principle of duality even before that and then take a special case of the multiplication theorem and come to this conclusion, we have done hard work. But is it saying something that we do not already know in a smaller dimensional space? Let us take a two dimensional space. So, there we go. Let us take just these two vectors. Let us have this two dimensional space, span by this page below here. And let me have two vectors. Let us call them v 1 and v 2. This is the origin and I have two vectors. Let us call them v 1 and v 2 and I can draw two bases. One basis I will show in red. One orthogonal or orthonormal basis, a basis. What is an orthonormal basis? A basis of orthogonal unit vectors. What is a basis? A collection of vectors which spans that space. That means the linear combinations of which would create all the vectors of that space. So, let me draw one orthonormal basis in red, u 1 cap, u 2 cap. And let me draw an other orthonormal basis in green for the same space. Let us call it u 3 cap, u 4 cap. Now, we can express v 1, v 2 in terms of any of these bases. Let us express them in terms of u 1, u 2 and let us also express them in terms of u 3, u 4. So, of course, you know that v 1 can be written as v 1 1, u 1 cap plus v 1 2, u 2 cap and similarly for v 2. And we can also write them in terms of u 3, u 4. v 1 is v 1 3, u 3 cap plus v 1 4, u 4 cap and similarly for v 2. Now, we can query what is v 1 dot v 2 in terms of the dot product as we understand it in two dimensions. Of course, this is extendable to the inner product in higher dimensions. What is this dot product? So, to calculate the dot product, you would have to take one basis at a time. So, you will agree with me that that dot product v 1 dot v 2 can either be written as v 1 1, v 2 1 plus v 1 2, v 2 2 or it can also be written as v 1 3, v 2 3 plus v 1 4, v 2 4. In fact, if you go back and look at v 1 and v 2. Here, let me in fact use a different, let me put an angle between them in green. So, there is an angle between them, let us call it say theta. So, v 1 dot v 2 is essentially magnitude of v 1 times magnitude of v 2 times the cosine of the angle between them cos theta and this is independent of the basis. This is important. The dot product between two vectors is independent of the basis. We do not have to look at the basis and then calculate the dot. The dot product could in principle be conceived of without even having an underlined basis there. It is and it is a concept above the basis, it is not dependent on the basis. This is not difficult to appreciate in two dimensions and that is what we are writing down in the expression that we have got here. So, whether I take the dot product in basis u 1, u 2 or the dot product in basis u 3, u 4, it is the same. The dot product is independent of basis. Now, what is the consequence when we take the Fourier transform? See, in the context of the Fourier transform, where does this whole idea come? Where are the basis here and what independence are we talking about? Now, recall that we had said something about the perpendicularity of two rotating complex exponentials. We had discussed that in great depth. If I take two complex exponentials, they are essentially perpendicular. If I take the dot product over a limited time, so I have a complex exponential of angular frequency omega 1 and another complex exponential of angular frequency omega 2 and I first take the limited length dot product. So, I evaluate the dot product from minus t to plus t, then I can of course calculate a clearly closed form expression for the integral and I see what happens to that integral as t tends to infinity. We have learned this. We saw that that inner product starts going towards an impulse. In other words, that inner product vanishes as the two frequencies start differing when you take the interval going to infinity and it of course, you know, causes a divergent integral as the frequencies coincide, but that is the whole idea of an impulse. So, in a sense what we are saying is that when you take two different angular frequencies, you are talking about two perpendicular orthogonal functions. Let us write that down clearly. So, we have seen that e raised to the power j omega 1 t and e raised to the power j omega 2 t are orthogonal for omega 1 not equal to omega 2 in the limit when all t is considered. So, essentially what we are saying is when you write down x t is, let me repeat that. So, I will repeat that. Essentially what we are saying is when I write down x t is essentially 1 by 2 pi integral 1 minus 2 plus infinity capital X omega e raised to the power j omega t d omega. We have written x t in terms of an orthogonal basis of e raised to the power j omega t. So, x t has been written as a linear combination of e raised to the power j omega t. This is essentially one basis for expressing x t. And where is the other basis now? The other basis is what we learnt in module 1. I repeat that the other basis is what we learnt in module 1 namely with the impulses. So, x t is essentially minus 2 plus infinity x lambda delta t minus lambda d lambda. A linear combination of impulses and impulses are anyway orthogonal. Why are they orthogonal? Because they are non-overlapping. After all, two impulses located at different locations essentially confine themselves to a point. And these two points are different, they are non-overlapping. So, the impulses are trivially orthogonal. So, when we write x t like this in terms of a linear combination of impulses, this is an expression in another basis, the basis of these impulses. Now, here these are the coefficients in that linear combination, coefficients with respect to the basis. And as far as the first basis that we looked at was concerned, these here are the coefficients in the basis of the rotating complex numbers. Now, we can draw a beautiful parallel with the two-dimensional explanation that we gave. What is Pascal's theorem say? Pascal's theorem says essentially when I take an inner product of two functions, it is independent of the basis. It does not matter whether I take the basis to be the set of impulses or the set of complex rotating phases. You take the inner product by multiplying corresponding components with respect to that basis and integrating. And it does not matter if you choose the basis to be the time basis or the frequency base as simple as that. We shall see much more in the next session when we deal with a few more properties. Thank you.