 A warm welcome to the 39th session in the second module of the course signals and systems. We continue in this session to discuss further the idea of orthogonal components as evinced in passable theorem and make an intuitive explanation of the same. So in the previous session we had seen the following, we had identified that different frequencies, different erase the power j omega t are orthogonal and we had reasoned it in the following way. We started with an interval from minus t to plus t and we established that erase the power j omega 1 t and erase the power j omega 2 t are orthogonal over this interval with omega 1 equal to k times 2 pi by 2 t and omega 2 equal to L times the same thing. And we said as t becomes larger and larger as t tends to infinity omega 1 omega 2 spread all over the angular frequency axis essentially saying that you know the larger the interval when we consider the entire time axis, entire time axis implies all angular frequencies are orthogonal. Needless to say k is not equal to L here. So now if we look at passable theorem, if we look at the frequency side of passable theorem or if you like angular frequency side, it essentially says minus to plus infinity capital X omega capital Y omega bar d omega and we know that x t is equal from the inverse Fourier transform to be capital X omega erase the power j omega t d omega and so to Y t. So essentially what we are saying is that these are like the perpendicular components or orthogonal components and these are like the orthogonal vectors and therefore this is like an alternate representation for the dot product or the inner product. Now what is important is that we once again review the idea of the two dimensional space that we talked about. In two dimensional space if I have two vectors and I have two different perpendicular bases, I would get the same inner product no matter which perpendicular bases I chose and that is really what passable theorem says. And in fact taking further from there, passable theorem as I said is a very deep statement of the signal itself being an object above its representation. Let us write that down that is a very important idea. So what we are saying is that a signal is much more than its representation. A signal is an entity above its representation. Meaning the representation may change from time to frequency or vice versa but the signal does not change and the inner product of two signals does not change. So you know this is a statement which adds to our idea of a change of paradigm or a change of world view which a transform gives. It says that once you look at the transform domain you see the signal in a broader light as compared to when you see it in its natural domain. A signal could be a function of time but you can now also think of it equally well as a function of frequency. That is also a complete representation in a wide class of signals. You can reconstruct the signal from that representation in another domain which is nothing other than saying that if I already have the signal expressed in terms of one basis I can always express it in terms of some other basis. That is not something so special. So when we take recourse to this vectorial perspective of signals a lot of things become clear. And now we will make one more thing clear by invoking the passable theorem when we have xt and yt identical. So suppose xt is equal to yt. Let us see what happens. So passable theorem when xt equal to y. Of course we get minus infinity to plus infinity xt xt bar dt is equal to integral from minus to plus infinity capital X omega capital X omega bar d omega divided by 2 pi of course. And xt xt bar you can very easily see is essentially mod xt squared. And so to x omega x omega bar is mod x omega squared. So how a very beautiful interpretation of the square integral of the signal in time and the square integral of the signal in the frequency domain. And in fact we should give this a name. We call this the energy in the signal. And it is not surprising that we call it the energy. Because after all if xt were a voltage across a one ohm resistor. So if you had a one ohm resistor and if xt were to be the voltage across it as a function of time. Then the total energy consumed by the resistance or energy dissipated rather. Indeed integral mod xt squared dt over all t. So that is the justification for calling this the energy. By the way xt could also be the current. xt could be the current or the voltage. The current of course would be an ampere and the voltage involves. That is a minor point. But what Pasteur's theorem says and that we go back now to see is that the energy that you see as calculated in time as you would normally do. So you know we must qualify this. When you say this is calculated in time is equal to the energy as calculated in frequency. That is a new concept we have been. This is the energy calculated in frequency or calculated from the frequency axis. Now also we can now give a significance to this integrand itself. This integrand has a significance. This integrand is the density of energy on the frequency axis. Now if you think about it, what here is the meaning of density? Suppose you had a mass and you said the density of the mass. Of course masses are very rarely uniformly dense. Many masses are non uniformly dense. Density is a point point in three dimensional space for a mass. So how would you get the total mass? If you knew the point wise density inside the mass you would integrate that density over the mass. The same thing is being done here. Look at the expression here. This is the energy density or the density of energy on the omega axis. We are integrating that density all over the omega axis to obtain the total energy as calculated from the frequency. So it makes a lot of sense. It makes for this quantity to essentially be mod x omega square which we will now call the energy spectral density. And when we integrate the energy spectral density over all omega with of course a normalizing factor of 1 by 2 pi, we get the total energy. If one really wants to be symmetric, one could call mod xc square as the energy time density. See I mean you do not have to do it. This is more commonly used as a term. This is less common though we could use it to be symmetric. So you know mod x omega square the magnitude square of the Fourier transform has a very important significance. It tells us how the energy is distributed over the frequency. Mod x omega of course has the other significance of telling us how that particular frequency is going to get modified. So of course mod x omega and the angle of x omega. That is going to tell us how the frequency that particular whatever part of the signal has the ea is the has the rotating phasor with angular velocity omega. It is going to get modified according to mod x omega and angle x omega. That is one aspect of the Fourier transform. But the Fourier transform also says something about the way the energy is distributed over the frequency axis and this is a consequence of possible theorem. So in fact, this also tells us that if you look at the linear shift invariant system which has a frequency response, then you can see how the energy is going to get affected in the output. In fact, now let us look at that aspect in little more detail. So if you have a linear shift invariant and it has an impulse response h t and this has a Fourier transform. Let us call the Fourier transform capital H omega. If you give it an input x t which also has a Fourier transform called capital X omega. And of course you know that the output Fourier transform y of omega is equal to x of omega times h of omega and mod y squared omega is also equal to mod x squared omega times mod h omega the whole squared. So this tells us that there is a relationship between the energy densities. The energy density at the output is equal to the energy density at the input multiplied by the energy density of the frequency response. The multiplication to which convolution transforms in the Fourier transform causes this to happen. So mod h omega squared has that physical interpretation. It tells us how the energy is affected at different frequency components. We will see more in a minute. Thank you.