 A warm welcome to the 23rd session of the second module of the course signals and systems. We are now in a position in this module where we can deal with more general signals from the point of view of decomposition into sinusoidal frequency components and analysis as to what happens when you pass through a linear shift invariant system. To do that we should try and recapitulate some of the important conclusions that we had drawn in the previous session which will now help us in generalization. So let us look at some of the important conclusions that we had drawn. What we had seen is that if we have a linear shift invariant system let us call it S with impulse response h t and to it we give a periodic input, a periodic signal input, periodic with period t and of course let us assume that the Dirichlet conditions are obeyed that is not demanding too much so we could decompose this into its Fourier series. Now the output can also be written as a Fourier series and the interesting thing is the output has a quantity that depends on the impulse response which I am now writing down here. This is the quantity that depends on the impulse response and in fact we know what that quantity is. The quantity is obtained like this. Now we need to look a little more carefully at what this quantity is and what interpretation we can give it. We had kind of started in the previous session to give an interpretation but we now need to complete that job and in fact then to take it to the input as well if the input is not periodic. So let us interpret this quantity. So if you look at the quantity capital H of omega integral for minus to plus infinity h t e raised to power minus j omega t dt we can clearly interpret this as a dot product or inner product. It is an inner product between impulse response h t and the rotating phasor e raised to power j omega. Now this interpretation makes a lot of sense. Now what does an inner product do? An inner product calculates a projection or a component. So it is as if you are trying to find the component of the impulse response along a rotating phasor rotating with an angular velocity capital omega and at the specific values of omega given by each of the Fourier series components you evaluate this dot product and use it to modify the Fourier series coefficients. Let us write that down formally. Let us focus our attention on how those Fourier series coefficients are modified in going from input to output then we will be clear what is happening in fact the interpretation is very simple. So the output Fourier series coefficients are Ck times h 2 pi by t times k essentially you have substituted the value of capital omega here. So it is essentially the product is the input Fourier series coefficient at that frequency multiplied by the component of the impulse response at the angular frequency of frequencies corresponding to those coefficients. So it is essentially as if you are taking component by component and multiplying by the corresponding component of the impulse response, component of the input multiplied by component of the impulse response, simple enough. Now let us look at how you calculated the input Fourier series coefficients. There was also an inner product there. In fact we do not need to go through the whole derivation again. You will recall that in the input the Fourier series coefficients were calculated by taking an inner product but remembering that you need to take only one period of the input. You took only one period of the input and found the inner product with the corresponding harmonic or the complex exponential rotating with that particular multiple of the fundamental frequency. So here we assume the input was periodic. What if it is not? Can we generalize this is a question that we need to answer. If we look at this quantity which we will now write down very clearly, can we think of this quantity h omega as a new transform or a new way of doing things with the impulse response in its own right? So let us answer that question. So let us look at the quantity in detail once again. Now we have h omega is minus to plus infinity ht e raised to the power minus j omega t dt and this is a function of angular frequency. Now this quantity will be called the name just like we called the decomposition of a periodic function into its sinusoidal components as the Fourier series decomposition of the function. Remember I told you Fourier is the name of a very well respected mathematician. You could call him a mathematician or a researcher in dealing with differential equations whatever it might be. But basically all these ideas of decomposition into sinusoidal components of thinking of a wild class of signals as a combination of sine waves is attributed to Fourier. So what we have done here is now to go from a discrete frequency axis. You had discrete values of the angular frequency all multiples of a fundamental frequency including the 0. We are going from discrete values of capital omega to continuous values. Let us look at this quantity here. Here omega spans the entire real axis. So essentially we have gone to a continuous frequency axis and therefore this is like a transform. It is a transform of the impulse response and in fact we shall give it a name. Obviously we will call it the Fourier transform of the impulse response. Now you know we must spend a minute in understanding what a transform is. A transform is a way of recasting a signal in a different domain and if the transform is invertible as is the case with the Fourier transform you can come back to the original domain. So the signal has all this while been described in terms of its values as a function of the independent variable time perhaps or whatever the independent variable was. Now what we are doing is to think of the same signal as comprised of sinusoidal components with angular frequency ranging all over the real axis. So what we are saying effectively this is a different way of describing the same signal and it is adequate if it is invertible, adequate in the sense you have not lost something. So if you look at this transform carefully you know if we look at the expression this is like an inner product we have interpreted that before. So it is not too difficult to imagine how we can go back from the transform to the original function can we go back is the question. What I mean is can you go back from capital H omega to HT and the answer is yes. How do we do it? Well the answer is very simple use vector intuition again. So let us use vector intuition in two dimensions first. So for example suppose let us say this was the two dimensional plane in which you had a vector let us draw that vector v and these are the unit vectors here which I am going to show in green now u1 cap, u2 cap and we decompose v into its components it will show in blue this is v1 times u1 cap and this is v2 times u2 cap and v1 clearly is the inner product of v with u1 cap and v2 is the inner product of v with u2 cap. We can reconstruct v as v1 u1 cap plus v2 u2 cap this is the important idea here. How do we reconstruct a vector from its orthogonal components? Now you know here I have assumed orthogonal components so there is an issue here we do not know if these different sinusoidal components are orthogonal or not we knew it for restricting them to a period that we reasoned out at that time but we have not reasoned in general if I take the complex exponential the rotating complex number all over the time axis as I am doing here is there the same notion of orthogonality or is there not? It is an important question that we need to answer but we will do that later. At the moment let us take a finite dimensional space as we have here and we will look at what we exactly are doing to reconstruct the components essentially reconstruct the vector from its components this is the component and this is the unit way and you are taking component times the unit vector and adding it over all the components. So what we are doing is we are taking a sum summation over all dimension so to speak component in that dimension multiplied by a unit vector in that dimension. Now the question is can I use the same principle to reconstruct a signal from its Fourier transform and the answer of course is yes I said we should use vector intuition and this is the intuition that we are going to use the only thing is now we cannot do a summation we will do an integration because it is a continuum of omegas and in the next session we shall complete our discussion on how we can do this reconstruction and understand this Fourier transform a little better thank