 A warm welcome to the 19th session of the second module in the course Signals and Systems. We have seen in the previous session how we can decompose a periodic waveform into its complex exponential components as opposed to sinusoidal components. There we saw how we could go from the sinusoidal components to the complex exponential components. Now we shall do the converse. How do we go from the complex exponential components to the sinusoidal components? So, one thing is to be noted. Let me in fact make that point clear by reviewing what we did towards the end of the previous session. You will recall that we had already identified if xt is periodic period t. You will notice I am writing this again and again just to put the context in clarity and we have a sinusoidal decomposition. So, as I said we will separate k equal to 1 to infinity and k equal to 0. You know you do not really need to write a 0 cos phi 0 though that is of course a consequence of the expression. You can simply call it a 0. You can always modify phi 0 accordingly you see. So, this can now be decomposed. So, you have summation k going from 1 to infinity a k by 2 a k by 2 erase the power j phi k erase the power j 2 phi by t k t plus same thing, but with a conjugation and then plus a 0. Now, notice as I said that this is equal to c k for k positive and this is equal to c k for k negative and notice that for k positive and k negative the corresponding k positive and k negative you have complex conjugates let us look at it. So, if you look at this take the corresponding k here, here and here. If you take the corresponding k you have complex conjugates there. So, we will write this down formally c k is equal to c minus k complex conjugate for k going from 1 towards infinity and of course c 0 is you know it is real and in fact it is expected to be whatever it is essentially an average and that is equal to a 0 actually. So, this is true if x t is real and that is what we have assumed when we made a sinusoidal decomposition as assumed, but this is where we want to be a little more general now. We do not want to confine ourselves to real wave forms in which case this can be violated. So, c k may not be equal to c of minus k complex conjugate if x t is not real let us write that down clearly. So, will be violated if x t is not real complex and not real you see take the very trivial example of one single complex exponential. If you have one single complex exponential rotating with a period of t, you need to take only one term in this complex exponential expansion corresponding to that particular angular frequency of rotation and all the terms are absent that is an very simple example of a complex signal which does not have a sinusoidal decomposition. Now, how do we go from the complex exponential decomposition back to the sinusoidal decomposition? So, let us look at that complex exponential decomposition to the sinusoidal decomposition for real x t very easy right x t you have assumed it to be real of course in the form of summation k going from 1 to infinity c k e raise the power j 2 pi by t same summation but with minus k instead of k and c 0 and c 0 has to be real as average is real you know how to calculate c 0. c 0 is essentially 1 by t integrate x t d t over an interval of t. Now, we know what to expect for c k and c minus k. So, you know that these are complex conjugates and therefore, let us take them together. So, x t is c 0 plus summation k going from 1 to infinity c k e raise the power j 2 pi by t k t plus c k complex conjugate and the complex conjugate of that rotating phasor 2. So, you will notice that this is essentially the complex conjugate of the whole term. So, we have a sum of two complex conjugates when you have a sum of two complex conjugates it gives you essentially two times the real part of any one of them. So, let us write that down. Now, how do we write down the real part? So, real part is essentially the real part where you write c k in polar form here you have written c k in polar form magnitude and angle and that gives you mod c k cos 2 pi by t k t plus phi k where phi k is essentially the angle angle of c k. So, overall what we have is x t is then c 0 plus summation k going from 1 to infinity 2 times mod c k cos 2 pi by t and here essentially we have a sinusoidal decomposition. So, essentially here a k is 2 mod c k and phi k is of course, as it is. Now, once you have so you know it is very clear for a real signal you can go from the sinusoidal decomposition to the complex exponential decomposition and you can go back from the complex exponential decomposition to the sinusoidal decomposition very simple. So, I do not need to illustrate these separately is that right? We have already taken a few examples what I would encourage you to do as an exercise is to take the same examples decompose them using the complex exponential decomposition by the formulae that we have written and then go back to a sinusoidal decomposition and verify that they are the same just to satisfy yourself that would be a consider. Let me just take an example. So, what I am saying for example is you could take for example the square wave. So, exercise decompose the square wave take the let us take an asymmetric square wave for variety and let us take only a part of the cycle to be present from 0 to t 1 into its sinusoidal. So, this is the periodic. So, of course this is of this is x of t shown between 0 and t, but x of t plus t is equal to x of t for all t decompose it into its complex exponential components and its sinusoidal components go one from the other as discussed and verify this is the exercise I would recommend you do to understand these two decompositions very well. Thank you. We will see more in the next session.