 A warm welcome to the 42nd and the last session of module 2 in the course Signals and Systems. In this session, we are now going to review a few ideas that we had discussed in the previous session and then we are going to summarize all that we have learned in this module. So, as far as recap is concerned from the previous session, we had looked at the triangular pulse. So, we said that if we took a triangular pulse from minus capital T to plus capital T, it was easy to compute its Fourier transform if we use the differentiation property. So, the Fourier transform of this pulse is equal to 1 by j omega times the Fourier transform of a pair of rectangular pulses which was very easy to compute. Now, here in this particular case, you might argue that we can also think of the triangular pulse as a convolution of two rectangular pulses. In fact, let me write that down for you. So, you could think of this triangular pulse essentially from minus T to plus T with the height of A, a symmetric one of course, as the following convolve with itself. Now, of course, here you know you would have to rise and then fall. So, how would you rise and fall? You would have to have some rectangular pulses here. The exercise I am going to give you is fill in those rectangular pulses so that if we call this whole thing as let us say y t. So, y t is this whole this pair of rectangular or this set of rectangular pulses which I have asked you to fill in so that this triangular pulse so that the triangular pulse x t, x t is equal to y t convolved with y t. This is the exercise for you to do. Now, if you carry this exercise out, you will realize that finding out the Fourier transform of those rectangular pulses is not difficult at all. So, again it is very easy to find out the Fourier transform of the triangular pulse because convolving a function with itself means multiplying the Fourier transform of that function with itself. So, you could find the Fourier transform of this pair of rectangular pulses and then multiply it by itself to get the Fourier transform of the triangular pulse. That is another way of doing it. What I am trying to illustrate to you from this example is that there are multiple ways in which you could use the properties of a Fourier transform to calculate more Fourier transform from simpler Fourier transform. In fact, in this case you could do it nicely by two approaches, but I will show you an instance when it is much easier to do it using the differentiation approach. That is particularly the case when you have many linear segments and the linear segments are not quite symmetric. Let me take an example and I will give it to you as an exercise to do. So, I am saying if we take this set of linear segments piecewise linear segments. So, let us say you know without any loss of generality we could begin at 0 and you can always shift and so on you know that is all right up to T1 and then up to T2 and then another one up to T3 and all have you. Let us say it first rises to a height of A and then it falls to a value of minus B here and then rises to C and maybe at T4 it falls to 0 and it is 0 everywhere else. So, this is the function next here, piecewise linear function. The exercise is to find its Fourier transform and the hint is do it with the differentiation property. You see, let me give you a hint. When you differentiate this function you are going to get rectangular pieces. Finding the Fourier transform of the rectangular pieces is not difficult at all. Of course, in this case if you try to write it the convolution is going to be quite messy. Maybe it is not easy to do that at all. But if you take the differentiation property it is very easy to do. So, let me in fact write let me draw the derivative of this function. So, how will the DDT or DXTDT look here I am showing it in green. This rises to A over an interval of T1. So, you have a positive rectangular pulse here of height A divided by T1. Here this falls from A to minus B over an interval of T2 minus T1. So, you need a negative pulse here and the height of this negative pulse is well you can say minus B minus A. You see it reaches minus B from A minus B minus A by T2 minus T1. And here of course, you have a positive pulse again. It goes to C from minus B between T3 and T2. And finally, you need a negative pulse here again. It goes to 0. So, it falls from C to 0 over an interval of T4 minus T3. So, this is the derivative and it is very easy to calculate the Fourier transform of this derivative. Once you calculate the Fourier transform of the derivative, then you know the relation between the Fourier transform of the black function here, this black function and the green function. And therefore, once you find the Fourier transform of the green function, you can find the Fourier transform of the black function. So, much so. Now, we have seen several different properties of the Fourier transform. We have also seen a very important principle of duality in the Fourier transform. What we have done is to introduce in this module, this whole idea of Fourier transformation, its properties and some examples. This is not the last word on the Fourier transform. In fact, in this module, we have seen several properties of what is essentially the continuous time Fourier transform. We have not said too much about what happens when we take a discrete sequence. We have not discussed the context of discrete systems and discrete sequences in any great depth in this module at all. That is because we need to relate continuous and discrete a little better. We need to understand how to get discrete sequences from continuous signals under what circumstances that is possible and what would happen in the transform domain when we do that. So, this session brings us to the end of this module 2 and also to the end of the first course that we are offering on signals and systems, where we have looked at signals and systems in their natural domain in module 1 and signals and systems essentially only continuous independent variable signals and systems in the transform Fourier transform domain in module 2. We have seen several properties of the Fourier transform. Again, as I said, I am emphasizing again and again, we have restricted ourselves to Fourier transform in the context of a continuous independent variable. We have not taken the case of a discrete independent variable in any depth at all in this module. So, where do we go from here now? We need to bring these two kinds of signals and systems together. Let us put down where will be the next set of steps in our journey of signals and systems which we will take in the next module which will come in the next course. Let us put them down clearly. Where do we go from here? First, relate continuous independent variable systems to discrete independent variable systems by the principle of sampling. I would not really say 0.2 separately, but as we do so, bring relationships in the Fourier domain. Now, when we have brought those relationships in the Fourier domain, establish discrete variable versions of the Fourier transform. And the fourth is to generalize the transform domain. For the continuous variable case, we generalize to the Laplace transform. For the discrete variable case, we generalize the Z transform. And having done all this, we acquire the ability to deal with a much wider class of systems than we can at the moment. So, where are we right now? We have looked at signals and systems in the natural domain. We have looked at continuous independent variable signals and systems in the Fourier domain and we have established certain important properties of the Fourier transform. What were those properties? First thing is linearity. The second thing is time shift and modulation. The third thing had to do with duality and the con… And that was a huge set of properties and so on. Convolution, convolution, you know, the consequence of convolution and the duality of convolution and multiplication and the consequence of the multiplication properties. You know, there was a branching out. Multiplication property leading to the passables theorem and there were so many ramifications of the passables theorem, it carry an interpretation for mod x omega squared and the differentiation, multiplication by independent variable duality there. And of course, there are other properties too, but these were the main ones that we discussed in some depth. This is certainly just a beginning in our whole study of signals and systems and that is why we would now encourage you to take up the next module, module 3 and the module after that module 4 in the next course to understand better some of the things that I have just outlined in the last few minutes. Thank you so much for attending this module and hope to see you in the second course and in module 3. Thank you very much.