 So, warm welcome to the 10th session and in this session I am going to begin with a challenge for all of you. That is one thing I like to do when I teach a course, I like to challenge your imagination and I am going to do it quite a number of times in this course beginning with this time. The challenge is all this while when we have been taking examples, we have been looking at examples of system which are either both additive and homogeneous or neither additive nor homogeneous. So, it looks like additivity and homogeneity come together as a package. Is this always the case is the challenge that we want to address and what I am going to do since this is the first challenge that I have posed before you, I am going to give you a little hint. I am going to write down a couple of systems for you and you will notice that to answer this challenge I would actually make use of systems which have complex outputs or inputs. So, input x of t would be a complex function of the time t and the output y of t could also be a complex function of the time t. We do not want to restrict ourselves to real functions anymore. And with this little background, let me now write down two system descriptions for you and ask you whether these systems obey additivity, homogeneity and shift invariance and then probably the answer to the challenge would lie there. So, here we go. Let me put down system 1 where I describe the input-output relationship by y of t is the real part of x of t. And remember y of t, x of t are both complex functions of t. I must emphasize this. They are complex signals, not necessarily real. And I also put down another system, system 2. Here y of t has two descriptions depending on the past value of the input. So, it is equal to x of t times x of t minus 1 divided by x of t minus 2 if x of t minus 2 is not equal to 0 and y of t is equal to 0 if x of t minus 2 is 0. Now, here again we assume in general that x t and y t are complex. But you know a little explanation is in order in the second system. In the second system, in fact, let me point out to you what we are trying to do here. You know, you will notice that there is a dependence on x of t minus 2. So, essentially, you look at a past value of the input. If it is 0, you just transmit 0 to the corresponding point on the output. If it is non-zero, then you bring in division by that point. You need to introduce this little item of care as I might call it because otherwise you would have an undefined expression there. x of t into x of t minus 1 divided by x of t minus 2 would become undefined if x of t minus 2 is equal to 0. Now, why I emphasize this point is because you do need to have these little ends being tied when you describe system. This is an example. When describing system, you have to be careful to ensure that our system descriptions are meaningful. In fact, now let me answer a few questions or rather ask a few questions to you which would help you answer them. The questions are, is system 1 additive homogeneous shift in range? When answering this, the care that you need to take is remember the constant can be complex that is when you consider homogeneity and similarly for system 2, a system 2 additive homogeneous shift in variance. Now, that is not all. You will in fact find that both these, I am giving you the answer. You will find that both these systems are shift invariant. I am not giving you the answer to the other two. But now my further question to you is and then I am writing it down here. How do we make these systems not shift invariant? Can we do something to make them not shift invariant? And again I give you a hint, bringing a time dependent operation there. Now, you know, these three properties together is what shall be our next point of interest. Why do we take these three properties together? What is so special about them? And in fact, that is going to form a huge segment of this course. Systems which have all these three properties together and of course, we have seen examples of them. Your RC circuit is an example, it obeys all the three properties, additivity, homogeneity shift invariance. There are several. In fact, take any electrical circuit which has pure resistors, inductors and capacitors. I am assuming that the resistors, inductors and capacitors are ideal. Similarly, you could take mechanical systems where there are masses, frictional elements which have linear force velocity relationships and so on. And in fact, you could take such systems to any domain, a civil engineering domain for example. And here again, I would encourage you to try and construct an example. In the civil engineering domain, in the hydraulic domain, the fluid flow domain, I could think of a tank, a tank which stores fluid like a capacitance. And I could think of a pipe which carries fluid like a resistive element. So, there are interesting parallels. And one exercise and this whole discussion is meant to invoke thought in your mind. So, one more question that I am going to pose to you in this discussion is, we have seen analogous system. We have seen a mass with, you know, obstruction to its motion as analogous to a resistive capacitive circuit. Now, I want you to build an analogous system to both of these by using hydraulic system. Let me pose the problem formally. So, what I am saying is exercise or challenge to you. Build a capacitor like model for a tank of fluid. You know, so here, you have to decide the quantity that you should choose as input-output relationships. And the relationship should be like a capacitance, should be a derivative relationship, isn't it? So, you have something like B of t is some constant, say kappa 0 times dA t dt. So, what B and what A should we choose? Second, build a resistor like model for a pipe of fluid. So, what it means is when fluid flows through a pipe, it is assumed that it suffers resistance or obstruction to flow from the boundaries of the pipe. So, build a resistor like model. And now, finally, build an analog to the RC circuit, this RC circuit with tanks and pipes. Well, so much so for challenges. I shall leave you to the next one these challenges for a few sessions before we talk about them. Anyway, in the next session, we are going to answer this question that we have raised briefly in this session. Why are we so keen on these two properties or rather these three properties? Additivity, homogeneity, shift in balance. We will meet again, so on. Thank you.