 So, welcome to the second discussion that we are going to have on the contents of the first week. So, I had some very interesting questions from my teaching associates here in the previous discussion and they weren't able to exhaust all their questions and comments in the first discussion. So, here we are we have several more inputs from them in this discussion and let me hand over to them. So, they could present their inputs and questions one by one. Yes, over to you. Sir, in week one it was mentioned that the signals are reasonable functions. I don't get it. Can you explain a little bit what are reasonable functions? Well, you know reasonable was an informal term I used. Let me give you an example of what I call a reasonable function and what I call an unreasonable function. Right. So, I will write it down. Reasonable function or reasonable signal which I can or rather reasonable function which I can call a signal. Pick a simple exponential. Right. So, you have x of t is equal to e raised to the power minus say t by 2 for t greater than equal to 0 and 0 for t less than it is an exponential it looks like this. 0 all the way up to 0 as a function of t and then rises to 1 at t equal to 0 and then decays exponentially. Now, this is reasonable. I mean you would not understand why it is reasonable until I give you an example of an unreasonable function. But one thing I want to point out I have taken this example intentionally because here we have a discontinuity here. So, in spite of the discontinuity in these function I have still called it reasonable. So, reasonable does not mean that it cannot have discontinuity. In contrast, let me now take an example of what I call an unreasonable function. So, let me define x t in the following way. x t is equal to 1 whenever t is rational and 0 whenever it is irrational. Now, I call this unreasonable on several counts. You see take any interval any tiny interval as tiny as you desire let us say 0.51 to 0.5102 a tiny little interval of size 0.0002 isn't it? Now, in this tiny interval there are an infinite number of points where the function takes the value 1 and an infinite number of points where the function takes the value 0. In fact, it is an example of a function which is continuous nowhere in a certain there is there is no sense of there is no sense of you know the properties of the function are changing even over a small I mean I am speaking informally right now. I am not really being too rigorous mathematically, but in this tiny little interval of point 0.0002 you have so many changes taking place in the function and infinite number of changes in fact. We are not trying to talk about this kind this is an we do not really encounter these kind of signals in real life. And even if we wish to idealize real life situations is unlikely that we would like to take these kinds of signals as an idealization that is what I meant by unreasonable. Now, having said all this I must be clear that the word reasonable and unreasonable are actually informal and not making a formal definition. Broadly speaking reasonable functions are those where you are likely to be able to use them to understand real life situations even if by a little bit of idealization. And unreasonable functions are those which are unlikely to occur even by a large stretch of imagination when you try and model real life situations that is one way to understand the answer to the difference between unreasonable and reasonable. Of course, we will probably say a little more about this in subsequent videos. So, you could also wait that discussion little bit. Sir, so this is in reference to the RC circuit example you have taken the videos. In that context we discussed the properties of additivity homogeneity and hence linearity follows. So, my question is would an RL circuit also exhibit these properties and if so could you just demonstrate them or show an example of that? That is a very good question. In fact, this question helps me take two issues all at once. One is of course to show that the same principles that we develop for the RC circuit can also be carried to the RL circuit. And the second to illustrate how one very important principle in electrical circuits namely the principle of duality actually helps us make an abstraction or is ground to abstraction. Now, let me take the answer to this question first. So, let us take the RL circuit. Now, the first thing is you could have two kinds of RL circuits. You could have a parallel RL circuit and a series RL circuit. In both cases of course one can draw an analogy. Let us take a parallel RL circuit first. And let us take a series RC circuit for reference. I will first draw the parallel or the analogy here. Now, here I will write resistance capacitance. Here I shall intentionally write conductance which is the reciprocal of resistance and inductance. And here of course you remember that I had written an equation relating the input and the output voltage. So, I have the input voltage being VIT and the output voltage being VOT if you like. And I had related the input and the output voltage. In fact, I said that VOT that is the voltage across the capacitor plus the voltage across the resistor which is R times the current in the resistance which is R times C times VOTDT. This is the total voltage equal to the input voltage. Now, this is the equation that describes the series RC circuit. In contrast, let us look at the current input in the circuit. So, I have IIT if I would like to call it that input current to this circuit which branches into a current flowing in the conductance and a current flowing in the inductance. Let me call the output current as the current flowing in the inductance. And now I have two components to IIT. IIT is equal to of course that is simple essentially by using the Kirchhoff's current law at this node. Just like you use the Kirchhoff's voltage law here in this loop, you are using the Kirchhoff's current law here. So, IIT is equal to IOT and of course the voltage across the conductance. Now, how much is the voltage across the conductance? It is going to be the current in the conductance multiplied by the resistance. You can similarly write down an expression for the remaining part of the current here. The voltage across the conductance is the same as the voltage across the inductance here. And how much is the voltage across the inductance? It is essentially LDIOTDT that is the voltage and you multiply it by the conductance and get the current. So, this is the current contributed by the conductance. Now, look at the similarity of these two and I am sure you understand that essentially we are trying to solve one problem. You know, if you solve one problem, you have kind of solved the other problem already. Now, this is actually behind what is called duality in electrical circuits. So, duality is just one instance of abstraction in electrical circuits. Duality says resistances and conductances are dual. Of course, resistance and conductance are really the same object, but inductances and capacitances are duals. So, the roles of voltage and current can be interchanged. Now, that is just one level of abstraction. So, you solve one circuit and you have already solved another circuit. That is so useful to have. What we are saying in signal and systems is much more. You understand one system and you have understood many other systems which are of a similar kind. Now, I put an exercise to the class. I have taken the dual. So, I have taken a parallel RL circuit and a series RC circuit and I have drawn an analogy between them. So, suppose I took a series RL circuit, how would I put down a linear system description for that? I am going to put this as a challenge for all of you who are participating in this course and let us see if some of you can answer this on the discussion forum. So, very good. So, I am very glad that there are some interesting questions that have come up. We will now continue this second discussion and I just wanted to mention. Another example of the reasonable and unreasonable function and we have discussed to some extent this kind of example even in the regular videos, but I am I repeated it here because I want to make it clear what a reasonable unreasonable meant at least at an informal level. So, now with that let me invite a few more questions. Yes. In the videos we actually discussed about systems. So, there were systems which have electrical input and electrical output. So, my question is whether we could use something like a transducer as a system because it can have a mechanical or a temperature input and it gives out an electrical output. Absolutely. That is a very good question. In fact, it is important to know that one could have different kinds of input and output. The nature of the input and the output could be different. In fact, right now of course, we are confined to a context where the independent variable is the same. So, the independent variable typically is time. So, for example, you know you could have a temperature input. You could be measuring the temperature in the room as a function of time and it could be they could be a essentially a temperature measuring device which generates an electrical output and electrical voltage or an electrical current by whatever mechanism. So, you could very well have a temperature input and electrical output or the same pressure input could then generate an electrical output, a pressure transducer that is possible or you could have transducers that take you from one non-electrical domain to another non-electrical domain. In fact, I would like all the people participating in this course to come out with several examples of these kinds of systems where you have different kinds of input and output. That is the domain from which the input comes is different from the domain in which the output is. But of course, we would like to take situations where the independent variable is the same. The independent variable is naturally time in many contexts. In some contexts, it could be space. Now of course, I must emphasize that the independent variable could also be different. For example, if you look many of you might be familiar with a cathode rheosiliscope. In a cathode rheosiliscope, we give a voltage waveform for example, as an input. Now the voltage waveform is a function of time and it is displayed as a function of space on the screen. Now in a certain sense, many of the dynamic displays that you have are also examples where you are making a translation from a time-based input to a space-based input. That happens, think about it. So, it is not necessary that the independent variable needs to be the same. But I would like you to first in the discussion forum to come out with examples where the independent variable is the same and then to come out with examples separately where the independent variable is different. So, the answer to your question is yes indeed and I am in fact putting further questions to you to encourage you to think further. Very good indeed. We will have some more discussion in subsequent discussion sessions. We need to conclude this one at the present moment. Thank you.