 A warm welcome to this live session in which I would like to respond to some of the discussion that has been taking place on the discussion forum and also to express some of my own ideas in the context of what has been asked. In this session, I would also like to summarize what we are going to do in the following week briefly, just to give you a preview and also to recapitulate what we have completed in this week. Let me first recapitulate. So, in this week, you have been introduced to the idea of signals and systems and I think many of you have appreciated very well and also discussed very well. I have been seeing on the discussion forum, a very nice discussion on what signals are, what systems are and in fact, you also had the pleasure of watching Jatin Todeshyam in his discussion video, where he showed you an example of linearity and he also in fact showed you that linearity was somewhat a restrictive assumption in many situations in real life and he posed the question to you or he rather made you think about why linearity is meaningful at all. Now, let me respond first by answering that question, why is linearity important? Linearity and in fact, linearity and shift invariance in a philosophical sense is a very good way to understand many non-living systems. I would put it that way, so you know typically living systems or systems with consciousness or systems with life are seldom linear, they are highly non-linear. But non-living systems so to speak, you know other than animals and plants if you want to put it that way, those systems are often well modeled in a certain zone by a linear approximation and in fact, the linear approximation can be different for different zones. So, some of you might be familiar with how a transistor, a bipolar junction transistor operates or a field effect transistor operates and in that case, you know you will remember that a bipolar junction transistor exhibits linearity to increments, to small increments. So, it is incrementally linear, that is how you develop what is called a small signal model of a bipolar junction transistor or of a field effect transistor and in fact it is based on that small signal model that we derive many of the principles of electronic amplifiers. So, linearization in a certain zone of operation is often a strategy used to understand how systems behave in a certain zone of operation. Now, you have also seen examples of systems that are reasonably linear over a large range of operation. For example, the mass has a linear force acceleration relationship but there is nothing that changes too seriously there over a large range of operation and in fact, so also when you talk about resistance over a large range of voltage and current, it exhibits linearity, voltage is proportional to current. The same thing is true of capacitors and inductors over a reasonably large range of voltages and currents and over a reasonably large range of frequencies and here I must refer to some of the discussion that took place on the forum. If you by chance happen to have a circuit where the dimensions of the circuit are comparable to the wavelengths, see if you have alternating current and alternating voltage in a circuit then you should be working at low frequencies. In other words, the wavelengths associated with those electromagnetic waves corresponding to that frequency must be large compared to the dimensions of the circuit. So, for example, at 50 hertz or 60 hertz as you often have for power supplies, many of the circuits that we use are indeed very small compared to the wavelengths. So, as long as circuits are small compared to the wavelength, the all the circuit equations, the lumped circuit models that we use, a capacitor has its current proportional to the rate of change of voltage and inductor has its voltage proportional to the rate of change of current. All these things are reasonably meaningful in a certain range of frequency operation and of course, also in a certain range of voltage and currents but that range is reasonably large. So, I am trying to say that in many situations in real life, the range of operation in a linear sense is quite large and quite comfortable for us to make a linear model and build principles around a linear model. Of course, I do agree that we should be equipped to study and understand non-linear system that would happen in subsequent courses. Now, what is the convenience of linearity? In fact, we have talked about several properties in this particular week. We have looked at additivity, homogeneity, shift invariance and you must have realized that each of these properties has a certain meaning in our everyday life. Shift invariance in fact refers to this whole idea of systems behaving as they do for a long time period, time zone. So, for example, if I buy an electronic amplifier today, a sound amplifier, I do hope it will continue to operate as it does at least for a few years, if not a couple of decades. What is the real meaning of shift invariance? The invariance of operation of a system as a function of time. It does not mean the system gives the same output, it says that if you perform the same experiment on the system at different points in time, you see the same outcome. Consistency of behavior that is what shift invariance in some sense represents at one level. So, shift invariance again is not too unreasonable an assumption. Though again, with living beings and non-living beings, there is a distinction. Living beings are likely to be not shift invariant in their behavior. I mean, we know how humans experience changes of mental state, changes of mood and it is unlikely they are shift invariant in their behavior. And you know, in fact, you will listen to one of those discussion or you will listen to them where we have kind of exaggerated the so-called linearity of a human being and in fact, we have used it intentionally as an example to trigger in your mind this distinction between the non-linearity of living beings and the non-linearity of non-living beings and non-living systems. So, you will have one of my teaching associates discuss Hooke's law, you know, the stress strain relationship where the range of linearity is much larger and then I have pretended to talk about humans being linear and you will find it is so ridiculous, you know, human beings are terribly non-linear and one of my teaching associates has pointed that out. You will listen to that, but you know, you must respond. I am very glad, I am very happy to see that a lot of people are responding to our material and to our discussions and believe me, there can be small errors, there can be things that are not clear in the videos, there can be things that are not clear in the explorations and you should bring them to our notice that we can discuss them in the discussion forum and resolve your difficulties. Now, I am going to take up a few of the questions that were put on the discussion forum for further discussion. So, let me now, you know, in fact, see the discussion and I am looking at the discussion right here in front of me. So, I will pull out a few of them. I mean, I am talking about the technical questions. Now, you know, one of the interesting things that was asked was this question on abstraction. So, the question is, is it fair to abstract sound energy or wave pattern with light as both the forms of energy are patterned transmission in space and time? Also does abstraction have any fundamental principles or criteria to be followed before relating each other? And I can see my teaching associate has answered it very well, Ashwit has answered it very well. He has talked about abstractions as a simpler or an approximate way to understand something. He is right. So, you see, does abstraction have any fundamental principles or criteria to be followed before relating one another? See, abstraction in a way means being able to capture the essence of many situations and bring about a common description for many situations. That is the sense in which we use abstraction here. So, you saw the RC circuit and the mass friction circuit or the mass friction system and you saw that there was a common concept relating them the way they behaved, the differential equation that described them. So, when we extract certain things common from many situations and we try and deal with those situations together by virtue of the tools and techniques that we develop, that is what we mean by abstraction here. And then abstraction often occurs in the context of analogies. So, you could use a mass friction analogy for an RC circuit or an RC circuit analogy for a mass friction system. Analogies are in some sense subservient to abstraction because you have made an abstraction starting from the fact that many situations have something in common. I had a very interesting question on system transformation. Can anyone tell me, Hilbert transformation is a non-linear transformation. So, which property is not obeyed? And I am very happy to see that Prothiq, Pr0teq, I think he is Prothiq from Kanpur, he has answered the question very correctly, he said Hilbert transformation is a linear transformation. In fact, in a Hilbert transformation you know at this point it is a little too early to discuss the Hilbert transformation in detail, but for some of you who might have some exposure. The Hilbert transform is essentially a transform which works, it is easier to describe it in what is called the frequency domain. What it does is to bring a 90 degrees phase shift to all the sine waves that are present. And by the way, this operation is linear, it is additive, it is homogeneous. In fact, it is also shift invariant. So, it is a Hilbert transformer in the formal sense of the linear shift invariance system. And I am very happy, I must mention that this gentleman Prothiq from Kanpur has answered some of the questions very well, I am very happy to see that. And I can see an interesting discussion between Prothiq and Rick 3110, I do not know what the actual name is. Then we have a very, you know there was a set of questions which came on the RC circuit. So, you know one pertinent question in the RC circuit was would the RC circuit continue to be linear or would that represent a linear shift invariant system if there was an initial charge on the capacitance. And I am very glad to see that many people have answered this question and the answer is no. If there is an initial charge on the capacitor, the nonlinearity of, you see the system becomes nonlinear. In fact, let me write down something to explain this. You know, let us take the very simple example of an identity system or just a trivial amplifier system. So, y t is a times x t. Now, this is linear and shift invariant. The moment you write y t is equal to a x t plus b where b is a constant. This is nonlinear, but it is shift invariant. That is because if you look at this, here this system in fact obeys neither additivity nor homogeneity. That is very interesting. Obeys neither additivity nor homogeneity and I leave it to you to prove this. You can do it in your discussions. I have seen little bit has already happened in the discussions. This was one persistent train of thought that I saw. Of course, there are many other important questions that were raised. Now there was one question about whole idea of video color signals. Now color, when represented as r g b is really associating three functions of three signals to the space on which this color image is placed strictly, but in other thought about color is that color, say you know if you really look at what is called true color, it is represented by the wavelength of light with your v i b g y o r that really represents the wavelength of light. So, in that sense it is a linear scale. So there are different ways, see the way we represent color on a screen requires three numbers. So you would have to associate three numbers with each independent variable there. There are really three signals that come and of course, the r g b can be transformed in various ways. You can have different groups of signals associated with a color video. Essentially here there was also question, you know there were lot of people who asked questions about independent and dependent variable that is interesting. So I saw that some people asked whether you could have you know an interchange of the independent and the dependent variable. Now the independent and dependent variable normally is a matter of context. So the variable on which you have direct control which you can vary on your own, you know if you have some person manning the system and if the person is in control or is in charge of one of the variables, you call that variable independent. So independent and dependent are often contextual. Now you could have an implicit equation relating the independent and the dependent variables in that case just the equation does not tell you independent-dependent but the context can. So for example, if I could change the voltage applied across an RC circuit but I cannot directly change the current, the current changes are the consequence of the voltage applied. I call the voltage variable independent strictly that is so you know independent-dependent is sometimes contextual. Many times it is obvious in engineering system what is independent and what is dependent. However people have pointed out in some cases there could be a confusion and in that case then you have to decide which one you want to treat as independent and dependent if you want to think of it as a system. Well there are several questions actually I am very happy to see that there are some very good questions here see that the RC circuit is coming repeatedly as one of the questions. One person had asked whether you know if you have a non-homogeneous linear differential equation of nth order well even if it is non-homogeneous well see again in fact I think there is a I would not there is a good discussion which has followed that so you know I think I will leave it at that and I can see the discussion is going in the right direction alright. So do not think that I have really exhausted all the discussion that is there I am looking forward to much more discussion on what we have learned so far and what we are going to learn in future I just picked up a few things that I saw in the discussion I wanted it to be a live discussion between you and me we will do this in future too. Now we will close this for the time being but we will come back and say a little bit about what we are going to learn in the next week. Thank you.