 A warm welcome to this session in which I am going to have my teaching associates, Sushrut Thakar participate. I told you in one of the earlier discussion sessions of this course that one of the things we want to do is to connect to you through our teaching associates and Sushrut has very kindly agreed to do the needful today to bring before me certain questions which he thinks a lot of students would have and in fact all of you should ask questions on the discussion forum, you should write what doubts you have because then we can use them to frame and to plan these discussion sessions. We will look at a few other questions also in a discussion session. I am going to ask Sushrut now to tell me some of the questions and some of the points that he has identified which could be weak links in the video that we have recorded or which could also be somewhat difficult for people to understand or which require more explanation. So go ahead Sushrut, tell me what you have noticed. Thank you sir. So we talked about causality and stability in the lectures. So I thought that causal and stable systems are very important but are there systems which are non-colosal or non-instable which are also used in real life? That is very interesting, very good question. So let us take them one by one. Now you know what does causality really mean? In fact let me repeat probably some of the things that I have said in the session. Causal means obeying the cause-effect relationship. Now this cause-effect relationship is only meaningful in time, not in space for example. So you see what I am saying is that it does not make sense to say something coming from behind is a cause or something coming from the front is a cause. You see I mean somebody could push you from behind or somebody could push you from the front both of them are equally troublesome. So in space there is no causality in that sense but in time there is causality, you know till now we have not really had any meaningful ways of going back in time. So what it really means is that in time there is a clear sequencing in one direction. If you are dealing with real time implementations you can only deal with past samples and the present or past inputs and the present input. You cannot really deal with future inputs or future parts of the input. So in that sense at a philosophical level causality has a meaning in time and that too when one wants to implement real time. If one wants to store data and implement things offline then of course causality is not so important. In space causality does not have much of a meaning, you could go backwards or forwards. In fact if you look at images you could do one-dimensional processing on each of the rows of the image or each of the columns of the image. So you know essentially if I want to do two-dimensional processing you know when we deal with two-dimensional signals what we do is to process one-dimensionally on each of the rows and then process similarly one-dimensionally on each of the columns that is one way to do two-dimensional processing. But there again upwards or downwards or left or right is symmetric there is really no causality there in that sense. So in space there is no causality. Now coming to stability you see let me give you two examples of systems which are by design unstable in fact if they become stable then there is nothing left in them. So let us see two useful examples of unstable systems the first is a discrete system a bank account. Now you know let us assume that the bank makes a calculation of the current balance you know in the account every once in so much of time. Now you see we will assume that time is fixed so maybe you know somebody I remember on the forum and raised the issue of months have different lengths well let us ignore all those differences for the moment let us say you know there is a fixed time interval after which the bank makes a calculation or recalculates what is the current remnant in the account and so on you know including you know what you have deposited what you have withdrawn and the interest calculation all together right. Now suppose you operate the bank in the following way your account in the following way you deposit something in that account and just keep it there do nothing about it. So it is like giving an impulse input it is like giving an input if it is so you see you could think now from the point of view of the bank when it makes a calculation it looks at all the transactions in a certain interval of time together for the purpose of interest calculation for example you know it you know you do not keep calculating interest by the day or something you know you you look at a lump calculation over this period all these transactions are taking place this was the balance at that point in time you make an interest calculation no banks it is and calculates interest for every hour or every day or so it is not a continuous time function in that sense. So here you know let us let us look at the situation so you put in some money into your account at one point in time so it is an input which is non-zero at a particular point in time and it is 0 else. So let us write down the discrete system so you know you have x let us say without any loss of generality x of 0 is a one time deposit. Now what kind of an output would you like in your bank account you see would you like that deposit to just remain as it is forever would you like that money that you have deposited to grow albeit slowly or worse would you like that deposit to diminishing time what you like so should I would of course like the money to grow yes you want the money you know if you have put in 100 units of currency today you would like after maybe one year that it becomes 102 units let us say whatever be the bank interest rate you know and then maybe after that it will be 102 by 100 times 102 so you know each year you would like it to grow compound is it not typically the interest compounds so you would at least you know Nare I think most banks operate on that principle of compounding the balance every so much of time so you would of course like it to grow that is what your bank account would normally do in fact you need to do that to take care of inflation you see now look at it from the point of view of discrete systems what are we saying when we say your account your bank balance grows in time what are we saying about its impulse response you see what is an impulse in this context that impulse would be a one time deposit so this X of 0 where you have fed in something at one time is like an impulse so you know you it is like X of 0 times delta N this is the input applied impulse input and the output to this impulse is an exponentially growing sequence as a function of N this is the impulse response after all isn't it call it H of N now how would you test for the stability of the stability would require to check you studied this in this week haven't you the stability would require you to check summation N mod H N and you will agree with me that this diverges so the system is unstable by creation and that is what you wanted to be see even if the balance remains constant even so the system would be unstable and what you would definitely not like is a stable system here if the balance diminishes in time you would not put your money in that account definitely isn't it so there are situations where a stable system is not desirable like this one let me take another example from continuous time you see we have something called an oscillator in continuous time an oscillator generates a periodic wave form with a small disturbance input so you know I could of course describe the circuit of an oscillator but that would be beyond the scope of this course but when you could think of a very simple oscillator constructed out of inductance and capacitance so you know inductance and capacitance when they cancel out one another's impedance they could give you an oscillation now what does an oscillator basically do an oscillator generates a periodic pattern could very often if it is a linear oscillator could generate a sinusoid a sinusoid of a given frequency and that could happen just on a strike perturbation in fact the idea of an impulse has this physical meaning you just perturb the system slightly and generates a sine wave all of its own now obviously if one applying a small disturbance like what you might call a very narrow pulse which tends towards an impulse and you know you get a sine wave a steady sine wave as an output obviously that that could be looked upon as the impulse response of the system that impulse response is not absolutely some of it so obvious with a steady wave form periodic wave form is definitely not stable but you know an oscillator is very useful in communication systems in many electronic systems where you want to keep time or you want a periodic wave form to mark something out for you or you know you might want a periodic wave form to be applied to the display channel or there are so many reasons why you want might want to generate a periodic wave form that is another example where unstable systems are what you want if the oscillator was stable it would not retain its periodic wave form at all so it would not so you know there are situations you know by themselves you must understand the properties and one must not also put you know judgments on the properties immediately so being causal is often the case in time but non-causal systems have a place being stable is often what a system is designed to be but then there are examples of unstable system which are very useful very good so let me have some more questions sir we also talked about the associativity and commutativity of systems LSI systems specifically and I thought these are very basic properties of systems so are there any physical implications or are they physically useful very good that is a good question see let us understand the meaning of associativity and commutativity it is slightly deeper level so let us take commutativity first commutativity essentially means interchangeable of order now you know it is remarkable that only linear shift invariant systems guarantee commutativity so if you have two systems connected one after the other applying on to an input you are not guaranteed that if you interchange their order the output would behave in the same way but if the systems are individually linear and shift invariant yes you are now why is this important because in certain circumstances with the with the inaccuracies that are there in linear shift invariant systems it might be better to have one of them proceed the other so you know if you have a it could it could very well happen for example that you might have a system which is unstable and you have another system which is stable now for whatever reasons you have a choice of putting the stable system first or the unstable system first like you could put the stable system first and unstable system next or you could put the unstable system first in the stable system next and both of them could be linear and shift invariant it is always a more question which is better from an implementational point of view it is one of these configurations may have a merit over the other and of course when one goes into specific systems one realizes this now this can also you know this for example you may have this choice in a slightly different sense I have already told you about abstractions you know so you have an abstract model of two systems what they want to do to the input but their realization could be in different forms you could realize one of them as a mechanical system and the other one as an electrical system now you might very well have a situation where you have full freedom which part you want to realize in mechanical form and which part you want to realize in electrical form you know and if both of them are linear and shift invariant so you know if putting the you know if you know that putting the second system first will allow you to realize it more easily in a mechanical form and that is the form in which you want to do the first operation then that freedom is available to you the abstraction is the same so you know the interchangeability is the abstraction level also it may be true that the first system is more convenient to realize in mechanical form and the second system is more convenient to realize in electrical form so let me give you an example you want to do something related to the kind of force viscous system you know that we talked about that so you want to study some kind of a force you know acting upon a mass with a viscous context you know the viscous friction and so whatever so you know you want to study that context now you also want to do some operations on the output that you get so when you record the movement and then you want to convert that movement into an electrical form and you want to do some filtering on the electrical form to get some kind of a pattern now here you could in principle do the reverse so you know you could think of a filter which works the mechanical domain but that is not so convenient to do you might want to implement that mass force system in the mechanical domain and then implement the filtering operation in the electrical domain you know so of course you could do it the other way but then that is it is not so you know the order in which you place a system may have some other physical context of choice so at the abstraction level commutativity means that you know I could interchange the order to my convenience right so yes we should we will see some more questions after a couple of minutes yes so we will continue yes and associativity to yes I need to talk about associativity so let us let us break for a couple of minutes and we will then resume okay thank you.