 So, warm welcome to this session of response from me to all of you. I have been keenly observing the discussion on the discussion forum and I am very happy today to take a few questions, a few points that have been raised in the discussion. So, let me go to the discussion and identify those points. You see, I notice that there is a general concern about the link between discrete time and continuous time. So, in fact, in week 3, you have moved to discrete time signals and systems and in fact, one of these people also raised a question of definitions and properties and examples and so on. Now, I wanted to make a few remarks in that connection. I do understand that many of you are probably, you know, very well-versed and very skilled in mathematics as a formal subject, I mean, from the point of view of being a mathematics say, you know, having done a higher degree in mathematics or something of the kind. At points, you might find the discussion not as rigorous or as formal as you might like it to be and that is probably intended because we are addressing an audience, not all of whom are really mathematicians in that sense or mathematics majors or those who have done a higher degree in math and the aim of this course is actually to bring mathematics down to a context where it can be used to understand many systems at once. So, the aspect of being formal in mathematical proof, though we have tried to do as much of it as we think appropriate, that has not been the primary objective of this course. We are not trying to be too formal in the mathematical sense. Yes, but if you do find inaccuracies at places, maybe some people noticed what they thought were some inaccuracies at places, we will certainly want to correct them. This is the general remark I have that is, you are not trying to be too formal in a mathematical sense. We want to strike a balance between formalism, being formal, being accurate and also being a little informal at times because that helps us connect with a wider class of people, all of whom may not have had that formal training in mathematical rigor. So, do bear with us if you find that at points that rigor is not as much as you would like it to be, but if you find inaccuracies, please point them out and we shall definitely try and either clarify them or correct them as the case might be. Now, you see one thing I want to tell you, there was this example we gave about some kind of an economic system and so on to explain discrete system and that example was I mean only meant to explain what those discrete system equations mean, it is not meant to reflect any particular economic system anywhere in the world and it was not even intended. What I do wish to tell you however, is that a lot of people have used such discrete system analysis in understanding economic system, but that requires a much higher level of understanding of discrete systems. So, here we are trying to build a very elementary system and therefore, we had a very elementary kind of conceived of or you know contrived example just to bring the point home, but I think I will give you a better connection between continuous and discrete systems in the discussion now. You know why should we at all look at discrete independent variable systems, if we think we do not explicitly find them in the world around us. Now, the first thing we should question is do we find them in the world around us? My opinion is that we do, actually I took an economic example or a financial example, because all said and done in a certain broad sense financial calculations are done at discrete points in time. So, you know typically calculations of interest or calculations of tax are often done in many countries in many different economic system, they are done at discrete points in time whether they are uniformly spaced or not is a different issue and somebody has quite correctly pointed out that you know the intervals may not be exactly uniform, but you know instead of getting too much into the integrity of that what we do need to appreciate is that certain things are discrete, I mean for example you know decisions about allocation of budgets and so on in many parts of the economic system of the world are discrete, it is not done every day, it is done at a certain interval. So, these when there are phenomena which have an underlying continuous character, so you know for example an economic system has an underlying continuous character is always transaction going on in markets all over the world, but certain important events in that system take place at discrete time instance, you know discreteness comes in there, you have a continuum of transactions all the while, but some important things important events important calculations are done at discrete intervals. And then a natural question is if I study those discrete events which are central to that system to that context, can I come up with some understanding of the whole system or the way the whole context works. Now, in fact I want to answer this question informally at this stage, so I will give you another perhaps totally disparate example, different example, suppose I took a speech signal, now look at the old systems of recording speech with tapes, you know with continuums, so you record the speech as a continuum waveform, a continuum function of continuous time. And look at today's context, where we record digital audio, digital speech, there is a fundamental difference in paradigm, so when you go to the digital context, what you are saying is that you look at the speech at specific instance, so let me illustrate this to you, what I am saying is you have the speech waveform, let us say this is the time axis, you have the speech waveform, now the continuous time paradigm is look at the speech waveform as a continuous function of time, in contrast the discrete system paradigm is or discrete time paradigm is sample the signal preferably uniformly, so you know you take the sample value here, then you take the sample value here, by sample I mean look at the value of the speech signal at different instance as you see it here, and keep doing that, you know and record these samples, process the samples instead of the original continuous signal. So, what you are saying is you do not really need to look at that signal all the time, you look at it only at certain chosen instance, those instance might be uniformly spaced in time, and then you get a discrete signal, you get a sequence, because you can number the sample, sample number 1, sample number 2, sample number 3, in fact you can put, you can choose to put your 0 anyway, you can call this the 0 at sample for example, and then you have a minus 1, minus 2, minus 3, minus 4 here and 1, 2, 3 here, so you can number the samples with the integers, you can index the samples with the integers, and that gives you a sequence. Now whatever you are trying to do with the underlying continuous signal with the continuous time system, you can now do with the samples perhaps provided the samples keep everything that the signal had intact that is the catch line here, so when do these samples keep all the signal content intact, that is itself a big question, and we cannot answer it in this course, we will be answering it in module 3, so I do agree that you know at this point you would think of continuous independent variable systems and discrete independent variable systems as seemingly two different streams of thought, but I just wanted to respond to some of you in this video by telling you that there is a connection there can be an intimate connection between them, and that connection comes much more prominently module 3, you will understand it better because it requires us to build through all these ideas of module 1 and module 2 to be able to understand that connection thoroughly, but at this point I can tell you a little bit, I can and that is in fact quite intuitive, I can tell you that if your continuous time signal has a maximum rate of variation, so you know that the continuous time signal does not vary in any part by an amount faster than a certain rate, you know, so you know that there is a maximum rate at which the signal can change, and in fact this gets reflected in the Fourier transform, so you will study about the Fourier transform in module 2, so there are certain conditions on the Fourier transform, basically it amounts to saying there are certain conditions on how fast a signal changes to decide how frequently you should take these samples, and once you have taken a decision on that, once you have obeyed that requirement of taking samples frequently enough, then whatever you are trying to do with the continuous time signal, you can do with the corresponding stream of samples, the discrete sequence, so in that sense this gives you a motivation to deal with discrete systems as much as you deal with continuous systems, and you could for the moment think there are many circumstances in which you can deal with discrete systems and achieve what you wanted to with the corresponding continuous system, and in fact the advantage of dealing with discrete systems is that you have the power of computers behind you, it is not so easy to do continuous time operations in a computer, but it is fairly simple to do discrete time operations in a computer. So, the versatility, the flexibility and the ever growing power of a computer is available to you when you move from continuous to discrete and that is why we would like to study discrete systems along with continuous systems. This is a little motivation, because I noticed there was some concern in the discussion, why are we dealing with discrete systems is the kind of do we have to deal with totally different examples, but not really, actually a lot of continuous systems could be represented by equivalent discrete systems using the principles that I told you, if you sample the signals frequently enough and deal with the samples fast enough using a discrete system you could be doing what you would equivalently do with the continuous time system at least to a certain degree of approximation. So, to that extent you know you understand now why you need discrete system. The second thing you must have now realized is that discrete systems have certain conveniences, the impulse is much easier to understand in discrete systems. Now, let me come to some of the discussion questions, I would like to take a few of them. Now, one very interesting question I see here is from Santosh Kumar, it says to get a system function for discrete or continuous time system we need impulse response h n h t respectively, but before that we need delta n delta t practically how is it gentle, how is h n practically obtain very interesting. Now, first let us talk about continuous time impulses, you know it is very rarely that you would directly study the impulse response. So, for example, for the RC circuit the impulse is a concept to what would be an impulse voltage mean, you know for an RC circuit it would mean a sudden surge a sudden infinite voltage. So, to speak given for 0 time. Now, for example suppose you put an impulse of voltage into you see if you look at the capacitor for example, in a capacitor the rate of change of voltage is proportional to the current. So, C dv dt is equal to I let me write that down. So, you see sudden change in voltage means an impulse current. So, suppose for example, the capacitors you look at v t and it suddenly has a change like this from 3 volts to 5 volts, if you had this v t across the capacitor then here you would have an impulse current required. So, the idea of an input it mean practically practically it would mean 2 things the capacitor voltage cannot really change. So, suddenly it changes over a very short time, if it changes over a very short time it means there is a huge, but very short lived current that is present in the capacitor and that in the limit become an impulse. So, this is a practical situation where you might think of having impulse very sharp significant, but short lived pulses. Now, the question of how do you practically obtain impulse response? You see you do not necessarily apply impulses to obtain an impulse response. For example, if you look at the RC circuit you do not really need to obtain the impulse response by applying an impulse. You could apply a unit step and you know the relation between the unit step response the unit impulse response. You can differentiate the unit step response and get the unit impulse response that is a more sensible thing to do in many circumstances. It is a bad idea to apply an impulse and impulse literally would mean as I say giving the system a significant jerk for a very short lived period and that may not be a good thing to do. So, instead of trying to do that give it a steady change of input you know unit step and then observe its response to a unit step and that can then be used to derive the impulse response by taking the derivative that is. So, it is a very important question. Now, in discrete time how do you obtain the unit impulse response? There it is less difficult. You see in discrete time an impulse is not really a forbidden sequence. It is a nice beautiful sequence like any other. It is one at the point where the impulse occurs and 0 everywhere very well defined unlike the continuous time impulse which is not really a function. It is a generalized function. Here it is a nice beautiful sequence like any other. So, applying an impulse to a discrete system is much easier conceptually. It is not so difficult to understand and you can routinely study its impulse response. Well, I recommend that you keep discussing. I am happy to see there are many other questions here and I am also happy to see that there are answers being given. I cannot possibly discuss all of them in a live discussion, but I am intentionally discussing some things live because I want to encourage discussion on the forum related to of course, what we are talking about in this video, but also related to many other questions that you may have. We are listening to your questions and again and again we are telling you we would like you to put your questions on the discussion forum. We would like you to respond to one another's questions and of course, our teaching associates are also doing their best to respond when they think that you are not coming up with responses. So, we normally wait for you to come up with responses and then we put in our response, but we will of course, respond directly where we think it is necessary. Thank you so much.