 A warm welcome to the third module and the second course pertaining to signals and systems. Many of you might have already completed the first course E210.1X in which you went through two modules. In the first module, you looked at signals and systems in their natural domain, in the domain in which they occur. In the second module, you looked at the same signals and systems in what is called the Fourier domain where essentially there was an effort to express all the operations and all signals in terms of sinusoids, sine waves of all possible frequencies or rather one should say all necessary frequencies. For example, if the signal were restricted to a certain interval of time, then you could make a periodic extension and you could use a Fourier series. That means you needed only discrete frequencies, all multiples of the fundamental. If the signal were continuous all over the real axis, that means the signal is not restricted to a certain interval of time, then it is not adequate to talk about a discrete set of frequencies. You need to take the entire real axis of frequencies to express the signal. I am just trying in this recap to bring out certain issues about continuous and discrete subtly. All the while in the first module and in the second module, we looked at continuous independent variable systems in their own right, we looked at discrete independent variable systems and signals in their own right and we did not mix the two. We drew analogies perhaps, there were similarities in the way we described and dealt with them. For example, the idea of an impulse response can be in the context of continuous independent variable system, then it can also be in the context of discrete independent variable systems. But then there were certain fundamental differences. Now, what is also interesting is that in one domain, for example, if I took a signal restricted to an interval of time and the signal were based on the continuous independent variable in that interval of time, it would still be possible to use a discrete set of frequencies, meaning continuous in one domain but restricted to a certain interval of time as parallel to discrete in the other domain and all uniformly spaced on the real axis. We notice that there is a connection, you know, we cannot deny that these are not two absolutely independent ideas of continuity and discreteness at a philosophical level. That is one observation that we can make from module 2. Now, interestingly the same observation can be thought of in a totally different context and here I refer to a very practical issue. In fact, in the introductory video, I have said something about this connection but now let me make it more explicit. In the olden days, you know that audio signals were recorded on effectively media which thought of them as a function of a continuous independent variable. So, magnetic recording, for example, would essentially think of the audio signal as a continuous function of time, the time axis is continuous there. You are in fact expressing that signal as a function of a continuous independent variable and recording it that way too. Now, those devices had their merits. There was no question of losing information or losing something in the recording if you did the recording properly. But in today's world, we do not use those at least not to the extent that we did in the past. They are slowly being phased out. Most of our dealings with audio and in fact even pictures, video, images are based on what are called discrete independent variable representations. So, for example, today we do not really hear about audio tapes as much as we hear about audio compact discs, audio CDs. And how can we create an audio CD? Essentially by recording audio as a stream of samples. So, we do not make a recording of the audio as a function of a continuous independent variable anymore. In a compact disc, in an audio CD, the audio is not recorded. This is an important point. It is not recorded as a function of a continuous independent variable. It is recorded as a function of discrete time instance at which you look at the audio. It is a different issue that those time instance are spaced very close by any standards of human perception. So, for example, typically an audio recording or an audio CD might require something like 44,000 times of measurement in one second. That means in one second, we make uniform time intervals of size 1 by 44,000 roughly. And in each of those intervals, we take one value of the audio signal. Now, this process of recording only one value associated with every time interval of a signal is called sampling. Let us write that down. So, sampling, let me depict it pictorially. So, I have this continuous signal or rather continuous independent variable signal. Let that continuous independent variable be time. What do we do when we sample? We look at the value of the signal here and then we look at it here and then we look at it here and we keep doing this at every uniformly spaced interval. What we have here is what we call the sample interval and what we have here are the sample values. So, what we are saying in effect is instead of this continuous signal here, we have a discrete signal or sequence emerging from the samples. So, this is, you know, you could think of all these as the samples and you could index them by the integers. So, for example, you could decide to put n equal to 0 here, it does not matter where you put it and you could say then this is n equal to minus 1 and this is n equal to plus 1 and then n equal to plus 2 and so on. And of course, backwards it would be minus 1, minus 2 and so forth. So, what we are saying is in sampling, we are trying to create a discrete independent variable replica of the original continuous independent variable signal as we do for example, an audio CD and we are hoping that by some process, the experience that I get from the discrete independent variable signal or the sequence that I have created is not very different from the experience that I get in the corresponding continuous independent variable signal. I mean all of you will agree I think that with today's technology when we play an audio CD, it does not sound as if there are jerks after every sound, you know. We still hear in fact, sometimes on account of digital media being a little more robust, the audio CD might sound better than some of those older tapes, so it is interesting. Going from continuous independent variable to discrete independent variable actually brings in robustness although it seems that if we go back to the figure here, I am only looking at the signal at this point and then I am looking at this point and I had looked at it at this point and then I am looking at this point and so on and so forth. So, have I lost something in between? So, let me mark this as a, am I losing this for example, am I losing this part of the signal? That is a question that comes to our mind. Now that is the question that we are going to set out to answer in this module. So in a way, we are going to do several things not separately, but together in this module. Let us list them without being too formal at this stage. The first thing that we are going to do is to look at this basic issue. I recognize that it is not always convenient to deal with the continuous independent variable signal and so I must first ask whether it is meaningful as we would say to discretize it, to sample it. Of course, nothing stops you from sampling, you can sample it, you can look at the signal at only discrete points on the independent variable, but that is the issue is does it keep all the information, quote unquote information, whatever you wanted from that signal, is it all there in the samples or not? Have you lost something by this process of sampling? This is the important question that we need to set out to answer. Well, under certain circumstances, if we can convince ourselves that we are not losing anything at all, that is the best situation to be in or not so good, we are losing very little, then we can perhaps take advantage of the robustness in dealing with discrete independent variable signals at the cost of perhaps losing a little bit of information or needing to do a little more work, even if you are not losing information, if you are not losing something important in the signal, you might need to do a little work to reconstruct the signal as you would like to see it ultimately. So, with the agreement that we are willing to do that additional work and we need to identify what that additional work is, we set out to sample. And once we set out to sample, then we have agreed that we are marrying continuous and discrete time or discontinuous and discrete independent variable, we are bringing a relationship between them, we are bringing equivalences, we are saying how they are similar, how they are different. It is a profound subject by itself and it is a very important thing to do because in some sense one can get the best of both. Many signals in reality are a function of the continuous independent variable be it time or space for example, but then it is far easier to deal with it when we discretize, take the example of pictures. Today most cameras are digital cameras, so they record picture elements of pixels. That means they look at discrete points in space, maybe again as I said those points are very closely spaced, so you do not even notice there is that discretization. But either by virtue of the interpolating capacity of our eyes, our visual system or by virtue of some processing that we do within the reconstruction device, we are able to see the image or the picture just as we would as if that space were continuous. So, there you have an example of two-dimensional sampling bringing a relation between two-dimensional continuous space and two-dimensional discrete space. So, my purpose in this first session of the third module on signals and systems was to give you essentially a feel of what we intend to do in this module. What is the question that we are trying to address? We are trying to address what is equivalent and what is not equivalent between continuous and discrete. We are trying to address the question how might I sample a continuous independent variable signal without losing any information at all or perhaps in a more realistic sense losing a little bit, but not serious enough to cause damage. If so, if I am not losing anything, do I have to do some additional work to reconstruct the continuous signal from the discrete signal? What is that work I need to do? Under what circumstances can this be done? And underlying all this is the question, what are the relationships between continuous and discrete independent variable systems? It is a related question. So, with this little introduction to what we are going to do in this module, let us proceed. I look forward to meet you again when we come to the second session of this module. Thank you.