 But, can I do better is the question that is the challenge. So, challenge can one sample in this case at less than 10 kilohertz and still reconstruct the signal in this particular case. And of course, if you answer the question for this particular case, you will come up with a more general answer as well. I mean pay attention to the question. In general, if all the frequencies from 0 to 5 kilohertz are occupied then I have no choice at all. I need to sample at more than 10 kilohertz, but here I know the signal occupies the band only between 3 kilohertz and 5 kilohertz. So, actually it is really occupying a band of 2 kilohertz from the frequency axis. Do I really need to sample at more than 10 kilohertz with this knowledge or can I do with less? And if I can do with less, what are those smaller sampling frequencies that I can use? And if I do use such smaller sampling frequencies, how will I reconstruct the original signal from these sampled versions? This is the question before you as a challenge. I told you I love throwing challenges to the class and you will find that this happens in subsequent lectures as well. But anyway, let us not get you know carried away too far by this challenge. Let us now come down to business if you want to call it that. We have agreed that we understand what it takes to sample a signal adequately and we also agreed that if you do not sample a signal adequately then you are going to run into the problem of aliasing. So, we want to avoid aliasing. We have sampled a signal at least and in fact more than twice the maximum frequency component present in the original signal and we are agreeing to reconstruct the original signal by cutting off all frequencies beyond the highest frequency component in the original signal. And how do you do that? Put what is called a low pass filter which cuts off after FM on the frequency axis. Incidentally, we are going to do filters in great detail, discrete time filters. We will talk more about filters. But for the moment we understand what a filter does. A filter retains some part of the frequency axis on a signal and throws out the rest. That is how we will understand a filter for the moment. In other words, it modifies the amplitudes and phases of the sine waves present in any signal that is given to it in a certain way. What I mean by that is it does not matter what amplitudes and phases the signal that is given to it has. What matters is how it modifies them. So, whatever be the amplitudes and phases of the sine waves up to frequency FM, if you put a low pass filter with a cut off of FM, it retains all the amplitudes and phases up to FM as they are and beyond the frequency FM it makes all the amplitudes 0. This is how we should understand a low pass filter. Similarly, if you had a high pass filter with a cut off of FM, it would mean that after the frequency FM that this filter would retain the amplitudes and phases as they are and before FM it would make all the amplitudes 0. This is irrespective of what amplitudes and phases they are at after FM or before FM. It does the same thing to all signals irrespective of what those amplitudes and phases are. Now, you know this is the kind of system that we are going to try and design in this course. It is very easy to describe this system. In fact, it is so simple. In a few sentences I have told you what a filter is, but you will realize as you go along the course that you can never do this exactly and you understand slowly why. You can only approximately do this and the whole art or the science of discrete time signal processing is how well you can do this, how closely you can do this thing that which you ideally want to do. You remember in the first lecture I had talked about the problem of separation of male and female voices from an audio recording. Now, if you speak the language of frequency axis, what would that mean? That would mean that you have a cut off point. You know in fact let me give you numbers. Typically speech waveforms for example, do not go beyond 4 kilohertz on the frequency axis. Audio waveforms seldom go beyond 15 kilohertz and definitely not beyond 20 kilohertz. So, for an audio signal if you sample at more than 20 kilohertz you are doing a good job. For speech signals if you sample at more than twice of 4 which is 8 kilohertz you are doing a good job. Now, there again between 0 and 4 you may reasonably assume that frequencies above 2 kilohertz would have a predominance of female component and frequencies below 2 kilohertz may have a predominance of male component. So, if you want to separate the male and female components you may wish to break up the signal into its bands between 0 and 2 kilohertz on one side and between 2 kilohertz and 4 kilohertz on the other side. So, let me if you pose the problem in that language then you may ask can I exactly put the cut off at 2 kilohertz or can I exactly break the signal between 0 and 2 kilohertz on one side and exactly between 2 kilohertz and 4 kilohertz on the other and unfortunately the answer is no you can never do this exactly you can only do it approximately and if you ask me to summarize why we need a whole semester to design filters this is essentially the reason that though the task is easy to specify it is not so easy to do what are the hurdles that we encounter when performing this task the hurdles are first to describe a general class of systems which will do this do what do exactly the same thing to all signals irrespective of what original amplitudes and phases they had at different frequencies. So you know that the system should not be partial it should not look at how much of male component or female component there is in the audio waveform and then decide what it will do it should be impartial between 0 and 2 I am going to keep between 2 and 4 I am going to cut or between 2 and 4 I am going to keep between 0 and 2 I am going to cut it should be impartial achieving this impartiality means that the system need to have several properties and we are now going to understand what those properties are in fact we will see soon that to get this impartial behavior let me write it down to get this so called impartial behavior on the frequency axis we need linearity and shift invariance you know in many discussions on systems and signals we encounter these terms we encounter the term linearity we encounter the term shift invariance and we have possibly had a lot of discussion in our past curriculum or in you know in our past degrees on linearity on shift invariance and their consequences. It is useful at this point to reflect if necessary with the benefit of hindsight why we started talking about linearity and shift invariance in the first place and the answer is this see the whole picture of what you want to do you want to get this impartial system which does the same thing on the frequency axis to all the signals that are given to it and to get this impartial nature you need a linear shift invariance system I am stating this at the moment but in the subsequent lectures we are going to prove this now this is true whether you are talking about continuous time or the independent variable being continuous or the independent variable being discrete perhaps some of us may have been exposed to this idea when we talked about continuous independent variable systems now we are going to describe and then prove these results in the context of discrete independent variable systems but before that let us put down what we mean by a discrete system so next question that we need to answer is what is a discrete system a discrete system has a sampled input and produces a sampled output and if we accept the notation that we have introduced some time ago we shall use x of square bracket n you know you remember when I started with the discussion of a sinusoid being sampled I said that if you take samples at all multiples of capital T then you could essentially substitute small t by n capital T and you could call x of n t as x square bracket n so we will use that notation in future we have a sampled input we have a discrete system which gives you a sampled output and we will use the standard notation y of n to denote the sampled output here now you know you have the sampled input we will agree now hence forth that the sampled input has been sampled according to the Nyquist principle or the Nyquist Shannon with take a theorem you have made sure that you have ascertained that you have taken samples at more than twice the maximum frequency component present in the original signal that means you know how to reconstruct the original signal from its samples just put it through a low pass filter whose which cuts off at the maximum frequency component present in the original signal now of course you could put that x of n into a discrete system do what you want with the samples and then the output can also be put through the same low pass filter so you know if you were to take an analog system which had the original continuous time signal as the input with maximum frequency component fm if you were to do some operations on the frequency axis with that analog signal and if you were to look at the output now if you sample the input and sample the output at the same instance you would get what you are calling x of n and y of n in this case that is what I mean by equivalence you sample the input you have done something with those samples you have generated an output the input xn has generated the output yn xn are essentially samples of the input at the nth instance I mean n refers to the instant number n equal to 0 means the point t equal to 0 n equal to 1 means the point capital T equal to I mean small t equal to capital T n equal to 2 means the point small t equal to 2 times capital T and you can do this n equal to minus 1 means the point t equal to minus capital T and so on so forth so n is essentially the sample instant to the sample number right you have the sampling instance for the input and you have the sampling instance for the output you have a relation between them there is a discrete system which creates that relationship and we are assuming that the discrete system does exactly what the original analog system would want to do to take again the example of male and female voice separation if you have this up to 4 kilohertz signal which is a conglomerate of several male voices and several female voices you would have an analog separator which would take the frequencies from 0 to 2 kilohertz and put it on one side and take the frequencies from 2 kilohertz to 4 kilohertz and put it on the other and you would have a corresponding discrete time system which does exactly the same thing that means it would sample the original audio signal or you know speech signal it would put essentially output a stream of samples and if you reconstructed the output signal as you did you know as you would reconstruct the input signal by the same process that I described then you would get after reconstruction of the output the male voices and the female voices in principle in different baskets so much so then for discrete systems this is what a discrete system is it takes a discrete input gives you a discrete output now remember a discrete system and let me note this down before we conclude the lecture today a discrete system is a relationship between all the samples y n for all integer n I mean and all samples X n in general so you know you must not think of it as a point by point relationship in general y of 10 could be related to X of 10 X of 9 X of 8 X of 7 all the way and also X of 11 X of 12 X of in principle yes I mean that could happen or may be at least y of 10 might be related to a few samples X of 10 9 8 and may be 11 12 and in fact there could be a relation between a group of samples of y and a group of samples of X so you must remember a system is a relation between streams of samples 2 full streams of samples that makes a system so much richer in nature and now we need to start you know studying systems by going step by step we cannot deal with this entire reality all at once so we shall do that starting from the next lecture onwards thank you.