 A warm welcome to the 10th session in the third module of signals and systems. We have already now established an understanding of what happens when we sample a sinusoid. We now know what happens. What is sampling? We need to answer that question more formally. Let us do that first. What is sampling formally? So in other words, what is ideal sampling? Ideal sampling is multiplication of the waveform by a uniform train of impulses. Where should this train of impulses be located? This train should be located at the sampling instance. So if xt is being sampled, we are multiplying it by a waveform which we will call pt. pt has impulses located at every sampling instance. So these are the sampling instances. Now you see in this process of multiplication, what have we done alternately? Alternately we have multiplied the original function xt by the Fourier series of that periodic waveform. It is a periodic waveform. The only thing is now that waveform has generalized function, it has impulses. But we know how to deal with it. I mean after all we can always isolate one period and put the impulse in the middle of the period. Do not put it at the ends because that can create trouble. Put it in the middle of the period and then analyze for a complex Fourier expansion like we did in the previous session. So we know what we are doing. And in fact we have corroborated that by proceeding the other way too. But now we have to recognize a few things, let us enumerate them. The first thing we have to recognize is that sampling essentially creates new sinusoids where there are none. Or if you focus your attention on one of the sinusoidal components in the waveform, sampling creates a group, a discrete set of new sinusoids. So let us do that. Let us focus on any one frequency in the original waveform. And let us assume for the moment to make life easy, the sampling rate namely 1 by TS or the frequency at which we are taking samples, 1 by TS is much larger than this frequency in the original waveform on which we are focusing. So let us write that down. Let us focus on one constituent component of sinusoid in the original waveform. And now the variant that we are going to bring in is we do not have to confine ourselves to an original phase of pi by 4. It could be a general constituent component of the form A0 cos 2 pi by T0 t plus phi 0. Any phi 0 will do. And we will assume that 1 by T0 that is the frequency the constituent components frequency that we are talking about is much less than the sampling frequency 1 by TS. What happens due to sampling as far as this constituent goes? Sampling or rather ideal sampling results in this becoming some constant. Let us not worry about it because it also depends on the strength of the impulses, some constant. Let us call it kappa 0. Now, be careful attention. I have made a small generalization here. You know, we have been writing this expression rather frequently over the last few sessions. So you must almost have got bored looking at it. But each time I am bringing in something slightly new and proceeding a bit slowly because sampling is a bit of a difficult concept to most people who study this subject for the first time. So it is important to hammer in certain ideas again and again and proceed step by step with little steps in the beginning and of course once one is reasonably confident about what one is doing then it is alright to go a little faster. But at this point we are hammering in certain ideas again and again anyway coming back to this. So look carefully. The generalization that we have made here is that we have now accepted for general phase phi 0 not just phi by 4. So you will notice that as long as 1 by T0 is much less than 1 by TS there is no confusion these are all distinct frequencies. So 1 by T0 much less than 1 by TS makes these all distinct there is no confusion between them. Now in fact you know when you say much less how much less let us now graphically put down the situation and answer this question. What frequencies are we talking about? So let us take a concrete example 1 by T0 is let us say 1 kilohertz and 1 by TS let us say is 20 kilohertz. What frequencies are we talking about? So you of course have the original frequency that is 1 by T0 which is 1 kilohertz all frequencies will be in kilohertz now for L equal to 1 go back and see what are we talking about? We are talking about L by TS minus 1 by T0. So 1 by TS minus 1 by T0 and L by TS plus 1 by TS so 1 by TS plus 1 by T0 what frequencies are we talking about? We are talking about 20 minus 1 which is 19 and 20 plus 1 which is 21 and similarly now we can write down for L equal to 2 20 into 2 40 minus 1 so 39 and 40 plus 1 which is 41 and so on. Now how much larger does this need to be to keep these distinct? So of course here they are very distinct you have 1 and then you have 19 and you have 21 and you have 39 and you have 41 they are all very distinct. Now why are we so keen on there being distinct? We are so keen on there being distinct because we do not want imposter to get confused with the original waveform. I must be able to see those imposters clear as crystal. I must be able to say this is an imposter sinusoid I do not want it and I must be able to whatever using whatever mechanism later remove that imposter. So if I focus by attention just on this 1 kilo hertz constituent then how much of margin how much of leeway do I have in changing my sampling rate 1 by TS so that I am able to distinguish the imposter. In fact now once you look at it like this it is very clear let us write that let us identify what is happening here how small can we make 1 by TS well the original sinusoid is 1 by T0 the next sinusoid is 1 by TS minus 1 by T0 that is the next and of course 1 by TS plus 1 by T0 is definitely greater than 1 by TS minus 1 by TS. So therefore the answer is very simple 1 by TS minus 1 by T0 needs to be greater than 1 by T0 for distinguishability as simple as that what are we talking about here we are talking about 1 by TS be greater than 2 by TS so simple in other words the sampling frequency 1 by TS must be greater than strictly greater than certain they distinguishable this should not come in overlap with it. So greater than 1 by TS must be greater than 2 by twice this constituent frequency so simple as long as it is greater than twice now take for example this very situation you have a 1 kilohertz constituent suppose for example you do not want to make that 1 by TS as large as 20 kilohertz you simply make it say 3 kilohertz rather parsimonious. So 3 kilohertz minus 1 that is 2 kilohertz that is the job 1 kilohertz the next one was 2 kilohertz 3 plus 1 that is 4 the next one is 4 the next one of course becomes 4 2 times 3 times 2 I am sorry 3 times 2 is 6 so 6 minus 1 5 6 plus 1 7 so let us write down that example what we are saying is suppose we do not be not so generous. So we take 1 by TS equal to 3 kilohertz and 1 by T0 equal to 1 kilohertz as before. So all the generated frequencies are of course 1 kilohertz of course all of them are in kilohertz just write them down all in kilohertz 1 3 minus 1 3 plus 1 3 times 2 that is 6 minus 1 6 plus 1 3 times 3 that is 9 minus 1 9 plus 1 and so on. So what are we talking about 1 2 4 5 7 8 10 and so on so forth and even with this rather parsimonious choice of 1 by TS equal to 3 kilohertz I am able to keep these imposters away from the original sinusoid. Now make it even smaller from 3 kilohertz bring it to say 2.5 kilohertz you can see work it out yourself it will still be distinguishable in fact the limit is 2 kilohertz you cannot bring it to 2 kilohertz because if you bring it to 2 kilohertz then 2 minus 1 becomes 1. So the first imposter sits on top of the original sinusoid we do not want that. So anything more than 2 kilohertz more than 2 kilohertz is okay. You are doing very well in terms of identifying what we need for sampling not to create trouble for us. In fact the trouble that sampling creates or in other words the confusion caused by sampling is due to these imposters and in the next session we are now going to formalize this we have taken one constituent and we saw what this process of sampling does like one constituent. Now remember sampling is going to do this to every constituent sinusoid and all these constituent sinusoids are added to put in place for the waveform. Now you want this care to be taken for every constituent what does that mean it gives us a very important theorem which we will talk about in the next session. Thank you.