 a long welcome to the 22nd session of the third module of signals and systems. Here, we continue with building more examples of the more general form of the Nyquist sampling principle that we discussed in the previous session. Now, in the previous session, I explained to you the connection between what we had seen as the construct of the Nyquist sampling theorem and what is popularly known as amplitude modulation. I was saying that Nyquist sampling principle tells you that you do not really need to use a sinusoidal carrier if you really want to get amplitude modulation provided you are willing to use a little more complicated. Well, when I say complicated, I mean system with a little more components. But of course, you know components that might be easier to build. It is sometimes difficult to get an exactly sinusoidal generator. So, if that is the case, then what I am saying now makes sense. If it is easy to get a sinusoidal, you know it is not just sinusoidal generator, you need a sinusoidal generator and you also need a multiplier, a good multiplier and that could sometimes pose a challenge in electronic hardware. So, a rough and ready way to do that is do not quite use a multiplier, just use a switch. That is what you mean by multiplying by a train of pulses. See, let us give that physical interpretation here. So, the physical interpretation, what I mean by that is if we choose the periodic waveform p t to be the following with an appropriate height, you know you could take the height to be 1 by delta if you like. Then what are we really doing? Multiplying x t by p t in this case. So, you know let us sketch it. So, suppose you have x t that looks like this, this is x t for you and I am showing p t in green over that. Now, multiplying this green p t that you see here by the black x t really means retaining what I am marking out as the red parts of x t. It really means retaining parts of x t and scaling them appropriately and how do you retain some part of a signal? What is the hardware mechanism to retain some part of a signal and remove the rest? You know of course, scale it. See, scaling can be done by the input output mechanism of retention and removal. Retention and removal is essentially switching. So, we need just a simple switching device. You know, retain switch on for this interval and off for this interval. So, simple and that repeats in every period and although it is a little beyond the scope of this discussion right now to describe hardware devices that do this because everybody listening to this course may not really have an electronic hardware background. For those of you who have a hardware background I would encourage you to think about device which would do this and could build it simply with operational amplifiers or with simple field effect transistor switches by appropriately controlling voltages that are applied. It is not difficult to do. So, you must think you know for those of you who have that background not a part of this course, but you know to be able to understand this better from electronics perspective you could think of switches which do this for you and you have switched on and off that is all that you are really doing. This is much easier than trying to multiply by a sinusoid. See, that is what we notice. It is much easier than multiplying by a sinusoid and now you can see the spectrum, the spectrum of x t times p t. You know that already. We have discussed that so many times before. So, it will look something like this. So, let us take the same speech signal x t that we did before and let us take 1 by t s as before to be 10 kilo hertz and what I will do is I will draw this in terms of 2 pi into 10 to the power 3. So, in 2 you know 2 pi into 10 raise to the power of 3 in 2 that is the omega I am drawing here. So, only I am writing only the in 2 part everywhere. So, I have 10 here minus 10 there 0 there and you know you have 5 somewhere in between exactly and you would have 6 here. So, you would have these carbon copies coming. It is the carbon copies that you want remember now. So, essentially you would have capital X omega as it is here, but multiplied by 0 and then c 1 x omega minus 2 pi by t s and c minus 1 capital X omega plus 2 pi by t s and so on. So, now you know it is not the original spectrum that I want after amplitude modulation. It is the first 2 carbon copies. So, if I want to do amplitude modulation successfully here I must actually retain the first 2 carbon copies and throw away everything else. I want to throw away the original spectrum throw away the c 2 c 3 c 4 and c minus 2 c minus 3 c minus 4 all of them. So, what do we want to do? We want to retain these let us mark them in green. I want to retain these green copies and throw away all the rest. So, you know it is interesting the fact that carbon copies or aliases are created is not necessarily a bad thing always. In fact, here you can see in amplitude modulation it is the aliases that help us construct the modulated signal. Well, how will you retain those carbon copies and remove all the others? You would do it by using what is called a bandpass filter. So, an ideal bandpass filter would have a frequency response that looks like this. So, we would need a bandpass filter. What is a bandpass filter? It is a linear shift invariant system which has a frequency response and I am showing you the ideal frequency response in green right now. Again my axis is omega and I have already written 2 pi multiplied by 10 to the power of 3 multiplied by and what is multiplied by is being written here now. So, you know I would require a bandpass filter that isolates what I have just before 6 and just after 14. So, it has a frequency response that looks like this. So, it would be minus 6 minus 14 here it would end here. Now, I am saying just a little beyond because you would not want to miss what is at 6 and what is at 14 see and of course, all this is you know these are all vertical lines ideally. So, ideal the amplitude is 1 here and 0 everywhere else. Now, of course, how would you make it non-ideal or how would you make it more practical? You would of course have to allow 3 things to happen. You cannot allow quite a flat such band like this all over where you want to pass things. So, you need some variation there. You cannot have a brick wall. So, it needs to fall off smoothly and it cannot go quite to 0. This would be something which is more practical so to speak. So, what I have shown in red is a more practical bandpass response. Remember, you know there are 3 things that make a frequency response practical. Wherever you ideally would have liked it to be flat, you cannot make it flat. It must vary slowly. Wherever you want a brick wall, you cannot have a brick wall. You will have to have a smooth transition and wherever you want it to be 0 all along, you cannot have it 0 all along, but you can make it small. In fact, there are technical terms that I used for these things and I would like to state them now. This is called the pass band, the band that you want to pass. This is a part of what is called the stop band. So, by the way, the stop band is not only between minus 6 and plus 6, but it is also after 14. You must not forget that. And this part where it moves smoothly, this part, you know the smooth fall off is called the transition band. So, practical bandpass filter for the practical filter which has ideally responses that are flat in different regions. That is what you ideally want, but you cannot get. So, a practical filter would have almost a flat pass band, almost a 0 stop band and a small transition band as you can make it, but you cannot make the transition band infinitely small and you cannot make the pass band absolutely flat and neither can you make the stop band absolutely 0. What is the implication on the process of amplitude modulation? It means and let us go back and see. It means that you cannot avoid passing some of these other copies, you know. So, for example, in amplitude modulation, you would have liked to have only these green things pass, but some part of that black spectrum, original spectrum around 0 might go through. And some part of the other carbon copies might go through. So, this is one example of how you do amplitude modulation in its entirety by using a very simple train of pulse. Now, why am I saying? Now, you know, it looks like this seems more complicated. You could have just built a nice sine wave and multiplied it. The problem is building a nice sine wave and multiplying both are difficult to do often in hardware. It is easier to do it this way. So, a lot of simple amplitude modulators actually work like this. Very nice. Now, in the next session, we shall look at some other examples of periodic waveforms and in the subsequent sessions including the next, we will also begin our discussion whether we should just leave a switching on and off mechanism or whether we should also hold the sampled waveform for some time. You know, would you just like to leave the samples as pulses or would you like to hold them on until the next pulse comes? Both options are available and we would like to analyze what happens in each of those options. Thank you. We will meet again in the next session.