 A warm welcome to the 21st session of the third module of signals and systems. We have now established a much more general form of the Nyquist principle and it is high time we now state it formally and go beyond just pulses. We have seen trains of pulses, we have seen its Fourier expansion and in fact now we should say the following and let me write that down right away. Now here I see a more general form of the Nyquist sampling theorem of course to give due credit it should be Shannon with take a Nyquist and so on you know all of them are credited. So I do not keep saying all the names every time but we should understand that all of them are credited in the background. So the more general form is let Pt be a periodic wave form, periodic with period t, ts perhaps and let Pt have a Fourier expansion. So the Fourier expansion would be Pt is equal to summation over all k Ck e raised to the power j 2 pi by ts times kt. So we are writing what is called the complex Fourier series. Now we also take the wave form x t which we wish to in principle sample but you know the word sampling as you will notice is not really going to come in anywhere. So here we are just taking a periodic wave form and the periodic wave form could be a train of impulses, it could be a train of pulses but it could be any other periodic wave form and we will see some examples shortly. So let x t be a signal with the Fourier transform that is all that you need and I will write the signal Fourier transform as capital X of omega where omega is the angular frequency. We are saying the Fourier transform of x t Pt is summation k going from minus to plus infinity Ck capital X omega minus 2 pi by ts times k. So simple essentially the original Fourier transform translated by every multiple integer multiple of 2 pi by ts. So you know that integer multiple is the kth. So the kth multiple is 2 pi by ts times k. We will continue this you know now we need to continue this. So it is a long statement multiplied by the corresponding Fourier series coefficient Ck and then added over all integers k and of course the proof is very easy. The Fourier transform of x t Pt is essentially the Fourier transform of this product and then you know x t is independent of k. So we can pull it along with each of those exponentials. We can write this down as summation k going from minus to plus infinity of this and now our job is very simple we have done it before. So we know that the Fourier transform of this is capital X omega minus 2 pi by ts times k and then the result follows from the linearity of the Fourier transform. It is a very simple proof. In fact in a way we have already completed this proof before when we talked about the pulses but I just formally completed the proof once again because now we will take a very simple other example of a periodic waveform just a pure sine wave. So let us take a pure sine wave for variety you know. The sine wave is an angular frequency of 2 pi by ts as you see. So of course the Fourier transform of x t times Pt is essentially now you know it is very easy to write down a Fourier expansion of this. In fact it is already a Fourier expansion in terms of real sinusoids but you can break it down into two complex sinusoids. So it is ac by 2 erases the power j 2 pi by ts times t plus erases the power minus j 2 pi by ts times t and the Fourier transform of x t Pt is so simple it is just capital X ac by 2 of course comes out omega minus 2 pi by ts plus x omega plus 2 pi by ts. Now in fact this is precisely what is done in what is called amplitude modulation. So you know what we do in amplitude modulation for you a lot of you might have been familiar with what is called the amplitude modulation radio. We do hear of it even now although a lot of systems are switching over to digital radio and so on but you know we still hear of the traditional analog radio and AM radio FM radio it is still there it is very much there and amplitude modulated radio is exactly what we are doing. So for example x t could be say a speech signal. So let us take a situation let us take some concrete numbers here. So for example x t could be a speech signal and let us draw some spectrum for the speech signal. A speech signal typically has components not more than 4 kilo hertz. So you know on the angular frequency axis to be 2 pi times 4 into 10 raise to the 3. So some spectrum you do not take the spectrum too seriously some spectrum of course it is going to be symmetric around 0 it is a real signal conjugates symmetric and let 1 by ts equal to 10 kilo hertz this is an example. So you know we need not be again 10 kilo hertz is rather low actually but still it is alright let us just take it as an example. What would the spectrum of x t times p t look like let us say now it would look something like this you have to centre this of course around 2 pi into 10 into 10 raise to the power 3 and move the whole spectrum that lies around this. So 10 minus 4 10 plus 4 that is from 6 to 14. So here this would essentially be a c by 2 so you know it is scaled up by a c by 2 up or down does not depends on the value of a c of course and this is again the one shifted backwards. So if x t is a speech signal then this 10 kilo hertz that we took was actually what is called the carrier you know what we do typically in amplitude modulation we aim to multiply the original signal which is to be modulated with a carrier and a sinusoidal carrier is the most desirable and when we multiply it by a carrier the spectrum gets shifted forward and backward by the carrier frequency that is what is happening here and as you can see if the carrier frequency is large enough the shifted versions do not reach 0 at all they should not as you can see here they are beyond 6 6 and beyond good. So this is the spectrum of the amplitude modulated speech signal and of course you have a 10 kilo hertz carrier so this is a surprising form of you know of course a lot of books do not write it like they do not write this as an example of the Nyquist theorem I am just intentionally restating Nyquist theorem now a little more technically. So if I have a pure sinusoid which essentially has two complex exponentials what you are doing in applying the Nyquist theorem here is essentially to carry out amplitude modulation and in fact this also tells you how you could carry out amplitude modulation not by using a sinusoidal carrier but any other periodic waveform. So for example you could build an AM signal as follows you could multiply it by the train of pulses that we have been talking about all the while in the last couple of sessions and then you could use a system that isolates not the 0 frequency part but the part around the carrier. You see what I mean? Let me say a little more about this in the next session. Thank you.