 A warm welcome to the 20th session of the third module in signals and systems. In the last session, we finally used formal algebra or formal derivation to establish what we had seen in the Shannon sampling theorem before, namely let us summarize and let us state very formally now. If you have the wave form x t, now here I do not necessarily need to have a band limited wave form, you know any wave form x t with a Fourier transform. So, x t has a Fourier transform capital X of omega. Here of course, capital omega refers to the angular frequency and we multiply x t by a periodic train of pulses, we call that p t and p t has a Fourier expansion, a Fourier series expansion. Here the period of the train is T s. What is the Fourier transform of the resultant? We saw that last time. So, the Fourier transform is essentially summation over all k, over all integer k of C k times shifted versions of the original Fourier transform. I had written down what this equation says in words the last time, now I will show it graphically and this time I intentionally show graphically with x t which is band limited, but you have been disobedient to the Nyquist principle. So, you know let us draw an example of x t and let us take an example of 1 by T s which disobeys the Nyquist principle, we should be more general now. So, let us take capital X omega looking like this. So, you know let us make matters simple, let us assume X omega has 0 phase and what I am showing here is the magnitude directly. So, the magnitude is non-zero only between 0 and omega m on the positive side and 0 and minus omega m on the negative side. It is of course corresponding to a real signal. In fact, this would be a real and even signal x t and in fact just as an aside, you know the exercise for you is find x t. You know it is not too difficult to find x t from this Fourier transform and the hint is convolve a rectangular pulse with a rectangular pulse in the frequency domain. Anyway, this is just to put the whole meaning of x t in perspective. Now, let us take 1 by T s or T s in such a way. Essentially, we will take 1 by T s not more than twice of the maximum frequency component, but only say 1.5 times that. So, 3 by 2 times omega m by 2 pi instead of making it more than 2 times. So, you know we are disobeying Nyquist principle. What would happen? So, let us take a first set of terms here. Let us draw the first few terms and remember in the train of pulses not in pulses, we saw that C 0, C 1 had magnitudes that look like this. Recall from the previous session C 0 might be somewhere here, C 1 somewhere here. So, for the first few terms we agree that there would be a decreasing set of values in magnitude. So, what would happen? Now, I sort of compress it. So, I have omega m here and I have omega s here which is 2 pi by T s and that is 3 by 2 times omega m. So, I draw the original spectrum, this nice neat little triangle and now I draw its so called carbon copies. So, I have one carbon copy here. It is around omega s. It is simply carbon copies this black thing around omega s. The good thing is that it has a magnitude less than. So, this is of course C 0 times x omega, this one and then you would have C 1 times x omega minus omega s here, but C 1 has a magnitude less than C 0 and then again I have 2 times omega s and so on. So, I will just draw 2 of the terms to make things clear. You can see this will come up to omega m by 2. We can draw one or two more terms if you like you know. So, omega s is 3 by 2 omega m and you could take the next term which I could draw in blue around 2 times omega s that is 3. Now, you know this is omega well you have 3 by 2 omega m. So, this goes up to omega m by 2. So, 3 by 2 plus that makes it 5 by 2 here and around 3 omega m again. So, 3 omega m to 2 omega m somewhere here. So, it will be even smaller in magnitude and the same thing on the other side anyway. Now, what I am trying to emphasize is that because we have used pulses and not impulses, these carbon copies have decreasing magnitude. So, you know those carbon copies overlap you can see that you know this says is black original and there is the green carbon copy which is immediately troublesome and then the blue carbon copy and so on. And in comparison to the situation where you used an ideal sampling that is there was a train of impulses which all had essentially a width equal to 0 and height tending to infinity. So, your impulses instead of pulses in that case all these carbon copies would have the same magnitude, the same strength, but here the good thing is they decrease in strength. So, the green one although it interferes with the black one has a strength smaller than the black one and the blue one has a strength smaller than the green one. So, in fact in a way using pulses instead of impulses has been a blessing because these carbon copies have decreased in magnitude in a certain sense seems to be a blessing, but you know you have to be careful looking at it by itself it is a blessing. So, if I look only at the sampling process there is no problem it looks like a blessing. In fact now I can of course obey Nyquist principle. So, here let me go back to the figure if the Nyquist principle were obeyed the carbon copies would not overlap. In that case we could conceive of reconstructing the original from its samples. Now generalized samples with sampling waveform given by the pulse train not the impulse train and you have seen in a few discussions before that we could use what we call a crude or a simple reconstruct. You could make the reconstruct as simple as just an RC circuit. The only catch there is that if you want to practically realizable reconstructor I need the response not to be quite flat. So, I must equivalently use a very small portion of the response around 0 and if I am trying to use a small portion of the response around 0 frequency what I am saying equivalently is that omega s must be very far away from omega m. So, you know let me draw the situation to illustrate it to you. So, now let us consider the situation where we are actually obeying the Nyquist principle. So, we obey the Nyquist principle, but we use a practical reconstruct. What would be the situation? You would have this original spectrum here. Hopefully far away you would have its carbon copies. And as I said if we have used this train of pulses instead of impulses then we have decreasing amplitudes. And you have the frequency response of the practical reconstructor. And you want the frequency response to be reasonably flat up to omega m and then drop off. The frequency response needs to be reasonably flat here. What are we saying? We are saying that now omega m must be much much less than omega s. In fact, what we are saying effectively is that whereas Nyquist principle requires you only to have omega m less than omega s by 2 you know. So, the situation is that in the graph omega s by 2 would come somewhere here and you only need omega m to be less than this omega s by 2. But that is not enough because for a practical reconstructor you also want to put it in this flat region which is pretty far from omega s by 2. As you go towards omega s by 2 in a typical reconstructor you have the portion where the response falls off fast and you do not want that part. So, there is the compromise. The compromise is that if I use a practical reconstructor I need to use a much larger sampling frequency. So, omega s needs to be much larger than 2 times omega m. But of course the good thing is if I am using practical sampling with pulses instead of impulses even though that practical reconstructor never goes to 0 as omega tends to infinity you are at least blessed that some of those other carbon copies have decreasing magnitude. So, even if they do appear in the output they appear with decreasing magnitude compared to the situation of ideal sample an interesting combination. Well, we would now formally like to give a name to these carbon copies. The so called carbon copies that we have been talking about all this while carbon copies of the spectrum are often called aliases. You know the word alias has the meaning of assuming a false a different identity. So, very often a person uses an alias to represent himself or herself that means a different name that is what the original spectrum is doing here. The original spectrum is pretending to be something else pretending to assume a different name. It is pretending to be a spectrum which lies around different multiples of sampling frequency. And if the aliases interfere with the original there is confusion. If the aliases do not interfere with the original there is no confusion simple enough. So, Nyquist principle says do not allow aliasing. We will see more in the next session. Thank you.