 A warm welcome to the 30th session of the third module in signals and systems. We have now been getting more and more familiar with what we can do with discrete systems and let us just recapitulate where we were at the end of the previous session. We had agreed that we would now call the input sequence of the discrete system as x square bracket n. This is essentially the sequence of speed samples and we had constructed its discrete time Fourier transform abbreviated by DTFT which was essentially summation n going from minus to plus infinity xn e raised to power minus j small omega n and we had an interpretation for the quantity small omega. Small omega is essentially the normalized angular frequency. We know what that means by now. We also gave this an interpretation. It was essentially the inner product, essentially a projection of the sequence xn upon the rotating complex sequence e raised to power j omega n. So, in some sense a component. So, you know now we can ask the question. First, we understand that it makes sense only to talk of small omega between minus pi and plus pi. The maximum component that you would have on this scale of small omega is pi. In fact, you should not go all the way up to pi. That is the whole point to keep a margin and in our strategy, we said that pi corresponds to 10 by 2. That is 5 kilohertz and the maximum speech component that 4 kilohertz. So, there is a margin there. And the physical interpretation of this quantity which we write here, the DTFT which we also decided to denote by capital X of omega is that it tells you how much of e raised to power j omega n there is in the sequence xn. So, now we can ask the question in a slightly different way. We can ask what does this system do? What is this discrete time system do? To a particular one of these rotating complex sequences, you know, if you want to call them that e raised to power j omega n rotating with different normalized angular frequencies omega. Well, the answer is very simple. Just put e raised to power j omega n in and you know what to expect. After all, these are what are called the eigen sequences. They go in and come out. So, put e raised to power j omega n here and you would expect it to respond with a multiple of this. But let us actually calculate. What would that multiple? What would that output be? So, e raised to power j omega n plus e raised to power j omega n minus 1 into half simple. So, take e raised to power j omega n common and I have 1 plus e raised to power minus j omega. What are we saying here? We are saying that it comes out just multiplied by the constant 1 plus e raised to power minus j omega by 2 comes out multiplied by this constant. It is a complex constant of course. Let us study this complex constant in a little more detail. So, what is this half into 1 plus e raised to power minus j omega? I can take e raised to power minus j omega by 2 common here and write 1 as a product of e raised to power j omega by 2 into e raised to power minus j omega by 2. So, what I am saying is that e raised to power j omega by 2 into e raised to power minus j omega by 2 is equal to 1. And of course, the other one would just be left with a minus e raised to power minus j omega by 2. And you know what this is? This is very easy to write in terms of the trigonometric function. So, this is 2 times cos omega by 2. So, I have half of 1 plus e raised to power minus j omega is essentially half into e raised to power minus j omega by 2 times 2 cos omega by 2. Very interesting. So, this e raised to power that particular rotating complex sequence gets multiplied by this factor e raised to power. I have written in this form because now it is very easy to identify the magnitude and the phase for the region between omega equal to minus pi and omega equal to plus pi. This gives the phase change and this gives the magnitude change. Why am I saying this? Because for omega between minus pi and plus pi cos omega by 2 is a non-negative quantity. Cos pi by 2 goes to 0 and cos 0 is 1. So, the cosine function falls from 1 towards 0 on both sides from 0 towards minus pi and 0 towards plus pi. So, this contributes no phase. And then this contributes no magnitude because its magnitude is 1. That is why I am saying that this tells you the phase change, this tells you the magnitude change. So, what is the magnitude change now? The magnitude change would look like this and that is what is really important to us. It is a cosine function, is not it? And remember I am interested only between minus pi and pi. Now, you know the discrete time Fourier transform is bound to be periodic with a period of 2 pi. And let us connect this idea to this whole business of sampling that we have been doing all in the previous sessions. You know take the discrete time Fourier transform, shift it pi 2 pi. What does it give you? Very clearly this quantity is 1 for all n. So, this whole thing is equal to capital X of omega. A shift of 2 pi makes no difference. That is not surprising. After all, what is this discrete time Fourier transform? It is essentially the same spectrum of the sample, ideally sample signal written on a normalized angle of frequency axis and that spectrum of the discrete sample signal, discrete time sample signal, if ideally sample, it is bound to be periodic with a period equal to the sampling frequency. But now, we have decided to normalize it and make the normalized angle of frequency 2 pi. So, this periodicity with 2 pi is not a surprise at all. And in fact, what a discrete system would do would also be periodic with period 2 pi. So, it is interesting what the discrete system would do to the original spectrum? It would also do to all these carbon copies or aliases. So, we need to worry only about what it does to the original spectrum. The same thing will happen to the carbon copies. Now, just one point for those of you who want to think deeper. Suppose you are not doing ideal sampling. Suppose your pulse is, but you are still doing sampling with a periodic train of say pulses. What would be the difference here? The only difference is that now you have to think of all these carbon copies as having different heights. But otherwise, much of what we are saying here would go through there too. I am not going through the full discussion here because it could be a little advanced for this course. Let us see. If we get a chance, we will look at it in detail. But some of you who might be interested might want to reflect deeper upon this. What would happen if you are not quite doing ideal sampling, but we have realistic sampling in which we use pulses instead of ideal impulses? Could not materially change all this? There would be small changes to be made. Anyway, now let us come back to the main discussion. So, a particular component is multiplied by this factor and the magnitude of this factor looks like this. Let us now draw the magnitude focusing only on the normalized angular frequency between minus pi and pi. What is the interpretation now? Let us now go back. This is the normalized angular frequency as I have drawn it. Let me go back to the original cycles per second frequency. This is 5 kilohertz in the original cycles per second frequency. So, 4 kilohertz is 4 by 5th the way there and we agreed that notionally the male voices would be in this part of the spectrum and the female voices which I will show in blue would be predominantly in this part of the spectrum. So, what is the system doing? If you look at it in this part of the spectrum, the magnitude is higher. This is the part of the magnitude that influences the male voices and this is the part of the magnitude that influences the female voices. So, if you pass this particular combination of male and female voices in the speech signal through the simple system which simply takes an average of each sample with the immediately passed sample, it would relatively enhance the male voices and suppress the female voices because the magnitude response in the range of the female voices is slightly lower than the magnitude response in the range of the male voices. Now, can you describe an equally simple system which would do the reverse? Enhance the female voices and suppress the male voice. I am going to give you a hint and I am expecting in fact, I will just work it out for a couple of steps and I am going to leave it to you as an exercise to complete the work. So, let me do that. Let me now give you a system. The hint is suppose you subtracted instead of adding. So, you had y of n is x of n minus x of n minus 1. What would it do to erase the power j omega n? It would do the following. It would do erase the power j omega n minus erase the power j omega n minus 1 into half. How do we rewrite this? We rewrite this like this. Now, I leave it to you to work out the rest. What is the rest? Find out its magnitude and phase and tell us what it does. Interpret. This is an exercise for all of you. And with this then, I shall leave you to work out this exercise. Wish you all the very best and we will meet again. Thank you.