 Welcome to the 28th session in module 2 of the core signal and systems. We will now take a few examples of the calculation of the Fourier transform and then build upon the properties as we had promised to do. So, let us take one of the most traditional functions. Let us begin with what is called the rectangular pulse. We make the pulse symmetric, height a going from minus t by 2 to plus t by 2 on the time axis. We call this function h of t or x of t, whatever you like. We find its Fourier transform. So, the Fourier transform capital h of omega is going to be given by integral from minus to plus infinity, h t e raised to power minus j omega t dt and that is the same as minus infinity to plus infinity. Well, or a rather for minus t by 2 to plus t by 2 0 else, you know this is the function multiplied by e raised to power minus j omega t dt and that is essentially integral from minus capital t by 2 to plus t by 2 a e raised to power minus j omega t dt, a very easy integral to evaluate. So, you have a e raised to power minus j omega t by 2 minus e raised to power j omega t by 2 divided by minus j omega and we could multiply and divide by t by 2. So, let us simplify this. So, you have 2j sign, you know you can look at the numerator, numerator essentially is 2j times sign omega t by 2 divided by j omega t and now the 2 can be brought down. So, we have a t and the j can go away, sign omega t by 2 divided by omega t by 2 and this is a very special function. We can rewrite it in the following way, a t sign 2 pi f t by 2 divided by 2 pi f t by 2 where omega is essentially 2 pi f. So, you know f here corresponds to the cycles per second frequency and this is of course, the angular frequency as you know capital omega. So, the expression that we have here is a t sign pi, we will take pi outside, pi gamma let us say pi gamma divided by pi gamma where gamma is f t. Now, this function sign pi gamma by pi gamma is called the sink function. The sink function is an oft repeated and oft used function in the context of signal resistance. It would be a good idea for you to get very familiar with this function, understand it very well. Let us sketch this function now. So, this is how the function would look as a function of gamma. You can see that this will tend to 1 as gamma tends to 0 and it would have nulls at every integer. So, at 1, at 2, at 3 and so on. The function is also even. So, as you see the function decays in its oscillation as gamma increases both on the positive side of gamma and on the negative side of gamma. At every integer it has a null or a 0. At gamma equal to 0 it takes the maximum magnitude of 1 and subsequently the magnitude only keeps going down. I mean the magnitude as in the magnitude of oscillation and also the maximum value is never reached again. The maximum value of 1 is never reached again. So, you could call it a kind of damped oscillation, but damped in the sense of 1 by gamma. It is damped as the reciprocal of gamma. A somewhat slow damping as we understand in signals and systems. We are more used to what we call exponential damping. You know, natural circuits when they have damped responses produce exponential damping. This one is kind of slow compared to exponential damping. Anyway, it is a damped sinusoid of some kind. So, let us summarize the properties of this function. Let me in fact mark them right here. Even function damped sinusoid and nulls at the impedance. These are the three things that we need to remember. And therefore, we could now draw the Fourier transform of the rectangular pulse. Let us draw the rectangular pulse and its Fourier transform together. We must draw the Fourier transform as a function of f. So, we have its Fourier transform h of omega or h of 2 pi f, if you want to call it that. Now, let us make a few observations about this. See, the first thing is that the function is limited in terms of its occupancy of the time axis. It goes only from minus t by 2 to plus t by 2. On the other hand, the Fourier transform in principle lasts all over the frequency axis. Now, this is something, this is a function which is oft discussed and oft quoted to illustrate that because you want to make the function vanish all over the time axis outside a certain finite interval, you have to bring together frequencies of all magnitudes and signs except for a few nulls. Note that frequencies of all magnitudes and signs f going from 0 to plus infinity and also f going from 0 to minus infinity except for the points where there are nulls. Those frequencies are not required. All these frequencies need to come together in appropriate amplitude to form this rectangular pulse. Now, the beauty is h of omega or h of f is potentially a complex number, but here the Fourier transform turns out to be real. That means the phase is either 0 or pi. If it is negative, the phase is pi. If it is positive, the phase is 0. So, look at the Fourier transform here. The Fourier transform has a phase of 0 here and a phase of pi here and of course a phase of 0 here, pi here and so on. What is more? If we look at the Fourier transform itself, let us see how it varies as t varies, varying t. So, let me draw one for a certain t and then let me halve the t. Let us see what happens with a certain t. And now on the same graph, I will show what happens when I halve the t. If t is halved, this frequency is doubled. So, this moves here. Moreover, this now becomes a t by 2. So, in fact, this will also be brought down. Let me correct and draw the correct plot in a dotted form. Let us not be too fussy about the drawing. It is just indicative. So, what happens when we halve t? The nulls are doubly spaced and the central amplitude is halved. Of course, you can similarly see what happens when we double. If we double t, the nulls will be half-spaced. So, now you will have nulls at 1 by 2t and so on. So, you will have twice the number of nulls in the same interval. And of course, the central height will also double. Now, you can see there is some intuitive logic in all this. Let us look at this curve once again here. You see, take the black curve and the red curve here. In the black curve, let me mark or identify or kind of emphasize what I call the main loop. Let me emphasize that main loop. And let me emphasize the same main loop when t is halved. You notice that when the t is halved, when we halve the value of t, the main loop width is doubled and the main loop height is halved. So, in some sense there is a conservation of area if you might want to call it that loosely. Let us not be too fussy. Let us not be too strict on our interpretation. But it essentially says that the main loop tries to conserve area in some sense. And this is also true of the side loops. Let me point out the side loops. So, the other loops that you see here for example, this one is a side loop here. Let me mark it. This is a black side loop and correspondingly you have red side loops too. So, you notice something interesting. As you change t, there is a tendency for these loops, the main loop and the side loops to conserve their areas. It will lose asymptotic sense. What is interesting and important is that when you stretch the pulse in time, that means doubling t for example, the Fourier transform is compressed. And when you compress the pulse in time having t, you stretch the Fourier transform. This is a very important property of the Fourier transform. So, that is also kind of intuitive. We will see more properties of the Fourier transform as we go to the next session.