 So, one will come to the 30th session of the second module in the core signals and systems. We shall continue in this session to build up on the properties of the Fourier transform. We had seen one property, a very important property of the Fourier transform in the previous session, namely linearity. So, linearity essentially says that if I take a linear combination of two signals who have a Fourier transform, the Fourier transforms are also linearly combined in the same way. And I had left it for an exercise for you to do to prove the same thing for the inverse Fourier transform. Now, let us take up the second property, namely what happens when you shift a signal in time. So, property number 2, time shift, what happens when you shift a signal in time. So, if h of t has a Fourier transform of capital H of omega or capital H of f, remember we are going to talk both the language of angular frequency and in the language of cycles per second frequency. What is the Fourier transform of h t minus tau 0? Of course, tau 0 is a constant. So, this is the question that we ask, very simple. Let us derive it. All that we need to do is to find the Fourier transform h t minus tau 0 e raised to the power minus j omega t dt integrated from minus to plus infinity. Put t minus tau 0 is equal to lambda whereupon t is lambda plus tau 0 and dt is the same as d lambda. And when t runs from minus to plus infinity, lambda also runs from minus to plus infinity. And therefore, this integral simply becomes minus infinity to plus infinity h lambda e raised to the power minus j omega lambda plus t 0 tau 0 d lambda, which you can now separate. So, you can bring e raised to the power minus j omega tau 0 outside and leave the rest inside h lambda e raised to the power minus j omega lambda d lambda. And notice that this is essentially capital H of omega or H of f, whatever you like to call it. And therefore, the Fourier transform of h of t minus tau 0 is e raised to the power minus j omega tau 0 times h of omega or in the language of f, e raised to the power minus j 2 pi f tau 0 h of f. So, essentially the magnitude mod h omega is the same as the magnitude e raised to the power minus j omega tau 0 times h omega. The magnitude is unchanged and the role played by this quantity, this multiplying factor. This adds a phase of 2 pi f tau 0 or omega tau 0 to the Fourier transform. So, it is very interesting. When you shift a function in time, the magnitude of the Fourier transform is unchanged. What changes is the angle or the phase of the Fourier transform. And in fact, let us plot the change of phase of the Fourier transform as a function of the angular frequency, which sets a very interesting idea that we will see now. So, we will notice that the change of phase as a function of omega is essentially omega tau 0. In fact, minus omega tau 0 to be precise. So, it is a straight line passing through the origin with a slope of minus tau 0. This is what is called linear phase or linear phase change. This is very interesting. So, when the change of phase as a function of frequency is linear in frequency, the implication is that the signal is shifted. Now, let us understand this change of frequency being linear. What is the physical meaning of the same? Let us take a single sine wave and let us see what happens when we shift at all frequencies by the same tau 0. Let us take one sine wave first. So, let us consider a sine or cos omega 0 t plus phi 0. Let us make it a 0 and let us shift this by tau 0. So, it becomes a 0 cos omega 0 t minus tau 0 plus phi 0. Now, we can identify the change of phase here. So, I rewrite this as a 0 cos omega 0 t plus phi 0 minus omega 0 tau 0 and pull out this part as a change of phase. The change of phase is minus omega 0 tau 0. As you notice, this change of phase is linear in frequency omega 0. So, if I want all frequencies to be shifted by the same time, I need a phase change which is proportional to the frequency. That is why linear phase has an association with shifting the entire signal by the same time shift. Now, I emphasize this point so much because linear phase is something that we keep yearning for. You will hear this term if you go ahead in designing systems, particularly filters. If you talk about channels, if you talk about systems for communication, for control, whatever it is, you will often hear the term linear phase. And right at this point, you should be clear why people are so fond of linear phase. Linear phase has the interpretation of shifting all sine waves by the same time shift. Now, visualize a situation in which you have a signal comprised of many sine waves and if you want to shift the signal in time, of course, all these sine waves should be shifted by the same time. That means, the phase changes should be proportional to the frequency. If they are not, then different sine waves should be shifted by different amounts in time and then you have a phenomenon called dispersion. So, let us write that down. So, if for whatever reason the phase change is not linear in omega, we get a phenomenon called dispersion. Different sine wave components are shifted by different times. Now, dispersion is a term that you will often hear, for example, when signals are transmitted on optical fibers or on other communication media. Actually, dispersion is a suggestive word. You know, typically what happens is you have reasonably narrow pulse and you know what kind of sine waves come together to form pulse. We saw that in a previous session where you made a decomposition of a rectangular pulse into its Fourier transform. In fact, it has frequencies of all magnitudes and signs. That is what we realized, except for certain nulls and now if you subject this pulse to a system which does not have a linear phase character whose frequency response is not linear in phase, what is going to happen? Different sinusoids which come together to form the pulse are going to be shifted by different times. So, the pulse spreads effectively. That is why it is called dispersion. Anyway, you will hear this term. For those of you who go further, taking this course into a course on communication systems or a course in wireless communication or mobile communication or control, you will understand better the idea of non-linear phase and linear phase. But I thought it is expedient to explain the importance of this idea of linear phase right at this point in time. Now, let us look at a reversal of roles here and we will take that reversal of roles further in a subsequent session. Now, here you have multiplied the Fourier transform by a phase changing term e raised to the power minus j omega tau naught. Suppose, we reverse the roles of time and frequency. So, we multiply the time function by a rotating complex number. Let us see what happens. In fact, we will give it a name. We are now going to look at the next property. That means, where we modulate. So, let us multiply h t by e raised to the power j omega naught t. Omega naught is a constant and of course, real. And let us query what is its Fourier transform given that h t has a Fourier transform of capital and h omega. The answer is very simple. Indeed, integrate from minus to plus infinity e raised to the power j omega naught t times h t e raised to the power minus j omega t dt. And very easily you see that this is essentially h t e raised to the power minus j omega minus omega naught t dt which is simply h omega minus omega naught. And there you are. We have the answer. So, it is very interesting. When we modulate, modulate means we multiply h t by a rotating complex number rotating with an angular velocity of omega naught. The Fourier transform is shifted by omega naught forward. Now, of course, omega naught could be positive or omega naught could be negative. In either case, the principle is the same. If omega naught is positive, then indeed we are shifting the Fourier transform forward by omega naught. If omega naught is negative, then we are in principle shifting the Fourier transform backward by mod omega naught. Simple. In any case, we are shifting. Now, you know, you are beginning to notice something interesting here. I have a property, property 2 which talks about what happens when I shift a function in time. It causes a modulation in the frequency domain. You know, you remember it multiplied the Fourier transform by a term e raised to the power j omega tau naught. When I reverse the roles of time and frequency, the consequences also are reversed. That is interesting. So, I modulate the function in time by rotating complex number and the corresponding Fourier transform is shifted in frequency. Shift in time becomes modulation in frequency. Modulation in time becomes a shift in frequency. We are beginning to see something interesting here. And we make that formal in one of the subsequent sessions. It is a very beautiful property of the Fourier transform. We shall see more properties of the Fourier transform in the sessions to come. Thank you.