 Welcome to the fifth session in the second module in the course on signal and systems. We will continue from where we left in the previous session to talk about how we could deal with a phase change intelligently when we have a sinusoidal input. Recall that a phase change was the problem as compared to a magnitude change in a sinusoid. The phase change did not result in a pure multiplication of a sinusoid by a constant independent of time if one dealt with the sinusoid in its primitive state. But if one considered the sinusoid as a combination of two rotating complex numbers if we call phasors rotating in opposite directions you could focus your attention on any one of those rotating complex numbers effect the change of phase there and then deal with both of them together. Let me illustrate this by taking any one of those complex numbers. So, let us consider one phasor let us call that phasor a e raised to power j omega t plus phi. Now let us make a phase change of delta phi which means we have a e raised to power j omega t plus phi plus delta phi. Now this can be rewritten as a e raised to power j omega t plus phi multiplied by e raised to power j delta phi and this is the important thing. A change of phase is just a multiplying factor albeit a complex multiplying factor that is not really our problem but it is just a multiplying factor and that is why we could in the context of electrical circuits R, L, C circuits treat the voltage current relationship as constant independent of time if the excitations were sinusoidal. So, you could look at one of those rotating complex numbers. Now let me come back to this particular phasor here you see if you look at it in this phasor we have a e raised to power j omega t plus phi here you have anticlockwise rotation. Now let me take the corresponding complex conjugate phasor. So, let me take the you know what we may call the complementary or complex conjugate phasor which of course would be a e raised to power j omega t plus phi the whole complex conjugated or a e raised to power minus j omega t plus phi and here again let us effect a change of phase. Now the important thing is that when we make a change of phase of delta phi here. So, let phi be replaced by phi plus delta phi as before you would of course get a e raised to power minus j omega t plus phi plus delta phi and this is essentially a e raised to power the original phasor multiplied by a e raised to power minus j delta phi as you notice this is essentially the complex conjugate of the previous constant. So, the beauty is that if I have the same change delta phi in the phase in the two cases in one case the corresponding constant by which the original phasor is multiplied is the conjugate of the other. So, I need to look at any one of them. In fact, let us look at the sinusoid underlying sinusoid as we call it. So, if you take the underlying sinusoid here which we could write as well a cos omega t plus phi. Now you see the beauty is that you could write this as 2 a e raised to power j omega t plus phi plus e raised to power minus j omega t plus phi divide by 2 and that is a e raised to power j omega t plus phi plus a e raised to power minus j omega t plus phi and here now I could make a change of phase in this. So, let us change phi to phi plus delta phi and what is the change that occurs what we have seen. So, here this is then multiplied by e raised to power j delta phi and this one by e raised to power minus j delta phi. So, in short what I am trying to bring out is that if I make a change of phase in the underlying sinusoid then it is reflected as a multiplying factor in both of the phases which come together to form the sinusoid and I need look at only one of those phases to understand the effect of change of phase which is just a multiplying factor. The other one is just the complex conjugate the multiplying factor for the other phasor is just a complex conjugate. So, I do not need to worry about that too much. So, the work involved is not very different. I could have dealt with the original sinusoid here I am just dealing with a phasor it is so much more mathematically convenient to deal with the phasor both for amplitude changes and for phase changes and even though there is one more phasor rotating in the opposite direction I do not need to deal with it explicitly. If I deal with one of the phasors what is happening to the other phasor is known automatically. This is the beauty of phasors and now again I am emphasizing that is why we allow complex signals and complex constants in our discussions in signals and systems. You recall that I had postponed the reason for this when I talked about allowing complex signals and complex constants in the first module of the course. I had said that we have to allow complex signals and complex constants and now we understand the reason why when dealing with the very ubiquitous sinusoid it is convenient to deal with the corresponding phasor a complex signal as opposed to the original sinusoid as it is well so much so. Now the next question that we are going to ask is what happens when you put a sinusoid through a linear shift invariant system we have seen that sinusoidal signals have a special place in our minds for various reasons. Now suppose I have a linear shift invariant system and for the moment let us make our life easy by assuming the system is also stable well why am I saying the system needs to be stable because the sinusoidal signal sinusoidal input is bounded and if the system is stable at least I am guaranteed the output is going to be bounded so I do not get into trouble in that sense. So let us assume we have a stable linear shift invariant system and let us see what happens when I put a sinusoid into it. So here we go so sinusoidal input to a linear shift invariant stable system let us assume that I have this linear shift invariant stable system S script S here and let its impulse response be ht. Now of course the LSI system is stable assume ht is real and S is stable we will see how it plays a role. So I have here the input to a cos omega t plus phi being given to the system and we question what is the output. Now by itself of course I know that I could use the convolution integral let me write down that convolution integral. So output by convolution the input xt is to a cos omega t plus phi the impulse response is ht of course ht is absolutely summed absolutely in fact integrable to be more precise. Let us find the output the output is going to be yt and here I will use this form h lambda xt minus lambda d lambda which is minus infinity to plus infinity h lambda to a cos omega t minus lambda plus phi d lambda. Now you see all that I can say from the stability of the system is that yt is going to be bounded but beyond that if I look at the expression here this integral as we see it no sense can be made out of how I might simplify this integral it is very difficult to simplify. So it is not going to help for me to deal with the sinusoid as it is. In the next session we shall see how we can do much better if we just think of the sinusoid as two rotating complex numbers of phases. Thank you.