 I also see the advantage of dealing with phases. You see what I am going to do is, I am going to think of a sine wave as the real or the imaginary part of a phase and instead of dealing with the sine wave, I am going to think that it was the phasor which was applied to the discrete system and this would result in a phasor coming out with the same frequency but a possible change of amplitude in phase, right. And the beauty of dealing with the phasor rather than the sine wave is, this process of changing the amplitude and the phase amounts to multiplying that phasor by a complex constant. So, I can represent the action of the system as multiplication by a complex number at that frequency. I want to be able to describe a system like that. Now, this requires some thought. Now, there is another way of looking at it. The other way is, I can think of a, of course, I can think of a sine wave as the real or the imaginary part of but how do I get real or imaginary parts? You see, I must be able to get them with some kind of an operation. So, another way to think of it is, when I add two phasors with the same frequency in opposite directions, what I mean by that is, if I were to take two phasors, both of magnitude m with opposite initial starting angle. So, it is like mirror images. You know, the phasors are mirror images of one another. They rotate with a mirror image angular velocity. So, you have omega not for this one and minus omega not for the other one. How would I describe this one? This is of course m and of course both of them are sampled. So, I have m e raised to the power j omega not n plus theta not if you would like and here I have m e raised to the power minus j omega not n plus theta not. You know, they are complex conjugates. These phasors are complex conjugates. If they begin from the opposite starting angle, if they rotate with opposite velocity of the same magnitude and if they have the same magnitude m, then they are complex conjugates. They always remain as complex conjugates and I am adding them to two times m cos omega not n plus theta not. So, in other words, a sinusoid really hides or has in it two oppositely rotating phasors. That is another way to look at it. In many systems that I deal with in discrete time processing, it is sufficient for me to see what the system does to one of these two. They have the same frequency notionally, but with opposite signs. So, one of them has the frequency omega not, the other one has the frequency minus omega not. So, in most of the systems, in many, I would not say all systems. We will deal with some exceptions, but in many of the systems that we will deal with, at least when we implement systems in this first course. What would happen is, what that system does to one of these phasors is the mirror image of what it does to the other. So, if I take the phasor rotating in the anticlockwise. So, here I have this, you know, you go back to those two phasors. Think of this phasor being applied to the system here and then think of this phasor being applied to the system. When this phasor is applied to the system, what is going to happen is, out comes a phasor with the same frequency with a changed amplitude and phase. Apply this phasor, same thing, out comes a phasor with the same frequency, same frequency means minus omega not here and a change of amplitude and phase. Now, what is going to happen in most of the systems that we are going to deal with, at least implement is that the change of amplitude for both of these will be the same. So, they will both of them will be multiplied by the same m 1 by m. But the change of phase here, that means, if the phase starting phase goes from theta 0 to theta 1 here, the starting phase here would go from minus theta 0 to minus theta 1. So, the change of phase would be mirrored, that is what the situation would be in most cases. That means, if I know what happens to one of the phases, I know what happens. I do not need to analyze what the system does to both individually. And therefore, I can replace my analysis of the sinusoid by an analysis of what happens when I put one of those phases into the system, because I know what it will do to the other phase. And once I know what it does to the other phasor, I know what it does to a sine wave. So, what I am saying is, instead of dealing with sine wave, I prefer to deal with phasors. And what is the reason for that? Dealing with phasors means putting a multiplicative constant, multiplying that phasor by a constant. So, what is my requirement? Now, let me put down my requirement of the system. And then I am going to start discussing the properties that will give me that requirement. My requirement from the systems that I design, my requirement is, when I put in, in fact, let me do three experiments. I put in a phasor with frequency omega 0 and initial phase theta 0. Out comes a phasor, A 1, e raised to the power j omega 0 n note with plus theta 1. And therefore, the system can be described by just this ratio A 1 by A 0 times e raised to the power j theta 1 minus theta 0. I can describe the system just by this operation, just by this multiplication at that frequency. Isn't it? Experiment two, same system, but different input, B 0, e raised to the power j omega 1, some different frequency, plus phi 1, out comes B 1, e raised to the power same frequency, but of course, phase and amplitude can change. And therefore, at omega 1, at that different frequency, this is to be described by the ratio B 1 by B 0, e raised to the power j phi 2 minus phi 1. Experiment number two, experiment number three. So, same thing, but at frequency. Experiment number three, same system. Now, put in c 0, e raised to the power j omega 0 n plus phi 3, let us say, plus c 1, e raised to the power j omega 1 n plus phi 4, out comes D 0, e raised to the power j omega 0 n plus phi 5, if you like, plus D 1, e raised to the power j omega 1 n plus phi 6. And that is not all, D 0 by c 0 is equal to A 1 by A 0, D 1 by c 1 is B 1 by B 0. First, secondly, phi 5 minus phi 3 is equal to theta 1 minus theta 0. And phi 6 minus phi 4 is phi 2 minus phi 1. All this must be true. Just to remind you what these quantities are, A 1 and A 0 were the amplitudes in the first experiment. Theta 1 and theta 0 are the amplitudes in the first experiment. In experiment two, the amplitudes are phi 2, phi 1, I am sorry, the amplitudes are B 0 and B 1 and the phases are phi 2 and phi 1. What we are saying is that the way the phase changes and the way the amplitudes change remains the same when I perform the third experiment. So, what are we saying? We are saying something very profound. This is true for any omega 0, omega 1, any A 0, A 1, B 0, B 1 and any theta 0 etcetera. So, it is a very serious demand I am making of my system. In words, what is my demand? When I put in two complex phasors with different frequencies, what I must get is a sum of two complex phases of the same two frequencies. The way the amplitudes and the phases change should be the way they change when I give those as individual inputs. So, if I look at what happens to the first phasor and if I look at what happens to the second phasor in their own right without the other being present, I must be able to tell what happens when I put them in combination. Now, this is a very serious demand of the system and to be able to satisfy this demand, I will leave the system to obey many properties. I shall list the properties today and we shall tomorrow take those properties up one by one. Further this to happen, the system must be linear, shift invariant. These are only terms, these are only names at this point in time. I need to qualify what these mean. So, you see it is, I mean just to make a remark here, I am sure that many of us who might have taken a primary course in signals and systems would have heard these terms before, but what is important is to understand why we bring in these terms. It is often that we start talking about system properties without understanding why we should discuss them in the first place, why we need to discuss them in the first place is now clear. I want to build systems which have that requirement on phasors being obeyed and if that system must obey those requirements, the only way is when it can be linear, shift invariant and stable and I need to understand what these three terms mean. We will do that in the next lecture. Thank you.