 The sequences are essentially functions of an integer variable and the physical interpretation of the integer variable is the sampling index. I said functions maybe a little carefully, but we need to qualify whether I am talking about real functions or complex functions. Well, I am going to allow complex functions and again the natural question is what is the physical meaning of a complex functions. If I sample a real signal, I am certainly not going to have complex values. So, where would complex signals come from? Well, complex signals for the moment should be thought of as hypothetical extensions of real signals. So, you could think of them as two real signals, one for the real part and one for the imaginary part and we will use complex signals as a tool to deal with real signals. For the moment, let us think of it with this reassuring note. So, we do not really have complex signals in nature, but we will allow the sequence to be complex because sometimes it helps us to deal with real signals by bringing in complex signals for a while, doing some work with the complex signal then going back to the real signal. Now, in a minute we are going to use a complex signal. So, you see the complex signal comes in as follows. Suppose I were to take a complex number rotating a complex number which maintains a magnitude of m, starts with an angle of theta naught and rotates an angular velocity of omega naught. We know how to write this complex number in polar form. The polar form representation of the complex number is very simple. It is m e raised to the power j theta naught plus omega naught t. Of course, this is the continuous complex number. Now, let us assume that I have to sample this complex number with a sampling rate of 1. I would get m e raised to the power j theta naught plus omega naught times n. You see all the time I am making the convenient assumption that my sampling interval is unity. Let us get used to this assumption. I mean I am normalizing my time measurement with respect to the sampling interval. Now, you see the beauty of this representation is that if I were to change its amplitude or its phase. So, if I were to change the amplitude or the phase initial phase theta naught, if I were to change the amplitude m e raised to the power j omega naught n plus phi naught is of course, expressible as m 1 by m times e raised to the power j omega naught n plus phi naught. On the other hand, this is of course obvious. So, when I change an amplitude to m 1, it is equivalent to multiplying that number by m 1 by m, simple enough. But the more interesting thing is the phase. So, if I were to change the phase to theta 1, this could be rewritten as m e raised to the power j omega naught n plus phi 0 plus phi 1 minus phi 0. Of course, that can be rewritten as m e raised to the power j omega naught n plus phi naught e raised to the power j phi 1 minus phi 0. So, in other case, you have a multiplying factor here. Whether you take a change of amplitude or a change of phase, it amounts to multiplying the original rotating complex number by a constant, albeit complex. This is a significant observation. Unlike a sinusoid where when I change the phase, it is not tantamount, it is not the same thing as multiplying the sine wave by a constant. But in a sine wave, of course, when I multiply the amplitude by, when I change the amplitude, it is equivalent to multiplication by a constant. So, therefore, the beauty of dealing with this rotating complex number is that multiplication by a constant factor, albeit complex, is a correct description of change of amplitude and change of phase. And of course, change of amplitude and phase together too. If I change and they are independent, you know, if I change the amplitude to m 1 and the phase to theta 1 or phi 1, it is equivalent to multiplying by m 1 by m e raised to the power j phi 1 minus phi 0. So, you can take them together. That means, when I now put, it is very clear, if I have a system, instead of dealing with sinusoids now, if I were to deal with these, which we will call phasors. So, we call such a rotating complex number a phaser with a single angular frequency, it is denoted by omega. Now, why are phasors important to us? The real and imaginary parts of phasors are sinusoids with frequency omega, angular frequency omega. Is that clear? That is straightforward because we can even write down the real imaginary parts. Indeed, if you have a phaser, m e raised to the power j omega 0 n plus phi 0, this can be decomposed as m cos omega 0 n plus phi 0 plus m times j sin omega 0 n plus phi 0. This is the real part and this is the imaginary part. Each of them is a sinusoid of the same frequency omega 0. So, now, I see the connection.