 The other way of understanding this is through the notion of eigenvalues and eigenvectors. You see what are we saying in effect, we are saying here that when you put the sequence into the system what comes out is the same sequence multiplied by a constant. Now this is not going to happen for all kinds of sequences. If you had you know a sequence which is square like in nature, square wave like in nature, it is not going to happen for LSI system in general. But for complex exponential this happens. When you put in a rotating phasor into the LSI system with impulse response h n out comes the same rotating phasor but multiplied by a constant. Now such a sequence which goes into a system and emerges as the same sequence but multiplied by a constant is called an eigen sequence of the system. Sequence that goes in that is unchanged in form as it emerges saying that the action of the LSI system is decoupled for each of these sequences. You understand this better if we can do one more thing by whatever means now. You could express any sequence, you see remember you did this earlier. You expressed any sequence x n in terms of these impulse sequences delta n minus k. So you wrote any sequence x n as summation k x k delta n minus k. So you were able to express any sequence in terms of the unit impulses. Now suppose by whatever mechanism you are able to express any sequence as a linear combination of such rotating phasors, rotating with different angular velocities. Then what we are saying here is if I take any one of those phasors that comprise the input, the response of the system to that component of the input is only the same phasor multiplied by constant. So if I decompose the input along each of these phasors so to speak for different amegas, then what I am saying is the response to the phasor at a particular angular frequency omega has nothing to do with the angular at the phasor at some other omega. They are all decoupled, they can be treated separately. It is slightly deeper issue but we will understand this better as we go along. So in other words what we are saying is I mean let me try and bring this a little more, you know it takes a little time to absorb this. What we are saying is take the same system S, LSI, HM. Suppose instead of giving e raised to power j omega n you give a1 e raised to power j omega 1n plus a2 e raised to power j omega 2n plus a3 e raised to power j omega 3n. Out would come a1 h omega 1 e raised to power j omega 1n plus a2 h omega 2 e raised to power j omega 2n plus a3 h omega 3 e raised to power j omega 3n. This is not at all difficult to show from the property of linearity or additivity and the property that when you put in e raised to power j omega n out comes e raised to power j omega n multiplied by h omega. So what we are saying is the action of the LSI system on each of these components is decoupled. I do not have to worry about what a2 and a3 are when I am calculating the output to a1 e raised to power j omega 1n. When I am calculating the output to a2 e raised to power j omega 2n I would need to worry what a1 and a3 are. They are decoupling. Now carry this argument to its limit. So if you could here I am taking a very specific input which had only 3 such e raised to power j omegas only 3 such frequencies. Suppose I use a continuum I use all the frequencies omega from 0 to 2 pi or all the frequencies from minus pi to pi. Incidentally now we make a remark about what range of values omega can reasonably take. Omega is the normalized angular frequency in the frequency axis. There is elastic or there is repetition omega between minus pi and plus pi. You see the reason for that is very simple. What does omega equal to 2 pi correspond to omega equal to 2 pi corresponds to sampling frequency. And we have agreed that every sine wave is comprised of 2 oppositely rotating phases. 1 rotate if a sine wave has frequency omega it comprises of 2 phases. 1 rotating clockwise with frequency omega and 1 rotating anticlockwise with frequency omega. In other words if you take both to be in the clockwise or anticlockwise direction 1 with frequency omega and 1 with frequency minus omega. 2 phases come together to form a sine wave. So on the phase of angular frequency axis each pair of frequencies omega and minus omega come together to form a sine wave. And obviously there are magnitudes the 2 phases have to have the same magnitude and opposite starting angle or opposite phase. Now we see in that when you sample you take the original set of frequencies or original frequency axis whatever it is shifted by every multiple of the sampling frequency and these shifts are added. On this normal scale the sampling frequency is 2 pi. So you are going to take the original spectrum shift by every multiple of 2 pi and add up these shifted versions. Let us show that graphically. Whatever there is you are going to shift this by every multiple of 2 pi and you are going to add them. Now obviously if you do not want these shifts to overlap this must remain between minus pi and pi. And of course that is also obvious from what we have seen in the sampling theorem. The maximum component that you have in the original signal should not be more than half the sampling frequency. So that means the so called unique omegas that we can deal with are only between minus pi and pi. So when we take any input x n and ask whether it can be expressed as a linear combination of e raised to power j omega n we need to worry about the omegas going from minus pi to pi. The frequency axis is continuous. I am not going to be able to use a summation now. I need to integrate the limit of a summation as the variable of summation becomes continuous even integral. So what we are saying is suppose I mean I am trying to motivate the whole idea. Suppose I take the same LSI system once again, give to it a combination x omega e raised to power j omega n. You know what I mean by x omega x omega is the component of x n along e raised to power j omega n. So what I am saying is suppose you are able to decompose just like you decompose the impulse response. Suppose you are also able to decompose the input along different e raised to power j omega n. And you did this for all the omegas going from minus pi to plus pi. Then it means that you know since minus pi to plus pi is exhaustive putting those components back should give you back x n. So what we are saying is this is what we are trying to ask is what comes out here, isn't it? And we are saying in particular when you take x omega e raised to power j omega n, what will come out? So x omega e raised to power j omega n is going to give us x omega h omega e raised to power j omega. Is that right? And therefore if you integrate this, if you integrate over omega here, you can also integrate over omega here. How can we say that? That is because of the property of linearity. If for each omega I can do this, I can do it for the combination of the omegas. It will lead us to a very interesting property of this inner product. This inner product of a sequence with the rotating phasor gives us a new domain called the discrete time Fourier domain or discrete time Fourier transform domain. We shall see more of this in the lecture to come and build up in greater depth the whole idea of the discrete time Fourier transform in the coming lecture. Thank you.