 Now, we need to see a few more properties of the discrete time Fourier transform. You see, we now need to take a minute to see where we are in our whole strategy of dealing with signals and systems, discrete systems and discrete sequences. A sequence was a mapping from an index set. So, it could be a set of integers, field actually, here the field of complex numbers. Specifically, we are talking about the index set of integers and the complex numbers as the field. We also appreciate why we have introduced complex numbers because we do want to have rotating complex numbers and physically, we sometimes need to deal with electromagnetic fields where it is easier to deal with complex phasors to represent the fields and there are many other such situations where it is a natural choice to use the complex field to deal with the situation. Anyway, this is a sequence. What is a system? What is a discrete system? A discrete system is a mapping from sequences to sequences. So, you have a system. Where you have an xn input sequence and output sequence yn and this mapping between sequences. It is important to understand a system as a mapping from sequences to sequences. Of course, if the system is memory less, then it also becomes a point wise mapping, but it need not be a point wise mapping. You must think of it as a mapping of the whole sequence xn to the whole sequence yn and you do this for all the sequences which are permissible inputs. Now, we have come to a transform. What is a transform? A transform, at least the discrete time Fourier transform is an example. It is a mapping of this whole paradigm, of this whole setup, of this whole paradigm worldview. It is a different way of viewing the world of sequences and systems. What I mean by that is when we take, for example, the LSI system, the transform acts on this, the transform acts on this and the transform acts on this to give you x omega here, h omega here and y omega here and the relation between xn, hn and yn gets transformed into a different relationship between x omega, h omega and y omega. So, relationship also undergoes a transformation. y is equal to x convolved with h gets transformed to y is equal to h times x and of course, this relationship is much more elegant. So, transform to mean often offers us the possibility or the potential to deal with that same setup in a much more efficient manner. The discrete time Fourier transform is a strong example. We shall see another transform after a while. The discrete time Fourier transform is admissible only when x, h and y, all of them have discrete time Fourier transform. That means physically you could think of it at the dot product of the input, the input response and the output on each of those rotating phases e is the power j omega in converges for all these omegas between minus pi and pi. And you also see the transform in the inverse transform. This transform is invertible. You can go from the sequence to the transform, you can come back from the transform to the sequence. What we need to do is to look for a few properties of this discrete time Fourier transform. Of course, we have derived one property right away. We have seen that when we convolve two sequences, the discrete time Fourier transforms are multiplied if the DTFT of the convolution converges. But now let us focus on the DTFT of one sequence and make some transform, make some changes to that sequence. So, let us begin with some properties. You see the DTFT can also be thought just as we thought of convolution as an operation in its own right. You can think of the DTFT as an operation in its own right. So, Xm is operated upon by the DTFT to produce X omega. Of course, here the operation takes you from one independent variable to a different independent variable. So, in a transform, the independent variable changes and we say that the domain has changed from the natural domain to the transform domain. In this case, the transform domain is the frequency domain and the physical interpretation of the frequency domain is it corresponds to the set of angular frequencies, normalized angular frequencies of the phasors that come into picture when that sequence needs to be constructed. Anyway, treating this as an operation in its own right, we can ask is the DTFT linear? In other words, if I took X1 and 2n respectively and had their DTFTs, X1 and 2 of omega, what would be the DTF alpha times X1n plus beta times X2n for any constants alpha and beta. And the answer is very simple, it would be just alpha times X1 omega plus beta times X2 omega. I will leave it to you in exercise to prove this. So, that is easy to do. You just write down the expression for the discrete time Fourier transform of this linear combination. It is easy to prove. So, the discrete time Fourier transform is a linear operator. What happens when we complex conjugate a sequence? Or before that, let us make life a little easier by doing something slightly simpler. What happens when we time reverse? So, if I take X of minus n, what do I expect will happen? Now, here again we can answer the question by intuition first and then prove it algebraically. If you time reverse, what happens when you time reverse or rotating phasor? If you time reverse or rotating phasor, it rotates in the opposite direction. And therefore, if you essentially rotate all the phasors in the opposite direction, you get the time reverse sequence. What it means is X minus n should lead us to X minus omega. All phasors rotate in the opposite direction. But let us prove it algebraically. Let us also prove the same algebraically. Indeed, summation n going from minus to plus infinity X of minus n e raised to the power minus j omega n can be rewritten as summation n going from minus to plus infinity X n e raised to the power minus j minus omega times n, where n is minus n. You see, it is very easy to see. If you put n equal to minus n, when n runs over all the integers, so does n. Is that right? And this is simply X of minus omega. She approved. We need to see several such other properties of the discrete time Fourier transform. And we shall do so in the next lecture. Thank you.