 A warm welcome to the second session of the fourth module in the course signals and systems. We are now embarking upon a generalization of the transform domain and in the previous session we have given the basic idea that we are going to use in this generalization. The basic idea is to capture or hold the input or the impulse response or the output signal in general hold it down or capture its growth in such a way that it is then Fourier transform. It is very simple. We are not asking for too much. We are saying you just need to hold that signal down by a decaying exponential and if you do that then you can take a Fourier transform. In fact, this also leads us to a more general way of dealing with systems that are amenable to Fourier analysis. So, the methods that we developed here are not only the systems that are not amenable to Fourier analysis, but also to systems that can be analyzed using the Fourier transform. They can also be captured of course and then dealt with with the Fourier transform more general. Now, all this was nice. We gave an example to explain what we meant by capturing or holding the signal down or ensnaring it, but now we must make this whole discussion more concrete by putting down a process that we are going to follow. So, let us begin first with a signal where we do not have a Fourier transform. In fact, let us begin with the signals that we took yesterday as examples in module 1. And let us then use the same idea, capturing holding down, but let us make it a little more specific and say what are the functions that we are going to use to hold down and what conditions must they obey. So, let us come down to the two examples, one discrete, one continuous. For variety, let us begin with the discrete example this time. So, we have the discrete sequence. Let us call it x of n given by 2 raised to the power of n u n. So, essentially x of n is equal to 2 raised to the power of n for n greater than or equal to 0 and 0 for n less than 0. And obviously, it does not have a Fourier transform. It does not have a discrete time Fourier transform, but then we can use that strategy. We can capture its growth by using an exponential that decays faster. And when we do that, we can then identify what would happen to that captured sequence in the Fourier domain. So, let us do that. Let us now multiply x of n by let us call it r raised to the power of n. So, you have 2 raised to the power of n times r raised to the power of n. In fact, let us make it r raised to the power minus n. So, essentially r is a capturing factor. Now, under what conditions on r would we have a discrete time Fourier transform for the sequence? So, let us ask this question. Now, you know without any laws of generality, consider r to be a non-negative real, in fact, a positive real constant. Now, for it to have a discrete time Fourier transform, what are we asking for? We are asking for the following summation to converge. And in this case, we have course n runs from minus to plus infinity. In this case, the summation is essentially 2 raised to the power of n into r raised to the power minus n into e raised to the power minus j omega n summed over all positive values of n, which of course, can be simplified. Now, you can see this is a geometric progression. It is a geometric series of infinite lengths with a common ratio equal to 2 r inverse e raised to the power minus j omega. And if you want this series to converge, the common ratio must have a magnitude less than 1. How does that reflect then? We are essentially saying that mod 2 r inverse e raised to the power minus j omega must be less than 1, but then mod e raised to the power minus j omega is just 1. So, therefore, mod 2 r inverse is less than 1, which essentially says mod r is greater than 2. Now, in fact, this tells us the condition on r, but more generally, if we look back at what we did, let us go back to what we had done in terms of writing a summation. What have we done here in this expression? Let us combine these two factors together and let us write this as z raised to the power minus n, where z is r e raised to the power j omega. Now, remember z is complex. In general, z is complex. And in fact, now we have an interpretation for r. r is essentially the magnitude of z. So, what are we really saying here in this? We are saying the magnitude of z must be greater than 2. And how can we sketch this region in the complex? You see, how do you represent complex numbers? You represent them by drawing a complex plane. Just as you can represent real numbers on a real axis, you need to draw a two-dimensional plane is required to represent complex numbers. And you can show a region in this complex plane, which corresponds to this inequality. So, what are we talking about? We are talking about this complex plane here. Let us call it the z plane. We are talking about a circle of radius 2. And the circle is essentially mod z equal to 2. So, mod z greater than 2 is essentially the region exterior to the circle. Now, you know what is very clear is that this convergence or lack of convergence has only to do with r. It has nothing to do with omega. So, it is the magnitude of z, which determines whether this series or this sum converges or not. In fact, we give this region a name. This region, which is exterior to the circle where this converges is called the region of convergence. And we often abbreviate it by R, O, C for short. And now, we will say that we have a more general transform. We call this the z transform. So, it is the z transform of the sequence xn. And of course, to keep some consistency with our earlier notation, we shall write capital X as a function of z being summation over all n, all integer n, xn times z raised to the power minus n. This is called the z transform of the sequence xn. Now, the z transform is both an expression and also region of convergence. Both are important. So, in this case, this is the z transform, but we can rewrite it. And now, we know how to sum it provided the region of convergence condition is satisfied. So, in fact, this in total is the z transform. You know, the z transform now needs to be specified both with an expression in z and a region of convergence in the z plane. Now, let me illustrate to you why it is important to have a region of convergence along with the expression. Can we separate the region of convergence or can we ignore the region of convergence and yet remain specific to the sequence? We will soon see the answer is no. In fact, let us take a sequence that goes the other way. You know, so here you have 2 raised the power of n and it goes from 0 towards plus infinity. Let it go the other way. So, to explain the importance of the region of convergence, let us consider another sequence. Let us call it x 1 n which is minus 2 raised the power of n for n less than equal to minus 1. So, you know, it is complementary in some sense and 0 else. So, it begins at minus 1 and goes backwards and it has the value minus 2 raised the power of n. So, the same exponential sequence, but now it goes backwards starting at minus 1. Let us consider its z transform. Obviously, z transform needs a summation from minus infinity to minus 1 and you can expand this. You know, you can write this as 2 inverse z plus 2 raised the power minus 2 z squared plus and so on. So, it is very simple. It is a geometric series again with a common ratio equal to 2 inverse z and you need this common ratio 2 inverse z to have a magnitude less than 1 which means mod z now needs to be less than 2 less than 2 remember and now you can sum it you can sum this. So, x 1 z would then be minus 2 inverse z divided by 1 minus the common ratio and you can multiply by 2 z inverse numerator and denominator and there we are if we do that this would become. So, when you multiply by 2 z inverse be minus 1 divided by 2 z inverse minus 1 which is 1 by 1 minus 2 z inverse very interesting. So, x 1 z and x z have the same expression 1 by 1 minus 2 z inverse, but a different region of convergence that is very interesting. So, we need to deal with z transform with a little more care and we will see more in the next session. Thank you.