 Now, what can we say about the region of convergence? The region of convergence has nothing to do with omega, it has only to do with R. So, the region of convergence has always to do with circles centered at the origin. What is R? R is the radial distance of the complex number from the origin. So, the convergence or non-conversions of the Z-transform has only to do with the radial distance of the points that you are considering from the origin of the Z-plane. And therefore, the regions of convergence of the Z-transform are always essentially between concentric circles with the centers located at the origin. That is a very important observation. Regions of convergence of the Z-transform are always between concentric circles located with centers located at the origin. So, essentially a function of R, not of omega. Now, it should be noted that one of these circles could go all the way up to infinite radius and one of the circles can go all the way down to 0 radius. So, the circles do not need to have non-zero radius or finite radius. Also, it is a mood point whether the circles themselves are a part of the region of convergence. The circles that bound the region of convergence may or may not be a part of the region of convergence. That needs to be seen case by case. This is particularly a tricky point when you are considering infinite radius and 0 radius. There is an issue there. Let us take an example. Let us take the sequence h n is equal to delta n, the simplest possible sequence that we can conceive of, the unit impulse sequence. The Z-transform of this sequence is very easy to see. It is essentially 1 into Z raised to the power minus 0 which is 1 and this holds true for all Z. And therefore, the bounding circles have 0 and infinite radius. Essentially, the region of convergence lies between the concentric circles of radius 0 and radius plus infinity. The entire Z plane is the region of convergence here. However, in this case the two boundaries are also a part of the region of convergence. The circle of radius 0 and the circle of radius infinity are parts of the region of convergence. Now let us take the sequence h n is delta n minus 1. It is very easy to see that the Z-transform is Z inverse and it holds true for all Z, but Z equal to 0. Again, the region of convergence is bounded by circles of 0 and infinite radius. However, the circle of radius 0 is not included in the region of convergence, whereas the circle with radius infinity is. So, you see it is a very tricky point. The bounding circles may or may not be in the region of convergence. That must be carefully seen. Again, let us take another such example. Let us take h n is delta n plus 1 and it is very easy to see that the Z-transform is simply Z holds true for all Z, except Z tending to infinity. So, the ROC is again bounded by circles with 0 and plus infinity radius. Very interesting. So, here we have all of the Z-plane included. The same boundaries, the boundary Z equal to 0 or R equal to 0 is included in the region of convergence, but the boundary, the circle with infinite radius is not included. I must also draw the attention of the class to one difference between the notion of infinity for reals and infinity for complex numbers. The infinity in complex numbers is not a point or a set of points. It is a contour. In fact, it is several contours. Essentially, an ever enlarging contour takes you to the infinity on the complex plane. So, infinities on the complex plane are several ever enlarging contours. The circle is a possibility. As the circle enlarges and continues to enlarge, it takes you to the infinity on the complex plane. Any other contour, any other closed contour, as it expands indefinitely, takes you to an infinity in the complex plane. So, the notion of infinity in the complex plane is a little different, a little more general, a little more intricate than the notion of infinity on the reals. Anyway, this is essentially about the reach. So, you see the Z-transform. Now, this of course illustrates that you have boundaries which are concentric circles. The boundaries need to be checked for inclusion or exclusion. But now we need to explain why it is important to specify the region of convergence along with an expression. Although sometimes we take that for granted, it is not correct to do so. One must always specify a region of convergence along with an expression. Let us take the very same sequence that we had or let us take the very same Z-transform that we had a minute ago. We had the Z-transform 1 by 1 minus half Z inverse with mod Z greater than half. Now here of course, it is very clear that Z equal to half and that means mod Z equal to half can never be in the region of convergence here. That is because mod Z equal to half is a singularity, is a contour with a singularity. You see at Z equal to half, this expression diverges. This expression has no meaning. The denominator becomes 0. That is called the point of singularity where the expression is unevaluable. Now, you see when a contour holds a singularity on it, then that contour cannot be in the region of convergence. A contour which hosts a singularity cannot be in the region of convergence. And therefore, we have only two possible regions. You see the contour must be excluded when there is a singularity on a contour in the expression that contour cannot be in the region of convergence. So, the only choice is as far as the region of convergence must be, you see it must be what is called a simply connected region. You cannot have parts of it. You cannot have a region of convergence separate with some you know part which is in no man's land, no. It must be a simply connected region. That means, if you take any two points in that region, any contour joining the two points must all lie within the region. So, for example, in this case, it is very clear that we cannot allow the circle with radius half to be in the region of convergence. So, either you have the region of convergence within mod Z equal to half or outside of mod Z equal to half. And we have chosen the region of convergence outside mod Z equal to half, but we could choose it within mod Z equal to half 2. So, let us see what happens if you take the region of convergence to be mod Z is less than half. In that case, what we are essentially saying is we multiply the numerator and denominator by 2 Z. In fact, minus 2 Z if you please. And in fact, it is very clear that mod 2 Z is less than 1. Since, mod 2 Z is less than 1, 1 by 1 minus 2 Z can be expanded using the idea of a geometric progression. 1 by 1 minus 2 Z can be expanded as 2 Z summation n going from 0 to infinity 2 Z to the power of n as mod 2 Z is less than 1 by using the simple idea of a geometric progression. And we can write this out explicitly. So, we have 2 to the power 0 Z to the power 0. So, 1 plus 2 Z plus 2 squared Z raised Z squared plus and so on. And of course, we can now see H Z 2. H Z is very clearly minus 2 Z times this. So, it is minus 2 Z minus 2 squared Z squared minus 2 cubed Z cubed and so on so forth. And it is very easy to see that this is essentially the Z transform of the sequence H n is equal to minus 2 raised the power of n for n less than less than 0 strictly. And 0 for n greater than equal to 0. In fact, I leave it to you to verify that this can be written as H n is minus 2 raised the power of n u minus n minus 1. I mean the way we have arrived at that conclusion is that you have a summation here. You can identify Z is the term or Z is contributed by the sample at n equal to minus 1. Z squared is contributed by the sample at n equal to minus 2 and minus 3 and so on so forth. And of course, you can see there are no samples from 0 onwards. And of course, this is a matter in point. If you look at u minus n minus 1, u minus n minus 1 is expected to be 1 whenever minus n minus 1 is greater than equal to 0 or n less than equal to minus 1, is not it? And 0 else as we expect. So, the same expression 1 by 1 minus half Z inverse when associated with the region of convergence mod Z greater than half leads to one sequence and when associated with the region of convergence mod Z less than half leads to another. Now, in fact, we can have even more complicated situation. We can have more than two possibilities for the region of convergence that we see later. And then we have more than two possible sequences if we do not specify the region of convergence properly. But we will come to that later. We will come to that or we will see examples of that later when we look at a few more properties of the Z transform.