 a warm welcome to the 28th session in the fourth module in signals and systems. In the last couple of sessions, we have been dealing largely with the Laplace transform and we have been looking at what in the system function or the Laplace transform of the impulse response determines stability. In fact, we saw that the region of convergence of the Laplace transform is central to stability analysis. So, one contour is the most important in determining stability namely the imaginary axis in the S plane. Now, it is natural for us to question what would happen for the discrete independent variable case namely, if I have a discrete system a system where all the independent variables are discrete and if the system is linear in shift invariant and if the system impulse response has a z transform. Then what is the z transform determines the stability or otherwise of the system? For causality we have already analyzed. The extreme contour mod z tending to infinity needs to be included in the region of convergence for causality. So, what is a similar condition for stability? Can one look at the region of convergence and come to a conclusion about stability or otherwise? First, to complete the discussion let us again convince ourselves that it is the region of convergence which is going to be central to the analysis of stability. So, let us make the point very firmly here. So, essentially the context is we are looking at discrete independent variable systems. So, we have an input x of n and output y of n. The system is linear shift invariant and we shall also assume it to be rational. That means the impulse response h of n has a z transform capital H of z with the region of convergence r of h where h of z is a rational function in z. Now, what we need to convince ourselves is that it is r h which is central to the analysis of stability. So, towards that analysis let us take an example as we did for the continuous independent variable case. Let us take the example of 1 by 1 minus half z inverse with two different regions of convergence. In fact, we know how to put these regions of convergence down. So, we have the z plane, we have the point half, there is a pole there, you draw a circle passing through this pole, center at the origin and now we have two possible regions of convergence. One interior to the circle and one exterior to the circle respectively shown in red and green. Let us invert the Laplace transform in each of these regions of convergence. In the red region of convergence the inverse Laplace transform is minus half to the power of n u of minus n minus 1 and in the green region of convergence the inverse is half to the power of n u. Now, clearly this is not absolutely summable and therefore, the system would be unstable here. If this were to be the region of convergence for the system function this would not be stable. In contrast, this is absolutely summable and the system would be stable. So, essentially we have the same expression, but the region of convergence makes it stable or unstable. So, this substantiates our argument that it is the region of convergence which determines stability or instability. The expression of course has an indirect role. You see, what is that indirect role? Let us look at now another expression and look at the two possible regions of convergence. Let us look at h of z being equal to 1 by 1 minus 2 z inverse. Let us go through the same exercise here. We draw the circle with radius 2 centered at the origin passing through the pole. We once again identify the two possible regions of convergence. The red region of convergence and a so called green region of convergence respectively interior and exterior with respect to the pole. We can write the inverse in each of these regions. So, the inverse in this region now is minus 2 raise to the n u minus n minus 1 and the inverse in this region is 2 raise to the n u n and clearly in this case, this is absolutely summable and the system is stable. In contrast, in this case, this is not absolutely summable and the system is unstable. You saw that it is of course, again the same expression, but with two different regions of convergence and one of them made the system stable the other one made the system unstable, but now the roles of the regions of convergence were reversed. So, that is the indirect role which the expression plays. I changed the expression. I had 1 by 1 minus half z inverse first and then which region of convergence corresponded to the stable system changed. Now, again this particular example points to what we are trying to get at in this whole discussion. Like the case of the continuous independent variable, is there something central to the region of convergence which allows us to choose between stability and instability. We have to look for something specific and to answer that question, let us look again at what is it that made the system stable or unstable in each case. So, for example, let us look at the previous system again. So, here you notice that the pole had a magnitude less than 1. So, there is a pole at z equal to half. If I took the region exterior to the pole, the green one here, the corresponding sequence that resulted for the impulse response was a right sided sequence. And if I took the region interior to the pole, the corresponding sequence was the left sided sequence, this one. This is right sided and this is left sided. And that is true for this next example as well. Here also the pole is at z equal to 2. When I took the region exterior to the pole, that is this region, you got a right sided sequence. That was 2 raised to the power of nu n. When I took the region interior to the pole, I got a left sided sequence that is minus 2 raised to the power of nu minus n minus 1. And the question was whether the left sided sequence was absolutely summable or the right sided sequence was absolutely summable. So, you see in either case, you saw that we have a poly-exterm like before. Here the poles were simple. So, the poly part was trivial, it was just a constant. The exponential part was non-trivial. The left sided exponential was decaying if the pole was exterium with a magnitude greater than 1. So, remember it is the magnitude of the pole which is central. That is not surprising because the regions of conversions are essentially between poles. So, you see what kinds of regions of conversions do we have in the z plane? We are familiar with that. How do you determine all the possible regions of convergence? You draw circles if you are talking about a rational system. So, for a rational system, you would draw circles passing through all the poles. Let us put that down. Let us put down the steps. What are the possible regions of convergence? So, we draw circles centered at z equal to 0 passing through each pole. We begin from z equal to 0 and move outwards. Identify all clean discs between these circles. Clean means no poles inside. Each such disc is a region of convergence. So, now you have identified how you can get each possible region of convergence. Now, you also know that any particular region of convergence is such that a given pole is either to the exterior of that region of convergence or to the interior. Now, you know you are going the other way. You are saying now I have fixed my region of convergence. The pole can either be to the exterior of the region of convergence or to the interior. Now, how does the situation change? So, for a given region of convergence with the same expression, take a given pole. It is either to the exterior or to the interior of the arrows. Now, write a partial fraction expansion. Identify terms for that pole. Invert terms for that pole. So, you know if you focus on one pole, you can clearly invert the terms knowing whether the pole is exterior or interior. If it is exterior to the region of convergence, it contributes left sided terms. And in general, if the pole is repeated, you get a poly-X term. There is a polynomial would be non-trivial, there is a polynomial in N. So, you would get a poly-X term with degree more than just a constant if the pole is repeated, but it would be a left sided exponential. In contrast, if the pole is to the interior of the region of convergence, you get a right sided exponential with a polynomial in N multiplying it. Now, this is close enough for us to determine how that pole influences stability or otherwise and we shall talk about it in the next section. Thank you.