 A warm welcome to the 29th session of the fourth module in signals and so on. We have now given a few clues in the last session to how we can determine stability in the context of a discrete independent variable system. Let us review those clues. The first thing is of course, we are talking about rational systems. Let us be clear about that. We are not talking about irrational systems. Secondly, for rational systems, look at the region of convergence. Look at one pole at a time. So, let us put that down. So, for the rational system, look at one pole at a time, identify its contribution to the impulse response. How do we identify the contribution? The contribution will be a poly-x term. The x part or the exponential part would come directly from the pole. The poly part or polynomial part would come from the terms in the partial fraction expansion. Now, we are going to see that it is only the exponential part that determines whether this contributes to absolute summability or violates. So, first we will write down a lemma. We will say if any one term, if any one of these poly-x terms has a growing exponential, the impulse response is not absolutely summable. So, what it means is that we need to establish what is called the linear independence of poly-x terms. You see, if you have one poly-x term of the form, some polynomial in n, p1 of n multiplied by alpha to the power n, another poly-x term p2n multiplied by beta to the power n. One can show without too much of difficulty and in fact, I will leave it to you as an exercise to prove this. What we are saying is we can never find constant c1, c2 so that c1 times this expression plus c2 times this poly-x term is identically equal to 0 for all n. What does this mean? What we are saying is that these two terms cannot cancel one another. So, if there is an exponentially growing term, it can never be overcome or encompassed or engulfed by an exponentially decaying term. An exponentially growing term spoils the whole game. The moment there is one exponentially growing term, it can never be overcome by any other of a different exponential parameter. And it is essentially related to this linear independence of poly-x terms. Of course, poly-x now, there is a condition here. If you look at it, we are saying proof that we can never find constant c1, c2 so that this is true provided of course, alpha is not equal to beta. That is of course, just a safeguard condition. Otherwise, you know, you can merge those poly-x terms and make them one. So, distinct poly-x term that is what we are talking about. Now that we have come to this conclusion, let us now focus our attention on any one pole. So, you know, you have this one pole and therefore, you have a poly-x term corresponding to that pole and you can determine what that poly-x term is and now we also know that it will either be exponentially growing or decaying. And the moment you have one growing poly-x term finished, the system is unstable. Let us write that down. So, we are saying one exponentially growing poly-x term makes the system unstable. So, in other way of saying it is such a rational system is stable if and only if all the poly-x terms involved have exponential decay. Now what does that mean? That means there are two possibilities. We will consider a pole with magnitude greater than 1. For this to be decaying, for this to contribute a decaying term, it must give a left sided sequence which means the region of convergence must be to the interior of the pole. In contrast, on the same statement, let us make a multi-statement. So, if a pole is of magnitude less than 1, we are making a multi-statement here and you should read the statements. First reading the black one as one statement and then wherever there is a replacement by a red replace and read as a second statement. So, pole with magnitude less than 1. For this to contribute a decaying exponential, it must give rise to a right sided sequence and therefore the region of convergence must be to the exterior of that pole. So, let us get the situation clear. How would you determine magnitude greater than 1 or magnitude less than 1? So, let us draw the z plane. Magnitude refers to a circle implicitly with radius 1 and that is essentially called the unit circle. So, draw the unit circle in the z plane. Let us look at a pole which has magnitude greater than 1. Let us mark it in red. Let us consider another pole with magnitude less than 1 and let us mark it in green. Now, of course, you would draw circles passing through these poles centered at the origin and of course, you would have to exclude these circles. Now, if you consider this pole, the region of convergence means to be to the interior of that pole. So, you should go this way because it is of magnitude greater than 1 and if you consider this pole, the region of convergence needs to be to the exterior of that pole because you want a right sided sequence. In either case, you can see that the unit circle is always included and this is the central idea. The unit circle is the critical contour here. If the unit circle is in the region of convergence, there you are the system is stable. If the unit circle is outside the region of convergence system is unstable. In fact, now what happens when you have a pole on the unit circle? The system immediately becomes unstable because the unit circle is automatically excluded from the region of convergence. So, let us now put down a very important theorem. So, a rational discrete system. Now, remember when you say rational discrete system, you automatically mean the system is linear shift invariant, its impulse response has a z transform and the z transform is rational. All this is inherent in saying rational discrete. A rational discrete system is stable if and only if the region of convergence of its system function includes the unit circle mod z equal to 1. So, simple and so elegant and now we need to complete one important detail which we also did in the context of continuous independent variable systems. What happens if you need the system to be both causal and stable? Now, it is very clear that if you want the system to be causal, the only kind of poly x terms which will allow other right sided ones. You cannot have left sided poly x terms in a causal system because for a causal system, you need the impulse response to be 0 for n less than 0. So, you cannot have a left sided term there, left sided poly x term. So, you want only right sided terms. Now, you also want it to be stable. So, in that case, you want all those right sided poly x terms to be exponentially decaying as far as the exponential parameter is concerned. So, what are we saying? You see another way of looking at it is, you want the unit circle to be in the region of convergence because you want the system to be stable. You want the extreme contour mod z tending to infinity to be in the region of convergence because the system is causal. Everything in between now has no choice, but to be in the region of convergence. You cannot have poles between these two included circles. That means there can be no poles outside the unit circle. In fact, there cannot even be poles on the unit circle. And therefore, a causal and stable discrete system must have all its poles within the unit circle. Very simple theorem. Let us write it down and that gives us a condition for causality and stability together. A discrete rational system is causal and stable if and only if all its poles lie within the unit circle because all the poles must contribute a right sided decaying poly x term. We have come to a very important point in our understanding of the z transform and the Laplace transform. We have been able to characterize both causality and stability. So, now looking at the system function, of course, the moment you have a system function linearity and shift invariance is implied, but now we know how to deal with causality and stability as well. Thank you and we shall meet again in the next session to see more.