 So welcome to the 30th session on signals and systems. Recall that in the previous session we had started talking about the characterization of the stability of a linear shift invariant system beginning with its impulse response. And in fact let me put down what we had understood. So we had seen if h n is absolutely summable meaning the summation of mod h n over all n, all integer n is finite, then the linear shift invariant system with impulse response h n is stable. What does it mean? A bounded input bounded by m x that means mod x n less than equal to m x for all n for some m x greater than equal to 0 strictly less than infinity. So the bounded input x n bounded by m x results in a bounded output bounded by m x times m h where m h is the absolute sum of h. Now this is good. This is a sufficiency condition. You see what is important with the sufficiency condition is that it is also constructive. That means not only does it tell me that if that condition is true the output is bounded given the input is bounded it also tells me a bound on the output. So that the condition is constructive. So constructive condition it tells me clearly what the bound on the output is. Now let me also complete the same discussion for continuous variable system. In fact the discussion is very similar. You have y t is integral minus to plus infinity h tau x t minus tau d tau assume x is bounded by m x that means mod x tau is less than equal to m x for all tau of course here all real tau and m x is of course the quantity strictly less than plus infinity and greater than equal to 0. If h is now note you see we had absolute summability here we shall talk about absolute integrability that is mod h tau d tau integrated from minus infinity to plus infinity is finite and is equal to m h. Then we shall show that y t is also bounded. In fact we can find the bound on y t and we will do that in a minute. Indeed y t is just integral minus to plus infinity h tau x t minus tau d tau. Let us look at the magnitude of y t it is the magnitude of this integral and here again we use the generalized property of mod a plus b is less than equal to mod a plus mod b. See what we are saying here is that an integral is actually the limit of a summation. So an integral is essentially a summation over smaller and smaller and smaller intervals. So we could now take the absolute value operation into the integral sign and put an inequality using the limit of a summation idea and how do we do that? So we will have mod y t is therefore less than equal to integral minus to plus infinity mod h tau x t minus tau d tau and that is equal to minus to plus infinity mod h tau mod x t minus tau d tau. But then you know that mod x t minus tau is strictly less than or not really strictly is less than equal to m x for all t minus tau and therefore we have mod y t is less than equal to m x times integral minus infinity plus infinity mod h tau d tau and here again if this is absolutely sonnable. So this is equal to m h m h is strictly less than infinity plus infinity of course greater than equal to 0. Then mod y t is less than equal to m x times m h for all t and therefore y t is bounded. So we have completed the proof for the continuous variable case. Now I am going to open the discussion about necessity. We have clearly shown this condition is sufficient. If this condition holds given a bounded input we have a bounded output both for the discrete case and for the continuous case. However we have not shown that if this condition does not hold the system cannot possibly be stable and we are going to do it as follows. We are going to essentially make use of a very ancient story and let me narrate that story for you before we go on to making a formal statement. The story relates to a person who appeared in a king's court and who claimed to know many languages and he had challenged the courtiers to identify his mother tongue. So the courtiers were hard. You see he spoke very well. He was a polyglot. He spoke many languages and spoke very well. So what could the courtiers do to trick him into speaking something in his mother tongue and how can we use that principle to identify the necessity of this condition to establish stability. We will come back to this in the next discussion. Thank you.