 So we said that an input xn is bounded if now I introduce not a notation we will use this notation to write there exists. So there exists an mx which is greater than or equal to 0 such that mod xn is less than or equal to mx for all n for all n integer of course and of course mx is independent of n that is obvious. So it should be noted that we are talking about the boundedness of the samples here we are not asking that the input be absolutely summable to know we are just saying the input needs to be bounded right but what we said is that given that the input is bounded the output is also bounded in that case if mod hn summed over all n summed over all integer n is equal to let us call it mh and of course mh is less than plus infinity of course mh must be non-negative if the summation is mh then the output is bounded not only is the output bounded we know a bound on the output the output is bounded by mx times mh that is very interesting. So we said we have a constructive conclusion here not just an existential conclusion the constructive conclusion is that if I know the bound on the input and I know the absolute sum of the impulse response the bound the bound on the output is equal to the bound on the input multiplied by the absolute sum of the impulse response. What we were asking yesterday is whether this condition is necessary for this to happen and that is what we will now need to answer but with a little bit of hard work before we embark upon that question I would like to allow a couple of questions yes there is a question from here. So the question is why should mx be non-negative but you see mx is a bound on mod xn mod xn can never be negative. So the least value of mod xn is 0 of course mod xn must be non-negative. So we are saying it must be non the catch is not non-negative is trivial the catch is that it is finite is that right. Any other questions before we work towards the necessary part any other questions so far because so far we have drawn some important conclusions none at all none no questions at all. So then we now embark on the very important question of whether this condition is necessary. So let us pose the question first the question is is it necessary for the impulse response to be absolutely sumable if every bounded input if every bounded input produces a bounded output is that right. So you see now we are asking the converse we are saying let the system produce a bounded output for every bounded input. So if every bounded input results in a bounded out in other words the system is what is called bounded input bounded output stable. Now we spend before we start answering this question we would like to understand some of the terms that we are using we call a system bi, bo or bbo stable if every bounded input results in a bounded output. Now remember we are saying every bounded input results in a bounded output in a stable system every bounded input results in a bounded output. So it is not adequate to consider a specific example of a bounded input and look at the output and see that is bounded and conclude the system is stable. Now I always like to explain the idea of stability of systems by drawing a parallel to sanity of human behavior. So you may take a parallel between stable systems and sane humans right when would you say a human behaves sanely when in every instance where he has a sane behavior from another human being he responds sanely right. So you know if you behave sanely if you behave properly with a sane human being he or she responds with sane behavior that is what a stable system does give it a bounded input it responds with a bounded output. But you see now take the case of a sane human being suppose an insane person comes in front of a sane human being it is not necessary the sane human being would behave sanely he may or may have you see it at all depends on how insane that person is if the person is all out to attack him that person might also behave equally insane. So even if a system is stable and you give it an unbounded input it is not necessary the output needs to be bounded if you give it an unbounded input a stable bi, bo stable system when given an unbounded input could produce either a bounded output or an unbounded output right. So what are sane what are stable systems those that produce bounded outputs when they are given bounded inputs like sane human beings. Now what about unstable systems they are also like insane human beings insane human beings at times can behave very sanely even if you behave insanely with them. So you see insane human beings are unpredictable is that right give them a bound so similarly unstable systems are unpredictable. If you have an unstable system give it a bounded input you know nothing it may produce a bounded output it may not produce a bounded output give it an unbounded input it may produce a bounded output it may not produce an unbounded may not produce a bounded output. So it should be clear that one should not conclude that if a system is stable then giving it an unbounded input will necessarily produce an unbounded output or giving it an unbounded input will necessarily produce a bounded output all that we can say is the output is unpredictable that is for stable systems for unstable system nothing can be said at all. All that can be said in fact all that can be said for insane people is that there is at least one instance when somebody behaves sanely with that person and he responded insanely that is why you call him insane if in every instance of behavior a sane response elicited a sane input or sane behavior elicited a sane response you would not call a person insane you call a person insane because sane behavior elicited insane responses and the same is true of unstable systems and specifying this because sometimes these finer if but then are not properly obeyed. Now coming back then to the question of stability and instability we have shown that a stable system you know it is sufficient for the impulse response to be absolute to summable in order that an LSI system be BIBO stable but we have now to ask if a system is BIBO stable can we immediately conclude the impulse response is absolutely summable and again to answer this question we are again going to take the help of a famous story which occurs in many legends in many countries in various forms and the story goes that there was this gentleman who knew many many languages and only one of them of course was his mother tongue. Now you know one the king posed the challenge before a very witty and intelligent minister to find out in fact that man had posed the challenge anybody who can find out my mother tongue will of course have me as his servant for a certain period of time and of course this person was other than a polyglot very knowledgeable in many ways the king did not want to have him in his council of ministers if possible at least for a while but for that of course the condition was his mother tongue needed to be deciphered. So what this other wily minister did was to let that person rest very peacefully on a particular pleasant night right and then in the middle of the night this wily minister sent a couple of soldiers and through absolutely cold and unpleasant water on this gentleman rudely awakening him and of course you can quite guess the language in which he uttered the vindictive you know the vendetta and the unpleasant things that he wanted to do his mother tongue was very quickly known right. This teaches a very interesting lesson if one really wants to drive a system to the brink and a human system is no exception use a situation which is extremely troublesome and we will do the same for a stable system. If we wish to prove the impulse response must be absolutely summable drive that stable you know try and drive that stable system almost to the brink of instability like you drive a sane human being almost to the brink of insanity to reveal or to test the person sanity. Now here you drive the stable system almost the brink of instability by forcing upon it an input which is bounded in principle but which forces these absolute some of the impulse response to emerge that is about the worst that can come out is that right. So let us then see which input to a stable system would force the impulse response to emerge. So the proof of necessity proof that it is hinges on the principle of a troublesome input and the troublesome input is the following you know that for an LSI system with impulse response h n and input x n the output y n is summation over all k h k x n minus k power x k h n minus k as we please and consider in particular n equal to 0. Now we are going to give a particular troublesome input x n and we want the output of course if x n is bounded troublesome but bounded. You see even in that story remember you know I mean if the legend goes that after this gentleman was rudely awakened from his sleep he asked who had thrown the water and of course the minister very politely told him that it was raining and there was a little gap in the roof. Now what I am trying to say is that you know here we are going to drive the system to the brink of instability but we are going to do it with a sane input a stable bounded input. So we will keep to the principles of boundedness but we will still twist and turn our input to bring out the worst and you see if you when you say the if assume that you are able to give this troublesome bounded input to the system. If the system is stable the output needs to be bounded and if the output is bounded every sample is finite in magnitude and therefore in particular when you put n equal to 0 y of 0 must also be finite right. So we shall choose we shall strategically choose a bounded input x n the output must then be bounded if the system is stable and the output is bounded then of course mod y of 0 must be finite in particular and what is mod y of 0 mod y of 0 is summation k going from minus to plus infinity hk x of minus k modulus of this and now let us very wickedly choose x of minus k choose x of minus k in the following way x of minus k is equal to the complex conjugate of hk divided by the modulus of hk if hk is not equal to 0 and it is equal to 0 if hk is equal to 0. So whatever look at so what you are saying is look at the impulse response at every point of the impulse response if the impulse response sample there is non-zero take its complex conjugate now we are allowing for a complex impulse response in general. So take its complex conjugate and divide by its magnitude and put that in the location minus k of the input and if that impulse response sample happens to be 0 then simply put a 0 in the same negative location on the input. Now if you do this how much is the output y 0 clearly y of 0 is obviously summation all k such that hk is not 0 hk now x of minus k is hk bar divided by mod hk and this is simply summation all hk not equal to 0 of mod hk so of course you do not have to worry about its non-negativity and of course it is trivial that this is the same obviously I mean you know this condition that mod hk or hk is not equal to 0 is not serious at all because wherever hk is 0 anyway it contributes nothing to the summation is that right so this is the same thing so it is quite correct to write y of 0 is indeed the absolute sum k over all the integers mod hk because wherever it is not wherever it is equal to 0 of course it contributes nothing to the summation and we want y of 0 to be finite and therefore this absolute sum must be finite there we have the cold water thrown on the system is that right although I certainly recommend such approaches to identifying properties of systems I do not recommend that you take this as a lesson in dealing with people is that right so people need to be dealt with differently from systems one does not always have to drive them to the brink of insanity now coming back then to the question of stability and instability we have seen now that you know it is very clear that the necessary is for theorem let us state the theorem the necessary and sufficient condition for Bebo stability and unless otherwise stated in future we shall use the word stable to denote Bebo stable so for Bebo stability of an LSI system is that it is impulse response be absolutely sum up here again I must bring a word of caution for an LSI system we have drawn this conclusion for a system which is not LSI this conclusion is incorrect anything can happen it is quite possible that the impulse response be absolutely summable but the system be unstable and we do not have to go very far to draw a conclusion that this is not adequate take for example the system counter example take for example the system y of n is n times x of n now I ask all of you is the system be both stable please remember Bebo stability is independent of the other three properties now we brought in a fourth property of systems we had additivity we had homogeneity we had shift invariance and now we have Bebo stability Bebo stability is not dependent on the other three properties it is independent to the fourth independent property in its own right so we can ask for any system whether it is or is not be both stable and we can ask for this system as well is the system be both stable it is very easy to see it is not in fact all that you need to do is to give it an impulse at different places if you give it an impulse which is located a unit impulse sequence located at 0 the output is identically 0 give it and give it a unit impulse located at the location n equal to 10 to the power of 6 and you get 10 to the power of 6 at the point n equal to 10 to the power of 6 and 0 everywhere else and therefore any no bound simple bounded input like the unit impulse does not result in a bounded output in this system is Bebo unstable however it is impulse response however h n is equal to 0 for all n and of course it is absolutely summable if I give it a unit impulse sequence at the location n equal to 0 as the input the output is 0 for all n and that is obviously absolutely summable so absolute summability is not enough when the system is not LSI.