 So, welcome to the 23rd session where we continue with more properties of discrete systems. We had looked at homogeneity and shift in variance in the previous session. Now we continue on to causality and stability. So, causality again has a very similar connotation as in the context of continuous variable systems. In fact, causality refers to cause-effect relationships being obeyed and let us formally define it. So, we have this discrete system back again, call it S if you like with an input xn. I am emphasizing this again and again because you know we need to have this notation well fixed in our understanding. Now causality means I perform the following experiments on the system. I take an input x1n and another input x2n and make sure those inputs are completely identical up to some point in time. So, assume two inputs x1n and x2n so that x1n is equal to x2n for all n less than or equal to some n0. Now, for example, n0 could be equal to 0 that means that the two inputs are identical up to the point 0 they may or may not be different afterwards. They are not insisting that they be different afterwards, but we are allowing for that possibility. Now, the only question that we are asking is what happens to the output up to n equal to n0 and that is what we answer in the context of causality. So, causality would mean now I am going to emphasize for all such x1, x2 and for all such n0, we have y1 of n the corresponding output sequences y1 and y2, y1 of n is equal to y2 of n for all n less than or equal to n0. Now, here I have started with a formal definition first you are already familiar with the corresponding definition in continuous variables. You see it is good now to get used to that formalism quick in the beginning we were a little slow you know with building formalisms, but now we need to use formalism as a convenient language. You will notice this is nice concise and crisp, but we are saying a lot here. We are saying give me any two inputs which are identical up to a point in time and any two such inputs are acceptable and any such point in time is acceptable. Given those I would observe that the outputs are also identical up to that point in time. Now, of course, here I am assuming time to be the independent variable, but whatever be the discrete independent variable. You see what should be noted here is that this happens for any I am emphasizing this again any n0 any point in time up to which you have the two inputs identical and any two such inputs and you observe the outputs to be identical. How does it relate to a cause effect relationship? You see if the effect comes after the cause as long as the two causes are identical you should see identical effects that is what you are saying in causality as long as the cause does not change the effect does not change. Simple enough, but a slightly more informal definition which I will now also write down is that causality means that the input output relationship is such that the output only behaves according to the past and the current input. It does not respond to the future that is an informal way of saying what causality means. Alright. Simple enough, very similar to what you have in continuous variable systems, now we come to stability. So, stability is equally simple. In stability what you are saying is something very similar to what you said in continuous variable systems you have this bounded input xn and let us say it is bounded by mx and you ask what can we say about the output yn? Now what does boundedness mean? Boundedness means that the magnitude of the input at every discrete variable point is less than or equal to the bound which you have given it namely mx let us write that down formally. So, we are saying mode x of n is less than equal to mx and of course mx is strictly less than nodes strictly less than infinity the strictly part is important otherwise it has then of course that definition has no meaning. So, strictly less than infinity, so it is a of course needless to say mx has to be a non it has to be greater than or equal to 0. Now, given a bounded input what can we say about the output? If the system is stable the thing that we are guaranteed is that the output is bounded. So, stability means yn is bounded and this happens for all such xn this is important. So, for every bounded input you are guaranteed the output is bounded. What you mean by saying the output is bounded let us also say that formally. So, what that means is that there exists my so that my is of course non-negative and strictly less than infinity so that mod yn is less than equal to my for all n. Now, just a point to be noted even in the previous definition when we talked about boundedness I must emphasize that mod xn is less than equal to mx for all n of course you know when I write it like this it is implicit but I want to emphasize that and that is also true here. I am saying that mod yn is less than equal to my for all n so I mean you can visualize the strip between 0 and my wherever that is and all the magnitudes lie in that strip for xn they lie in the strip between 0 and mx for yn they lie in the strip between 0 and my that is for boundedness. So, boundedness you know you can visualize it. Now, what are we saying let us be careful we are saying that every bounded input results in a bounded output. Now, let us ask some questions which could you know tweak our minds a little bit for example what happens when we give it an unbounded input can we say something now here again I take recourse to exactly the same metaphors that I used in the continuous variable case namely let us link stability or let us kind of bring a parallel between stability and reasonable behavior in people. So, you say a person behaves reasonably if he responds reasonably when there is a reasonable behavior given to him or her right so you say a person is reasonable if you behave reasonably with him he or she behaves reasonably with you the person behaves reasonably giving a reasonable input to the person maybe you could think of reasonableness if you really want to be very very crisp and very very simple think of reasonable behavior with the person as speaking in a reasonable tone do not raise your tone or you know let your tone not grow to infinity let your tone be bounded. So, as long as you are speaking to the person in a reasonable tone the person responds in a reasonable tone let us think of this as a parallel or as a metaphor. Now, ask yourself what will happen if you speak to this person unreasonably in other words if you behave unreasonably with the person by speaking to him in a tone which is growing. Now, even if the person is reasonable and if you behave unreasonably with the person you are not guaranteed to the person will either behave reasonably or unreasonably with you it depends on the person so reasonable person could behave unreasonably with you or reasonably with you if you behave unreasonably with him or her in contrast if a person is unreasonably what does it mean it means that even if you behave reasonably at least there has been one instance when you behaved reasonably with the person and he or she responded unreasonably to you that is what you call an unreasonable person. But that does not guarantee that when you speak unreasonably to this unreasonable person he or she would respond unreasonably or reasonably so when you behave unreasonably either with a reasonable person or with an unreasonable person nothing can be said about what response will emanate and the same is true of stable systems or unstable systems when you give an unbounded input either to a stable system or an unstable system nothing can be said about whether the system would respond with a bounded input a bounded output or an unbounded output nothing can be said. On the other hand if you gave a bounded input to a stable system you are guaranteed the output is bounded if you give a bounded input to an unstable system well the output could be bounded the output may not be bounded what you are sure about is that there has been at least one instance of an un of an of a bounded input which resulted in an unbounded output that may not happen for all bounded inputs and of course nothing can be said beyond that. Un instability is provable by one instance of violation stability cannot be proved by instance it must be proved independent of instance for all bounded inputs good so you understood two more properties in the context of discrete system you will see a little more in the next section. Thank you.