 So, welcome to this session in which we now have this young man sitting with me who had posed a question in the previous session. And I would like him to answer the question with a little prompting from me at points. So, he had posed a very important question. We had looked at several properties in the context of continuous systems namely, additivity, homogeneity, shift invariance, causality, stability and the natural question that we have is, do these properties make sense in the context of discrete systems? I am going to ask him to explain what he thinks additivity would mean in the context of discrete systems. So, Pratik, what would you think? How would you try and map additivity as a concept in the context of discrete systems? Please make an effort and I will prompt you. So, extrapolating from the continuous systems case, I would expect the discrete systems to also follow the same logic. So, suppose you have a system which takes an input x of n and gives an output y of n. So, if x1 and x2 are two inputs and y1 and y2 are the corresponding outputs, then if we give the input x of n equal to x1 of n plus x2 of n to the system and if the output is y of n is equal to y1 of n plus y2 n for all x1 and x1 of n and x2 of n, then the system as should be. Very good Pratik. That is a very good explanation. In fact, Pratik is very rightly observed that you could take the same idea from continuous systems. So, you have three experiments really to be done. You apply an input x1 n and ask for its output y1 n. You input x2 n and ask for its output y2 n and then you input x1 n plus x2 n. Remember, in these three times you are inputting different signals. So, different discrete, it is not just points, it is not just numbers. In fact, let me write that down clearly. So, what I am saying essentially is firstly, we like to adopt a different nomenclature. So, signals in discrete variable we will call sequences. So, xn is the input sequence. Now, the word sequence makes sense because it is a sequence of values indexed by the integers. So, the word sequence, you understand Pratik? Sequence because unlike the continuous case where there is a continuum of values, you now have a discrete set. So, you can talk about the next one and the previous one. You could not do it in the context of continuous systems. So, therefore, you have xn as the input sequence and yn as the output sequence. So, we will use that word henceforth. So, as Pratik said, I have this discrete system, let me call it s as he did. I apply x1n, of course Pratik abbreviated it, but I would not do that. I will say x1n produced y1n and then in the same system x2n produced y2n and when the same system was given x1n and x2n added up as the input sequence, then what was produced was y1n plus y2n and this happens for all possible x1, x2. This is very important. This is something which Pratik said which we must stress and emphasize. You see, you cannot just look at one particular, I have told you this before. You cannot just look at one particular instance of an x1, x2, look at what happens and then come to a conclusion. When you want to prove a statement, it must be proved independent of context or example. However, when you want to disprove it, suppose you want to show a system is not additive, it is adequate to take one counter example. If you wish to prove that everybody in a state is truthful, by looking at one truthful person, you cannot come to that conclusion. But if you wish to prove that that is not true, then looking at one liar is enough. That property holds here too. Anyway, now Pratik, you have given a formal explanation. Can you also take the tax example, you know the state where there was a tax levied on a service and explain physically what this additivity property would mean? So the tax example was, let me explain the tax example here. The tax example was that you had a service being provided by the state and that service of course was applied, the tax applied for that service was true in all the states where it was given and the tax was applied for two intervals. In the first interval, alpha tax was applied, you know per person to whom service was provided and beta tax was applied for the second interval and for two intervals there was tax after that there was no tax, right? And x of n was the population in the nth instant or nth interval during which you are making a tax calculation. So you remember that we had this very simple relationship here, y of n which was the tax collected by the state was alpha times xn, the tax paid in the first interval times the population in that particular interval plus beta times x of n minus 1. Now please take this and explain what additivity means here. So let us consider two states, both of which have the same tax rules. If x1 of n is the population in the first state for the nth interval and x2 of n is the population in the second state for the same nth interval, then we can see this property in the following way. Suppose s is the system, that is the way of calculating. Sir you said s is the system that collects tax, that collects tax state wise. So if x of n is the total is the population of both the states together then that will be equal to x1 of n plus x2 of n. Now we want to find out what the tax collected will be for the same interval. So from the previous slide we can see that y of n should be equal to alpha times x of n plus beta times x of n minus 1. So putting in the value of x of n which is equal to x1 of n plus x2 of n, then we can see that we can split it up into alpha of x1 of n plus alpha x2 of n plus beta x1 n minus 1 plus beta x2 n minus 2, n minus 1. That's right, now aggregate the terms. So this is alpha of x1 of n plus beta of x1 of n minus 1 plus alpha of x2 of n plus beta x2 n minus 1. Very good, that's correct. Which we can see is. Now give this to me, what is this real? So this actually is y1 of n and this is y2 of n. So this actually is an additive system because given two inputs, if you sum them up, outputs also get summed up. Very good. So this is true for any x1 and x2 of n. That is right. So this s is additive. Very good, that's a beautiful illustration, it's all congratulate Pratik on this. So we will see more about these properties in the next discussion. Thank you.