 So, a warm welcome to the 24th session. In this session, we would now build on the linear time invariant concept or linear shift invariant concept in the context of discrete systems. We call that when you bring linearity and time invariance or shift invariance together, a lot of things get simplified. In fact, in the continuous time case, we saw that one single experiment on that system was enough to characterize the system completely. The same principle will now extend to discrete systems. So, let us now focus in this session on linear shift invariant systems in the discrete context. And let us explain what you mean by linear shift invariant. Of course, we have individually defined the components of linearity and shift invariance, but we need to aggregate them together. So, let us assume this discrete system S is linear and shift invariant. That means three things happen. S is additive, homogeneous and shift invariant. Additivity and homogeneity together constitute linearity, as was true for continuous variables. And now we need to ask the same question as we did for discrete systems. Can I perform one experiment on that discrete system and say everything that I want to about the discrete system? What does that mean when I say, I want to say everything that I would like to, what am I really asking for? I am asking for being able to give a complete input-output relationship. After all, what is a system? A system is a mapping from sequences to sequences, when talked about in the discrete context. So, what we are asking is, can I make one experiment on that system? Can I look at one input-output relationship and say everything about the input-output mapping that that system implies? And once again, the answer is yes and the answer is essentially the same as what we had for continuous variable system with a little difference, that it is easier to understand that answer here. In fact, that is the reason why some people like to begin with discrete systems in a discussion on signal and system. We have preferred to begin with continuous time system because it is easier to relate continuous variable or continuous time systems to the natural world, to the real world. But understanding is probably easier with discrete system and therefore, let us now define the simplest possible sequence that we can have in a discrete system context, namely the unit impulse. So, now we proceed to define what we call the unit impulse sequence. I do not need to say too much about impulses because you have said enough in the continuous time case. But what you would really want in a unit impulse sequence is just one non-zero point. So, in fact, we can define this unit impulse sequence very precisely in a discrete context. The unit impulse sequence delta n is defined to be 1 for n equal to 0 and 0 for n not equal to 0. So, it is very simple. It is a very well-defined sequence. At 0, it takes the value 1. At all other points, it has the value 0. So, you can see that this unit impulse is what its name suggests it should be. An impulsive non-zero point in one location and 0 everywhere else. You recall that in the continuous time case also, what an impulse really meant is a sudden very short-lived phenomenon. An impulse is exactly that. A sudden very short-lived phenomenon. And of course, for the continuous time case, we had to work hard to understand the impulse. We had to begin with a pulse and we had to make the pulse narrower and narrower, but preserving the area. All those ideas are not required here. Here, we have a very well-defined sequence. It is one at a point chosen, namely n equal to 0, 0 everywhere else. There is no mathematical complexity in the definition. It is a sequence like any other. Now, let us perform the experiment of giving this unit impulse sequence to the discrete system as an input. And let us observe the output. Now, in fact, let us take an example. Let us take that taxation system again as an example and ask some questions about its linearity and shift invariance and then look at its unit impulse response. So, you will recall that the taxation system which we had looked at in an earlier discussion gave you the output. Let us call it, you know, ST, taxation system if you like. And the system looked like this. It said that at any instance of measurement, maybe a periodic measurement takes place of population and the instance is indexed by n, you have a population of x of n as a function of time and the tax collected is y of n at that instant for a particular service. And you will recall that at any instance, the population has paid tax over two instances, a tax of alpha over the current instance and a tax of beta over the previous instance. So, for two instances that population pays tax. And let us ask this question. Is this system linear and shift invariant? Now, it is very easy to see that it is additive. We have already established that. You will recall that my student actually showed that the system is additive, already established. Let us quickly prove that it is homogeneous and shift invariant. So, let us multiply x n by a constant gamma and give that as an input. See, if you gave gamma times x n as an input to the system, ST, then you have alpha times gamma x n plus beta times gamma x of n minus 1 as the output and that essentially is gamma times the original y n, where y n is the output to x n. Therefore, the system is homogeneous. Now, let us also check its shift invariance. Let us replace x n by x n minus d and give it to the system s. Of course, you can see the output is going to be alpha times x of n minus d plus beta times x of n minus 1 minus d, which is the same as alpha of x n minus d plus beta of x n minus d minus 1. And anyway, this is the same thing as y of n minus d and true for all d and all x and therefore, the system is shift invariant, a formal proof. But after all, this also has a very simple physical explanation. Suppose, I were to have the same patterns of population as a function of instance and if I were to look at the taxation pattern, I would see the same taxation pattern shifted if I shifted the whole population pattern. So, as long as the taxation rule is the same, the taxation pattern remains the same, essentially the taxation pattern remains shifted by the same amount by which the input is shifted. That is what we are essentially saying. So, if I had a certain set of population measurements at succeeding instance and I preserve that pattern and I shifted that whole pattern by d samples and I looked at the taxation pattern after the shift was executed, then the same taxation pattern is seen, but shifted by d sample. That is what we are saying. It is good to get a physical interpretation. Of course, a formal proof is always nice and we must complete a formal proof anyway in all situations. So, this system is linear and shift invariant. Let us give it a unit impulse as the input. So, we have delta n being given to the system S t and we query the output. Very easy. The output is going to be delta n times alpha plus beta times delta n minus 1. This is very easy to understand. This is essentially alpha at n equal to 0 and beta at n minus 1 equal to 0 or n equal to 1 and 0 everywhere else. A very simple sequence. So, at 0, it takes the value alpha. This is the value of n. At 1, it takes the value beta. At all other points, it takes the value 0. So, once again, you have only two non-zero points here and this constitutes what we call the unit impulse response of the system. This is just to give you an example. Similarly, any discrete system can be associated with a unit impulse response. But what is important is to study the unit impulse response for the context of linear shift invariant system. That gives a great service to that linear shift invariant system. In fact, now I am going to state a theorem straight away which is very similar to the theorem that we had for continuous variable systems and I am then going to prove it. I am going to state the following theorem for discrete linear shift invariant systems. The unit impulse response characterizes the system completely. What do you mean by characterizing the system completely? It means that given any input x of n, I must be able to determine the output y of n given the unit impulse response. So, I need to complete a formal proof now to show that if I have this unit impulse response, I can write down yn explicitly in terms of x of n. I shall do that in the subsequent session. Thank you.