 So, welcome to this 20th discussion that we are going to have. Now, we had been talking about the impulse response for a long time and we have agreed that the impulse response characterizes the linear shift invariant system completely. And we are now going to put more properties on the linear shift invariant system and ask certain questions about the impulse response. But before we do that, we also need to look at a whole other class of systems and treat them together with this. Those are the class of discrete systems. Let me explain what discrete systems are. So, discrete system or discrete independent variable system are systems where the independent variable assumes only discrete values. Now, an example is if you have the time axis, instead of treating the independent variable as continuous time, you could put uniformly spaced instance on this time axis. And if you have a spacing of t, capital T between time instance, then you could always choose to designate some point as the 0th instance. So, you could call this 0 for example. And then any instance would be of the form n times t where n assumes all integer values. Now, to take an example, capital T could be one year, capital T could be one month, capital T could be one day. You can decide. And once you have made a decision about the interval that you want to adopt, uniform interval, as long as you have accepted the interval as uniform and you have made a decision about that interval, we do not have to keep writing that interval again and again. It is in the background. It is only the instant number n that assumes some importance and some value. So, let us record that point down. We are saying, so only instant value or instant number is important once T is decided. So, we could now think of a system where the input is recorded only as a function of this instant number. So, let us call it x square bracket n. Now, we are using square bracket to denote discreteness. And similarly, we will also record the output only at those instance. So, we have xn as the input and yn as the output and we use square brackets to emphasize discreteness. Let us take contrasting examples, one with continuous time, one with discrete time. For continuous time, of course, we have had ample examples based on electrical and mechanical systems. For discrete time, let us take two examples with some variation between them. So, let us take for example, a simple bank account. And here, we shall use T to be the interest calculation period, bank interest calculation period or calculation interval. So, maybe six months, you know, that is often one norm adopted or it could be one year whatever. And therefore, we shall use y of n, the output, note the output to be the balance in the bank account at the nth instance. Then I am going to use x of n, which is essentially the input to the system to be the depositor withdrawal in the nth such interval of calculation. And let us assume that this is a very liberal bank. It looks at the previous balance of for example, you know, suppose they have a six-monthly calculation, they look at the balance just before the current instant and they look at the balance one instant before. So, six months before and two times six months before, which is one year before. And they give a certain percentage of that balance as interest to you at this instant calculated from six months before and a certain percentage calculated from one year before. What would be the equation describing this? So, we have this bank account, we have this xn, the deposit and withdrawal collectively in that interval and yn, the balance of that interval. And effectively, what I am saying is y of n is equal to y of n minus 1 times some alpha, the interest rate for previous interval, for previous balance plus beta times y of n minus 2, where beta is the interest rate for balance two intervals before, two steps before plus x of n, that is the depositor withdrawal in this interval of interest. Now, let me give you another example. You see, this is one example of a discrete system where you have the output depending on its own past. It is an example of what is called a recursive system. The output depends on its own past. But then we do not have to have a recursive system. Let us take an example. For example, you could have a population paying tax. Paying tax, let us say for two intervals on a service provided. So, let xn, the input be the population to which the service was provided in the nth instance or the nth interval. You know, take an example, suppose the service, let us say is some network service. So, the network service requires you to pay tax, tax for one interval and tax for the interval afterwards. We have different amounts of tax. And the output that we want to calculate is the amount of tax that we have collected in this interval. So, there we go. So, the tax paid is some factor, let us say alpha times xn plus beta times x of n minus 1, where alpha is the tax paid in the first interval of service and beta is the tax paid in the second interval of service. Let the tax paid be the output here of the system and whereupon we have what is called a non-recursive system here. Here the output does not depend on its own past. It only depends on the input and prior inputs. The good thing is now we have several students listening in to this discussion and it is good to involve them. And I can see that one of them has an important question to pose here. Let me take that question. So, you talked about various properties for the continuous systems. How do you describe these properties for discrete systems? Ah, that is very good. You know, it is interesting. I have talked about one quality of discrete systems which was in fact not explicitly referred to in continuous systems, recursivity and non-recursivity. But then there were several other properties of continuous systems that we had looked at namely, additivity, homogeneity, shift invariance, in fact causality, stability and others. Now, a very valid question has been asked by Prateek here, my student who is listening in. Those properties that we talked about in the context of continuous systems, could they be extended to discrete systems and if so, how? Now Prateek, I am going to ask you to answer this question, try and answer this question. I will correct you if you are wrong. Try and answer this question for the first property that we talked about namely, additivity. And we shall take that in the session to come immediately after. Thank you.