 Welcome to the 32nd session of the first module in the course signals and systems. Let us now take stock of where we are in the whole question of looking at signals and systems in their natural domain, in the domain in which they are defined and in which they occur naturally. Firstly, we have looked at continuous systems and discrete systems by now. In other words, we have looked at signals and systems where the independent variable is continuous and where the independent variable is discrete. In both of them, we have assumed that the same kind of independent variable exists on both sides of a system input and output. So we have continuous variable input, continuous variable output and therefore we can talk of a linear shift invariant system if that be the case which has a continuous variable impulse response. And similarly in the context of discrete systems, we could essentially look at discrete independent variable input, discrete independent variable outputs and if that system be linear and shift invariant then we could also look specifically at the impulse response which will also be a function of the same discrete variable and the impulse response remember would characterize the linear shift invariant system completely. We agreed that we would focus on linear shift invariant systems and we would now like to review a few things that we have understood about linear shift invariant systems but looking together at the continuous variable context and the discrete independent variable context. So let us review these things before we go further to make a few more mathematical derivations. So let us look at the whole question of input-output relationships. Now we will specifically consider the context of linear shift invariant systems. The continuous independent variable, what is the context? We have a linear shift invariant system with impulse response ht, remember t is the continuous independent variable. So the input xt and the output yt are functions of the continuous independent variable t and the impulse response ht is also a function of the same continuous independent variable. So in fact let us identify this, this is the impulse response and we know the relation between yt and xt that is what we are trying to write down now the input-output relationship. So we have y of t is integral over all the continuous independent variable. So we write down xt minus tau d tau. So this is the input-output relationship in the context of a continuous independent variable system. Now let us look at the same in the context of a discrete independent variable system and trying to put the two of them together because we want to compare them and we also want to see certain things together in the context of both of them. So let us come to discrete independent variables now. So the input-output relationship for a discrete independent variable linear shift invariant system. So we have an LSI system. So this is the situation. We have x of n as the input, y of n as the output, h of n is the impulse response. Let us identify each of them. This is the impulse response and we have an input-output relationship between x and yn which is based on the impulse response given by y of n is summation k going from minus to plus infinity xk hn minus k. Now we notice that there is a great similarity between what we had in the continuous variable system and the discrete variable system. In fact, let us write both of them together and then we will understand what I am saying. So compare y of t is an integral here and y of n is a summation there. There is a great similarity. The only difference is that there is an integral here and a summation here overall the independent variable. But you know now what we want to do is to look at this operation in some depth. We are doing some operation between two sequences or two signals you might call them. The input signal or the input sequence and the impulse response sequence and we are producing the output. So we wish to look at this whole thing that we are doing to the input and the impulse response. We want to look at it essentially as an operation between two signals whether continuous independent variable or discrete independent variable. Now to do that we shall begin with the discrete case because that is easier to deal with to an extent. So we shall establish a neat and an independent way to do this operation between the two sequences of the two signals so that we understand the operation in greater depth. Now towards that objective let us look at the expression again. Let us look at this expression again here. Let us take the discrete case first. You know what are we really doing? You notice that for every n there is a different summation. And let us once again illustrate this whole idea by taking an example. For example, we will use this notation x of n let us take a finite length sequence. So let x of n be the sequence which is say minus 1, 9, 5 and 3 respectively at n equals I will make a table x of n as a function of n. Now there is a neat way of. See essentially what we are saying is that there are only 4 non-zero points in x of n at minus 1, 0, 1 and 2 and they are respectively minus 1, 9, 5 and 3 and x n is 0 elsewhere. There is a neat and a convenient way of writing such a finite length sequence. And I am going to introduce that notation in the arrow notation. So we will put an arrow and write down the value of n here, 9, 5, 3 on this side and minus 1. So this means the same thing. Now this is the same thing as writing minus 1, 9, 5, 3 and you can put the 1 there if you like or it is the same thing as writing minus 1, 9, 5, 3 and writing minus 1 here. So all of these are really the same thing. This is called the arrow notation. And it is a very convenient way of denoting finite length sequence. Now we will similarly write down a finite length sequence for h. Let us do that next. So we have and we wish to find out y n. So let us write the expression for y of 0 for example. y of 0 would be summation over all k in principle for minus to plus infinity x of k h of n minus k with n equal to 0. That summation k equal to minus to plus infinity x of k h of minus. And let us write down these two sequences and see how they figure on the k axis. We do not need to write all of k. We can just start from a few. So we will write x of k there and h of minus k here. We need to calculate summation k going from minus to plus infinity x k h minus k which is essentially similarly only k going from minus 1 to 0. You need only two points. All the other products are 0. So how much is that? That is simply minus 1 into minus 1 plus 9 into 1. That is 1 plus 9. That is 10. So y of 0 is 10. Now I have illustrated how to calculate one point. We shall continue this discussion and we shall calculate the whole of the output. And then we will establish a certain neat way of doing this more systematically in the next session. Thank you.