 Welcome to the 14th book in this course and here we need to go further to understand that narrowing pulse. So, let us come back that very narrow pulse that we had in the previous lecture. So, we had this very narrow pulse of width delta and the height of 1 by delta and recapitulate that the area was held constant even as the width goes to 0. Now, this is a new object, we should call it an object because it is not really a function, you know. You cannot call this a function anymore because you will notice that this height goes towards infinity. So, not really a function, we call it a generalized function and this is a new term that we need to understand. We are not going to be too formal about explaining a generalized function at this time. All that I want us to appreciate is that a generalized function is not a function. This ever narrowing pulse with delta tending to 0 is a generalized function, it is a generalized function because it has some properties of function and some not. For example, it is not appropriate when you take delta tending to 0 to query what is the value of that object, whatever it is at a particular point. You see, for a finite delta you can do that except at the point of discontinuity where the pulse drops, otherwise the value is either 1 by delta or 0. But for a delta tending to 0, the height tends to infinity and therefore it does not make any sense to ask what is the value of this so-called function or object at a particular point. I mean, let us go back to the figure to understand. So, if I were to choose a point somewhere here, let us say and if I were to ask what is the value as delta, what is the value here as delta tends to 0 does not make sense. That is what is lacking in thinking of this as a function. That is why it is called a generalized function. Now we give this particular generalized function a name. We call this an impulse and if you want to be very particular you should say a unit impulse. Here unit refers to the area that has been encapsulated in that impulse. So, an impulse is a very interesting object. It encapsulates a non-zero finite area but with 0 width and we have a symbol for it. The symbol that we are going to use is this. So, going back if you call the indexing variable t0 and if this point is t0 equal to t then delta delta t0 minus t with delta going to 0 becomes an impulse delta t0 minus t0. So, just delta is now an impulse. This is of course the graphical symbol that we have used here and this is the algebraic symbol for an impulse. Now let us interpret that property that we had namely that when you put many impulses together properly weighted you get back the original function. Let us write that property down. So, what we mean there is x at t0 is integral over all t x t delta t0 minus t. This is the precise equation. This is the precise relation. So, with delta it was only approximate but with delta not equal to 0. Now all very well why are we doing all this? Hidden in this description or hidden in this observation is the use of all the three properties that we had. Additivity, homogeneity and shift invariance. Now let me take a system which has all the three properties. Additivity, homogeneity and shift invariance. And let us assume that I have applied to this system an impulse as an input, whatever that means. Now if you want to visualize what it means to apply an impulse as an input what you should do is to visualize first applying a very narrow pulse and then visualizing that width going smaller and smaller and smaller and then studying what happens to the output, all right. So, we are now coming out with a very important idea here. The response of a system to a unit impulse, what is that, let us define it, the impulse response a very important idea. What is the impulse response of a system? The output when the input is a unit impulse and what does that mean? That means that if I give a very narrow pulse as input and study the output as the pulse becomes narrower and narrower with an ever-narrowing pulse, the density is given as the input to the system and the output we shall call the impulse response often denoted by H of t. This is a very standard notation used very frequently. Now, let me apply all the three properties and see what it gives me, let me take additivity first. So, suppose I apply two impulse to the system, we have the same system S here and you give it two impulses, one located at say t1 and the other one located at t2. What is the output expected because of additivity? The output should be respond to whatever happens when you apply an impulse located at t1 plus the response to whatever happens when the impulse is located at t2. At the moment we cannot say much unless we use the other properties. Now, let me bring in homogeneity. So, if I know the delta t results in ht, then I also know that when I multiply delta t by a constant, let us say kappa, it should result in kappa times ht. This is a consequence of homogeneity and in particular if kappa were to be essentially the value of x at some point, this is still true. Let us keep that in mind. Finally, I am going to bring in shift invariance. So, if I shift the impulse, so I have delta t minus t1 and delta t minus t2, let us go back to the previous drawing that we had. So, here when I wrote down delta t minus t1 and delta t minus t2 as I have done here, these two are related by a shift. In fact, what we hope is that if the system is shift invariant, once I know the response to this delta t, I should automatically know the response to this. It should just be shifted by t1 and I should automatically know the response to this as well. It should again be shifted by t2. So, let us write that down formally. What we are saying is ht resulting, I mean resulting from, let me draw it like this, delta t resulting in ht means delta t minus say lambda for a fixed lambda should result in h of t minus lambda. So, here I am bringing in shift invariance. We have brought in these three problems and now we are going to synthesize them. For that, we shall have to take another session. Let us meet in the next session to synthesize these three consequences. Thank you.