 So, welcome to the 15th session and here our objective is to synthesize what we had from additivity, homogeneity and shift value. We will get a very important consequence. In fact, I dare say that what we conclude from this discussion in this session is central in signals and systems. So, let us get down to business right away. So, what we said was that we first defined an impulse response for a system. So, we had a system S, we called it S, we gave it a unit impulse and observed the output. Now, please a few comments here, h t is a function of time. So, in fact, we will take an example later to see how h t looks, h t could be a very reasonable function, it does not have to be a generalized function. Unlike delta t, which is a generalized function, h t need not be a generalized function. Of course, it could be, for example, suppose you have a trivial system, one that just puts out what it comes, what it takes in, in that case h t would be equal to delta t. So, an identity system would have an impulse response equal to the impulse. So, generalized functions are not unknown in impulse responses, but it is not necessary that every impulse response have generalized functions or impulses in them. Anyway, coming back again to this. Now, let us in particular invoke homogeneity first or in fact, let us invoke shift invariant first and afterwards homogeneity. So, shift invariant, let us shift the impulse by lambda. So, I have delta t minus lambda being given to the same system. Shift invariant would imply that the output should be h of t minus lambda. Now, let us invoke homogeneity. Take any function x t, by function of course, I am talking about signals here. So, take any signal x, let us say t is fine and observe at t equal to lambda. So, I mean let me show it graphically for you. So, you have x of t, whatever it might be and then I pick a particular point on t equal to lambda and put an impulse there. So, this is delta t minus lambda, this is how you should appreciate, this is how you should understand an impulse delta t minus lambda and now multiply it by x of lambda. So, at t equal to lambda, this is the value. So, of course, here what I am now going to do is to multiply x lambda by the impulse located at lambda. So, x at lambda multiplied by delta t minus lambda, you know, just to be most general what I will do is, it looked in the previous figure as if the impulse was of height equal to x lambda, let us not do that, let us make them quite distinct. So, let us redraw the figure, I have x of t here, so I will just show a little segment of x t here and I will draw first a unit impulse at the point t equal to lambda. So, this is delta t minus lambda, a unit impulse place at t equal to lambda. Now, I scale this unit impulse by x lambda, what is x lambda? x lambda is this, so it should look something like this, perhaps let us draw it back in black now. So, this is x lambda into delta t minus lambda and this is what I am talking about as the input to the function. So, what does homogeneity tell me now? Homogeneity tells me when I apply x lambda times delta t minus lambda to the same system which is also shift invariant, I should get x lambda into h t minus lambda. Now, finally, I invoke additivity and I invoke additivity by essentially using integration in place of summation. So, what I am going to do is to think of the integral as the limit of a continuum of sums, let me show that to you schematically. So, in the system what I am saying is go back to that x t, let us take a small segment somewhere of x t, now pick a few lambdas there, let us pick lambda 1 here and let us pick lambda 2 there and let us pick a lambda 3 here and you have these tiny little impulses sitting at lambda 1, lambda 2 and lambda 3. So, this one is x lambda 1 delta t minus lambda 1, this one is x lambda 2 delta t minus lambda 2 and finally, this one is x lambda 3 delta t minus lambda 3. So, what he says when I put these together putting these 3 impulses together that means taking a sum x lambda 1 delta t minus lambda 1 plus x lambda 2 delta t minus lambda 2 plus x lambda 3 delta t minus lambda 3 results also in summing the corresponding out. So, when you put this into the system what you would get is x lambda 1 h t minus lambda 1 plus x lambda 2 h t minus lambda 2 plus x lambda 3 h t minus lambda 3 this is the consequence of additivity. Now, visualize taking this argument finally and you know over integral instead of a summation. So, visualize these points lambda 1 lambda 2 lambda 3 coming closer and closer and taking all the continuum of points all over the time axis at least you know if you are really fussy and finicky omit places where x t is discontinuous, but otherwise take all other points it hardly makes a difference. In all those continuums of impulses one or two points dropped here and there isolated points of discontinuity is not a problem. So, now we make a formal statement. Now, instead of adding over different lambdas when I bring these lambdas into a continuum and I integrate the integral is the limit of a sum and if I invoke additivity then I also get the corresponding integral in the output and that is the central idea. So, let us write that down. So, what we are saying is I have the same system I apply to it minus infinity to plus infinity x lambda delta t minus lambda d lambda as the input and the system is additive additivity is now invoked. So, I should get the output minus infinity to plus infinity x lambda h t minus lambda d lambda. Now, this is a very very important statement what is this? This is just x of t we made a very profound statement let us spend a minute on it. We have said give me any input x t which is reasonable by reasonable meaning I could think of it as a combination of impulses properly weighted and then brought together in a continuum. Give me any such x t and if I knew what the system did to one of those impulses just a unit impulse located at 0 I know what the system does to any x t. So, the impulse response the response of the system to a unit impulse tells me everything about the system it tells me what output would emerge when any input x t is given to it. A very profound statement indeed and as I said this is kind of central to the whole course on signals and systems we shall say much more about this in the coming session. Thank you.