 So, welcome to the 11th session. And now we answer the question, why we have looked at systems with all these three properties or why we have asked about these three properties in the first place, additivity, homogeneity and shift invariance. What is it about these properties that makes systems which possess them special? Well, if you go back to our first discussion, one of the things that we promised was to evolve certain general principles for first understanding systems by bringing in abstraction and then building systems. In a nutshell, the answer is when a system has all these three properties, either by design or by accident, the system is very easy to analyze and rather easy to build. Now, it will take us a while to appreciate. In fact, it will probably take us all through the course to appreciate this. But right now, let us begin in this session by looking at a very simple principle that at least takes us toward that direction to appreciate why these three properties give us something very unique. And to do that, let us first understand a very simple principle, namely that you could think of any reasonable, now I explain what I mean by reasonable, any reasonable signal as a combination of very narrow pulses. First, let me explain that idea to you informally. So, here we go. Let me show this to you graphical. It is much easier to understand graphical. So, you have the time axis there and you have what I call a smooth signal, reasonable signal if you might. You know, what I mean by a reasonable signal is one that does not have more discontinuities than we can handle in a given finite interval and something that does not have any great abnormalities like the signal I told you, which is 1 at the rationals and 0 at the irrational. We do not want to go for those kinds of signals. Otherwise, also the signal is continuous almost everywhere except at some isolated points. That is what I mean by reasonable. So, let us look at a region in which the signal is continuous like you have here. So, signal is continuous in this region and we take this interval which I am marking here in which the signal is continuous. And I am now going to use this principle. I am going to build this principle of pulses. So, I divide this signal into very narrow regions on the time axis, very narrow indeed, as narrow as you could make it. Let me do that. So, first let me take intervals of size that one can discern and then we could imagine that interval becoming smaller and smaller and smaller. Let me take any one interval here. Take this interval for example and visualize that interval going smaller and smaller and smaller. Look at it carefully. You know, if the interval were to grow smaller with a continuous signal, smaller interval tends to almost constant function. So, for a continuous function on smaller and smaller intervals, the function is almost constant. So, essentially what I am saying is on a very small interval and please understand this carefully. We can think of the function. Let us say the interval size is delta. We can think of the function as a pulse. Essentially I am saying I can almost think of the function like this. You can visualize delta become smaller and smaller. And the height of the pulse is equal to the value of the function in that interval. So, of course, in brackets quote unquote constant, constant value of the function in that interval. Now, go back to the previous drawing that I had. So, here what we are saying essentially is that in each interval we have a pulse. Essentially now you can visualize. I am going to draw it in a minute, but you can now visualize. You have this little pulse, you could fix its height to be whatever. You know, you could fix it in such a way that the height is proportional to the inverse of the interval size. You will see why after little while. So, if the interval size is delta, I will fix the height to be 1 by delta. So, the area contained under the pulse is fixed to unity. You will realize later that it is the area which is critical ultimately not so much the height. So, you have this little pulse. You shift that pulse by different shifts which are multiples of the interval size. And each such shift you multiply it by the value of the function in that interval which is assumed to be almost a constant. And when you put all these together, you get something very like the original function that you had. Now, let us draw this to understand. So, what I am saying is you have this continuous, let us take a small segment of the continuous function. Let us take this basic pulse. And let us see if we can construct this continuous function out of this basic, out of this little basic pulse that we have drawn here delta and 1 by delta in height. Well, we will see that in the next session. It will take some explanation. Thank you.