 So welcome to the fourth session in this course and now it is high time that we did a little bit of formalizing. So we have been informally talking about imports, outports and systems all this while and we have talked about the importance of abstraction. By now I hope you are all convinced that abstraction has its merits. So we need to complete that process of abstraction by making certain formal definitions. So let us do that before we go ahead. What is a sigma? A sigma is a reasonable, now here I am being a little loose just in this part of the definition but I think we all understand what it means. A reasonable function of an independent value. Let me give you an example of an unreasonable function which I just alluded to in the first lecture. So x of t is 1 when t is rational and 0 when t is in rational, t is of course real. Now you know for a mathematician there is nothing unreasonable about this function. But for a signal and systems engineer there is in practical life and tell them going to come out with a function like this. No system will really generate a function like this. And also this function has many demerits, it is nowhere continuous and we will own the question of different demerits and so on. Although these functions have their merits in mathematics we do not want to bring in such functions in our discussion at least for now. By and large functions that you would encounter in real life or functions which could be called idealizations of those that we encounter in real life are what we will accept as signals. I know that is a little way but I believe that is operationally adequate at this moment as a definition of a signal. What is a system? It is a mapping from signal to signal. So now let me put down the system and signal together. So signal is a mapping typically from the set of reals. You know if you are talking about a continuous signal, continuous variable signal. It is a mapping from the set of reals to the set of complex numbers in general. Now please here I am jumping a few steps. You know all this while I have never really explained why I need to map time to a complex number and right now I shall not do that. But I just remember that real numbers are after all a subset of complex numbers. So even if I have a signal which maps time to a real number as you do for example in a voltage signal or in a force signal or in a pressure signal, you could also think of it as mapping time to a complex number except that the imaginary part of the complex number is 0. After a while we will make that imaginary part non-zero with good intentions but we will postpone that for a while. However we keep that possibility over. So we will say a signal is a mapping from the reals to the set of complex numbers. So in that sense now if you look at both these definitions here, the system is a mapping from signal to signals and the signal is a mapping from the reals to the set of complex numbers. So in some sense a system is one mapping above another. A system is a mapping of mappings. A system is a mapping on mapping, one level higher. Of course we have also talked about what is called a system description. A system description now we call it a little more flexibly an explicit or implicit relation between input and output of the system and what are the inputs and outputs? The two parts of the map. So we have said the system is a mapping on mappings. The input signal is a mapping from reals to complexes, complex numbers. The output is also mapping from reals to complex and the input and the output as composite entities have a mapping between them which we call the system description. That mapping can be implicit or explicit. Let us take examples of both. An explicit mapping which says that y t is let us say for example 5 times dx t d. Clearly you know what the output is and we shall use this convention in future. We will use when it is not otherwise specified, we will use y t to denote the output signal and x t to denote the input. This is an explicit map. Why is it explicit? Because if I give you x t there is a direct mechanism for calculation of y t. I can read off y t from x t being given. Let us take an example of an implicit map. An implicit mapping this relationship though it is specified is not a read out kind of relationship. It is something like 2 d squared y t dt squared plus y t plus 3 dx t dt plus 5 x t is equal to 0. Now although you know here the relation between x t and y t there is no read out. It is not easy to read out. Once I give you x t even for all time or whatever you need to do more work. You need to either solve an equation or work on an equation to come to a conclusion what y t is. Also this is implicit for another reason. You need to do a little bit of work to separate the input and the output. Of course here the job is very easy because if you look at this equation you can easily separate the input part and the output part. There is almost no effort there. You could simply say 2 d squared y t dt squared plus y t is minus 3 dx t dt minus 5 x t. But then suppose you had an equation something like this. You had d y t dt plus y t times x t plus x t squared is a constant let us say 3. This is also implicit but this is inseparable. It is very difficult to separate y t and x t here. It takes more trouble. Now we do not want to go any further to make distinctions on these description types. We just want to appreciate that there are all these possibilities in describing systems and system descriptions and we shall be content if one of them is given to us. It tells us about the system. What we are trying to say is we do not want a situation where I have to tabulate every x t y t pair that we do not want. But that is what ultimately a system is. A mapping from the whole x t itself to the whole y t itself. So in principle you should have every possible x t tabulated and the corresponding y t tabulated. That is not what we want. We want something more compact than that and that is what we are calling a system description. It could be explicit, it could be implicit. It could be easy to separate into input and output parts. May not be so easy to separate into input and output parts. Now once we have a system description, we can start asking some questions about that system. For example, the first question that we can ask, suppose I perform three experiments on the system. In one experiment, I give it the input x 1 t and generate the output y 1 t. In the second experiment, I give it the input x 2 t and generate the output y 2 t. In the third experiment, I give it the input x 1 t plus x 2 t. In other words, I add the two inputs that I gave in the first two experiments and I query what is the output? Now for a large number of systems, it is possible that the output also has the same additive law. So suppose we have a situation where for every such x 1 and x 2, if x 1 t plus x 2 t gives you y 1 t plus y 2 t, we then have the first system property, first characteristic of a system being established. So you see at this point, before I conclude this session, I want to bring in the notion of a system characteristic or system property. This is the first system property called additivity. The system is said to obey additivity if this is true for every such x 1 and x 2. If x 1 t plus x 2 t results in y 1 t plus y 2 t, we say the system obeys the principle of additivity. Similarly, in the next session, we shall look at more questions that we can ask about the system and the system properties that we can infer based on the answers. Thank you.