 So, welcome to the 8th session in which we shall first answer the question that we posed in the previous session, namely if we look at the system y of t is x of t plus 5 is the system shift invariant and the answer indeed is yes, because if x of t minus t 0 is given to the system, the output is x of t minus t 0 plus 5 and that is of course equal to y of t minus t 0. So, this is also an example of a system which is shift invariant but neither additive nor homogeneous. Now, all these ideas are very well until we keep to theory as we are doing all this while but we would like to relate them now to real life and let us relate them to the example that we saw earlier on in this course. For example, let us relate these ideas to the RC circuit or the resistive capacitive circuit that we have built earlier. For indeed let us recall that circuit it was a resistance and a capacitance in series connected to a voltage source. You could denote the voltage source like this with the polarity as shown and the output was taken across the capacitor. So, the capacitor voltage was the output, the input was the voltage source itself, the voltage input here and of course we can ask whether this system is linear or not or rather it obeys additivity and homogeneity or not. We know the equation of this system. We call that v t it is or v in t if you want to call it that was equal to r times c dv output dt plus v output t or in other words x of t you call x t the input is r c d y t dt plus y t. This was the describing equation of the system description. Now we can ask whether this system is additive and homogeneous or not and the answer is simple. Let us write down two different equations for two different inputs x 1 t and x 2 t. You have x 1 t is r c d y 1 t dt plus y 1 t and x 2 t is r c d y 2 t dt plus y 2 t. In fact, now we shall bring in a little variant on the notion of additivity and homogeneous and we shall bring in one test for both of them called superposition. Let us introduce the principle of superposition. We ask whether a linear combination of t in this so alpha times x 1 t plus beta times x 2 t also results in the same linear combination of the outputs or not. So take this equation and call it 1 and take this equation and call it 2 multiply 1 by alpha and 2 by beta and add and we get alpha times x 1 t plus beta times x 2 t is equal to r c alpha times dy 1 t dt plus alpha times y 1 t plus r c beta times dy 2 t dt plus beta times y 2 t. And after rearranging this it is very easy to see that alpha x 1 t plus beta x 2 t results in r c times d dt of alpha y 1 t plus beta y 2 t plus alpha y 1 t plus beta y 2 t and therefore we conclude that alpha x 1 t plus beta x 2 t results in alpha y 1 t plus beta y 2 t and this holds for all possible x 1, x 2, alpha and beta. When this holds for all possible x 1, x 2, alpha and beta we say the system obeys the principle of superposition. This is a very important concept the principle of superposition. In fact we spend a minute in understanding the meaning of the word superposition. Literally superposition means putting one on top of the other and that is exactly what you are doing. You are doing something slightly more. You are first scaling the two inputs and then putting one on top of the other and you are asking whether the output suffer the same fate. They each get scaled by the same constants and they get superposed as do the inputs. Do they suffer the same fate and does this happen for all possible inputs in all possible linear combinations with constant alpha and beta? If so the principle of superposition has been obeyed. Now we show that the principle of superposition subsumes both additivity and homogeneity. So, if the principle of superposition is obeyed then the system is additive and the system is homogeneous. Let me first show the principle of additivity based on superposition. In particular choosing alpha equal to 1 and beta equal to 1 gives the principle of additivity and choosing alpha as the scaling constant with beta equal to 0 gives homogeneity. So, superposition means additivity and homogeneity both together. We shall see more about these principles in the next discussion. But before we go to the next discussion I would like to pose a question before you. What changes might the system suffer which would destroy this principle of superposition? The electrical system I mean. What are the lucky say you know changes that that electrical system needs to suffer? What changes do I have to respect in that a possible changes to make the system non superposable or where it disobeys the principle of superposition? But do we need to have some changes in the element the way the elements behave? For example, well we will come back to it in the next session. Thank you.