 So, welcome to the ninth session and in this session we shall continue to answer the question that we had raised in the previous session, namely in the RC circuit that we saw as a real-life example to test additivity, homogeneity and shift invariance, what little changes do we need to introduce to make a difference to these problems. So, first of course, let us complete our discussion on the additivity, homogeneity and shift invariance of that circuit. Let us look at that circuit. We had this RC circuit there and we agree that the output yp would be the voltage across the capacitor and the input would be the input voltage to the circuit, r and c are the resistance and capacitance values respectively. And we had the system description given by x of t is RC dy t dt plus yt. We had reasoned that the system obeys the principle of superposition meaning that x1t resulting in y1t and x2t resulting in y2t implies that alpha x1t plus beta x2t results in alpha y1t plus beta y2t for all possible x1, x2, alpha and beta. So, the principle of superposition automatically gave you additivity and homogeneity and we now need to ask what is it that would destroy this superposition in terms of the circuit behavior or elemental behavior. You know, suppose you took the resistance, what little change could you make in the resistance so that this superposition principle is no longer true. Suppose the resistance were nonlinear. In other words, suppose instead of the resistance obeying the relationship voltage being proportional to current, there were a tweak of nonlinearity there. The voltage is not quite proportional to the current, but to some power of the current. Now, you might wonder why we are considering an example like this in real life. Well, when you go down to microelectronic devices, resistances do behave nonlinearly at times. So, if you were to construct an RC-like circuit in a small little semiconductor situation, if you were to construct it in a microelectronic situation, you might have to worry about this. You might have to worry whether the so-called resistor that you have built in an integrated circuit has a linear relationship between the voltage and the current or not. In other words, the first situation which destroys superposition is if Vt and It are nonlinearly related. For example, suppose Vt is equal to R times some power of I. Let us say It to the power of let us say some gamma. Gamma is a positive number. Now, of course, you know, when you write something like this, you have to be careful about interpreting it for negative I t. So, for negative I t, you would probably need to interpret this to the power of gamma in such a way that it has physical meaning. Whatever it is, when you do something like this, you make the circuit nonlinear and then it destroys the principle of superposition. In fact, there is another situation in which the superposition principle would be destroyed and that can be done in a very simple physical way. We can essentially evoke what we learnt in the last few discussions, the principle of a DC offset. So, suppose you put a DC offset in this circuit. In other words, you construct the circuit like this. You have a resistance, you have a capacitance, but then you also have a voltage source here, a DC voltage source. Let us say this voltage source is 2 volts with the polarity as shown. And of course, you have the input voltage here which we have called X of t or the input X of t with this polarity. And now you have the output voltage or the output which is the capacitor voltage plus 2 volts. There is a bit of trouble here. What is the describing equation? What is the current in the circuit? The current in the circuit is of course, again c times dVct dt where Vct is the capacitor voltage. Now with this, we can write down the total input Xt is equal to r times the current plus Vct plus 2 volts and that is r times c dVct dt plus Vct plus 2. And Vct as you see it is Yt minus 2 or in other words, Y of t is Vct plus 2 volts. Therefore, dy t dt is of course, equal to dVct dt, that is true. So, interestingly, here you have the relationship Xt is rc dy t dt plus Yt and this also obeys the principle of Soville. But now write the equation back in terms of Vct. So, here we have continued to obey superposition because it is the same equation essentially. But now write the equation back in terms of Vct and call Vct as the output. In other words, note that Xt is equal to rc dVct dt plus Vct plus 2 and call Vct as Yt, now or Y tilde t, a new output. You have the equation Xt is rc dy tilde t dt plus Y tilde t plus 2 and now you have trouble. You will notice that this is very like that DC offset that we had in our system description. And I leave it to you as an exercise. Let me write that exercise down for you. Show that this system does not obey superposition. Now let us ask the other question of the system. Is it shift invariant? Let us try to answer the question informally. What do we mean by shift invariant? You would essentially mean that if I shifted the input by a certain shift, let us call it t0, the output should be shifted by the same shift. Now look at the system description. Of course, here let us avoid that 2 volts DC offset. So, let us go back to the original system and ask the question first. This system only. Remember, if this were Xt and Yt, the system description, we will just quickly write it again, is Xt is rc dy t dt plus Yt and then X of t minus t0 is clearly produced by Y of t minus t0 because you will realize that dy t minus t0 dt is the same as dy lambda d lambda with lambda substituted by t minus t0. That is not too difficult to see. This is indeed true. That means, essentially we are saying that the derivative operator itself is shift invariant. So, because the derivative operator is shift invariant, when I take a derivative, when I take a sum, these two operations together do not destroy shift invariants and therefore, this whole system is shift invariant. Of course, that is not too difficult to see even by looking at common sense explanations. If I were to give a certain voltage pattern to the circuit and look at the output of the capacitor and if I were to shift that same pattern in time and look at the output of the capacitor again in time, the only change would be that the output of the capacitor would be shifted by the same amount in time. There would be no other change. So, the system is shift invariant. Now, I leave you with a question to be answered. I have that offset system which I have constructed with a voltage source of 2 volts and I have looked at both cases, taking the output of the capacitor and the source together and taking the output only across the capacitor with the 2 volt source present. Which of these systems is shift invariant? One of them or both of them or none of them and why? I leave you with this question until we meet again in the next session. Thank you.