 Welcome to the next discussion on system properties. You will recall that in the previous discussion that we had, we had identified one important system property namely, additivity and in additivity, what we did was to perform three experiments on the system. In the first experiment, we gave one input to the system and recorded the output. So, we had an input x1 and an input x2 respectively in the first and the second experiments. In the third experiment, we gave the input x1 plus x2 and here I mean x1 and x2 to be functions of time. So, each of them are signals in their own right. The corresponding output y1 and y2 in the first two experiments are recorded again as a function of time and in the third experiment, we query whether the output is also y1 plus y2. If it is and if this is true for all possible x1 and x2, then we say the system is additive. Now, let us take an example of a system description which is not additive to make the point clear. So, suppose for example, you had a system description where the output yt was equal to the square of the input xc, xc is the input. Clearly, this system description is not additive. That is very simple to see. If x1t produces y1t which is of course x1t the whole squared and similarly x2t produces y2t which is x2t the whole squared, then x1t plus x2t would produce x1t plus x2t the whole squared. And this is of course equal to x1 squared t or x1t the whole squared plus x2t the whole squared plus 2 times x1t x2t and it is this term that makes the system non-additive. If that term were absent, the system would be additive but that term makes it non-additive. Let us take an example of a system now, a system description corresponding to an additive system. I first came an example of a system description which was not additive because I want to bring out what is required to make a system additive. What you notice is that the moment there is a non-linear operation in the system description the chances are the system would not be additive. Let us take an example of a system description y of t is x of t plus x of t minus 1 minus x of t minus 2. We show this system with additive. So, very clearly if x1t is input to the system the output would be y1t given by x1t plus x1t minus 1 minus x1t minus 2 and similarly for x2. So, x2t would result in x2t plus x2t minus 1 minus x2t minus 2 and x1t plus x2t would result of course in x1t plus x2t plus x1t minus 1 plus x2t minus 1 minus x1t minus 2 plus x2t minus and you can see very clearly that this is equal to y1t plus y2t. The system is indeed additive and let us declare itself. Now here you also notice what makes a system additive. When there are essentially linear combinations of the input coming both from the current time and some other time the system is essentially additive. But let us also take an example now of a system which might deceptively seem additive but is not. So let us take the following system. You have y of t is x of t plus 5 a displacement so to speak or what you call a DC shift system if you mind like clamping the system to a higher DC value. Now here let x1t result in y1t then y1t is of course x1t plus 5 and similarly let x2t result in y2t where upon y2t becomes x2t plus 5 but then when x1t plus x2t is applied to this system what results is x1t plus x2t plus 5 and not x1t plus 5 plus x2t plus 5 which would have been equal to x1t plus x2t plus 10. So system is not additive clearly as I said it is deceptive because one might have been tempted to think that it is additive because there are linear operations all over the system description linear in the sense that you know you have sums of isolated terms so you might be tempted to think that it is additive but it is not additive. So essentially a DC shift in that sense is not additive so so much so for the property of additive we shall keep saying more examples as we go along but now let us bring in one more property that we need to discuss and that property is called homogeneity or scaling. So here in homogeneity or scaling as the name suggests what we do is to scale the input by a certain constant and ask whether the output is being scaled by the same constant and no other change is being seen in the output and whether this is happening for all possible outputs that and of course inputs that you can give the system. Let us first define the property formula. So here we go we first say Xt results in Yt by virtue of the system this is the input to the system and this is the output of the system remember both of them are sigma in entirety we multiply the input by a constant scale the input by a constant meaning multiplied by a constant which is called C and we query what the output is. Is the output equal to C times Yt for all possible that is very important for all possible X and C if so the system is homogeneous or it obeys scaling the system is homogeneous or it obeys scaling a very important definition. Now here I must utter a word of caution I mentioned briefly in one of the previous discussions that although the independent variable is taken to be real that is T for example is real T could be time it could be space the dependent variable could in general be complex we have not justified why we should allow complex numbers right now but we are kind of accepting it and we are noting that real numbers are a subspace or a sub they are a subset of the complex numbers in fact the real field is a sub field of the complex field. So therefore there is nothing terribly unacceptable in extending our outputs to complex numbers or inputs to complex numbers as a function of the independent variable and we shall do so the reason why I am emphasizing this is that the constant C here can be complex too as we go along we understand why we have to accept complex numbers but for the moment we will just take it for a given. Now we shall soon take up the next discussion where we look at homogeneity with a few examples but meanwhile I encourage you to look at the examples that we saw for additivity and answer whether they are homogeneous or not an exercise for you before we meet again for the next discussion thank you see you soon.