 Welcome to the sixth discussion that we shall have. In the fifth discussion we have given a few examples of systems both some of which were additive and some of which were not additive. Now we look at those very examples and ask the question of homogeneity about those examples as I asked you to do when we concluded the fifth session. I am assuming that you must have thought about it before we embark on this discussion. So let us look at the first example that we took there y of t is x of t square. Is this homogenous or does this obey homogeneity? Let us ask. Multiply x t by a constant c. So let us input c times x to the system. Clearly the output is going to be c x t the whole squared which is c squared x t the whole squared. Now although the output is also being multiplied by a constant that constant is not the same as the constant by which the input is being multiplied. The output is multiplied by c squared not c and therefore the system is not homogenous. I started with a negative exam as before because it is important to emphasize the small errors that we often make. It might look like the output is also multiplied by a constant but for homogeneity you need the output to be multiplied by the same constant. Now let us look at the other examples that we had y of t was x of t plus x of t minus 1 minus x of t minus 2. And we query what happened when c times x t is given to the system. C times x t would result in c times x t plus c times x of t minus 1 minus c times x of t minus 2 which is of course c times you can take c common and make it c times x of t plus x of t minus 1 minus x of t minus 2 which is clearly c times y t. And therefore the system is clearly homogenous. Now notice again that the fact that you had essentially a linear combination or a sum of inputs taken at different times gave you a homogeneous system. This is one sure short way of creating a homogeneous system. In fact this kind of a construct will be useful to us in many contexts as we will slowly see when we proceed through the course. But for the moment let me introduce one more new example here before we take the third example that we had in the previous discussion. So let us take the example of the system y of t is t times x of t. What it means is the input is multiplied at every instant. When I say instant I mean instant of the independent variable. So if the independent variable is time we are saying at every instant of time the input is multiplied by the value of the independent variable. Let me make a remark on this system what are we doing in this system. For example if the independent variable time what are we doing? We are looking at the time and we are multiplying the input at that time by the value of the time. That means an input which comes at time t equal to 10 is multiplied by a number larger namely 10 than an input which comes at t equal to 5. So let us look at an input output relationship or an example of an input output relationship in this particular system. So let us assume x t is the famed unit step that we had. We are introduced this unit step earlier. Now we will formalize it and give it a name too. So unit step is equal to 1 for t greater than equal to 0 and 0 for t less than 0. The unit step is often denoted by u of t and if we give this input unit step to the system what would the output be? The output would essentially be a ramp t times u t to be precise. Now you see what I mean. Whereas the input was constant from t equal to 0 onwards at the value 1 the output grows as time grows and we will take this as an example to analyze the next property that we are going to see but first let us complete the discussion on the current property namely homogeneity. So we ask is this system homogenous? Let us look at it. Well give c times x t to the system and what do you get? You get t times c x t and very clearly this is c times t x t that is very easy to see that c times y t and therefore the system is clearly homogenous. Now here it is interesting you have done something in the system which changes as a function of time in the sense that what you are doing at time t equal to 5 is not the same as what you are doing at t equal to 10. In spite of that the system is homogenous but it will reflect this discrepancy between what we do at different times will reflect in a new property which we will see soon after this one. Before we go to that property as I said let us complete the last example that we had as far as additivity was concerned namely consider the system y of t is x t plus 5 and ask is this system homogenous? Now apply c x t to the system and you can see the output is going to be c x t plus 5 that is not the same thing as c times x t plus 5 in general except for the trivial case c equal to 1. So clearly the system is not homogenous in fact therefore it is neither additive nor homogenous. However I have placed both these examples on the same page here with an intent that they must be investigated for the next property that we are going to see and let me introduce you to that property in this discussion and then take up the examples to analyze the property in the next discussion. The next property of a system that we are going to query is what is called shift invariance. Let us first formally define shift invariance. Shift invariance as the name suggests asks what happens when you shift the input without making any other change. So suppose you had x t being given to the system to produce y t as an output. What would happen when you gave x of t minus t 0 to the system same system of course if the output is equal to y of t minus t 0 and this holds for all x and t 0 that is very important again. It must hold for all x and for all t 0. If this is true then we say the system is shift invariant. Now before we conclude the discussion and ask the question that we are going to do before we take up the examples let me give you an informal meaning of the term shift invariance. Let us say you have a laboratory apparatus and you performed an experiment today on it and you see the cycle of outputs that you got today in the apparatus. You know so you gave it some kind of sequence of inputs and you noted the trail of outputs today and you came back to the laboratory tomorrow set up the same apparatus and perform the same experiment. If you expect to see the same results tomorrow and this should happen no matter which day or at which time you begin to conduct the experiment. Then we say that apparatus is shift invariant except for shift in time there is no change in the output provided you make only that change in the input namely you have only shifted the input in time. You have not changed the input otherwise except to shift. Shift invariant is essentially a statement of weather systems behave as they do now at any point in time and we shall see more of this in the next discussion. Meanwhile think again about those examples that we have to answer whether they are shift invariant or not. Thank you.