 So, welcome to the seventh session and recall that we did examples of homogeneity in the previous session and we defined the idea of shift invariance. What was the meaning of shift invariance? When I shifted the input by a certain shift or a certain quantity on the independent variable, the only change in the output was to shift by the same quantity on the independent variable. Now, let us take all the examples of previous session and ask the question of shift invariance on those. So, let us begin with the example of y of t is x of t the whole square is the shift invariant we ask. And of course, we reason that if x of t minus t0 is given to the system what comes out is x of t minus t0 the whole square which is indeed equal to y of t minus t0. So, the system is indeed shift invariant and you now summarize this system is shift invariant but neither additive nor homogeneous. Now, we ask the question about the second system that we saw namely y of t is x of t plus x of t minus 1 minus x of t minus 2 and we apply x of t minus t0 to the system clearly the output is going to be x of t minus t0 plus x of t minus t0 minus 1 minus x of t minus t0 minus 2 which is of course equal to y of t minus t0 that is not too difficult to and therefore, the system is indeed shift invariant. So, here is a beautiful example of a system which is shift invariant and also additive and homogeneous it has all the three properties that we have discussed so far additive homogeneous and shift invariant. Let us now take the third example that we saw you see you will now understand why I introduce those specific examples in the previous two discussions because you will see a distinction now when you look at these three properties. Let us come back. So, let us look at the third example which I had introduced in the previous session with the intent of investigating this particular property t times x t. Now, this is a tricky one when I give x of t minus t0 to the system what comes out is t times x of t minus t0 and not t minus t0 times x of t minus t0. So, let me give you an example let us take that very unit step input. So, we give the unit step input to the system 1 for t greater than equal to 0 and 0 for t less than 0 and query what the output is. The output is of course a ramp as we saw. So, from t equal to 0 onwards it give you t times u t essentially a growing straight line here. Now, let us see what happened if you gave it for example u of t minus 1 unit step shifted by 1 forward. So, essentially you have from t equal to 1 onwards you have the input being 1. So, you know we write this as u of t minus 1 being given as the input here and we query what the output is. Now, please note that the output would of course begin to be non-zero from t equal to 1 as you expect. So, it would be 1 at t equal to 1 1 times 1 and from there it is going to grow linearly. So, it is essentially going to be 1 multiplied by t from there onwards you know that is interesting. So, for example at t equal to let us say this is at t equal to 1 but let us say I take t equal to 5 at t equal to 5 the output is going to be 5 when this output as we see it this output is not the same not the previous one shifted by 1. This is very important it is not the previous output shifted by 1 forward and therefore with this one counter example we have shown the system is not shift invariant. You see you have to be very careful when you try and establish a property or when you try and contradict a property you will notice that here it was enough for me to use one counter example to contradict the property. I took an example of an input I shifted the input by 1 step and I asked whether the output was shifted by the same amount same shift and I found the output was not shifted by the same amount and therefore I could conclude the output is not the same as the input shifted by whatever shift the input had for this particular counter example and that was adequate to say the system was not shift invariant. However, I cannot use the strategy to conclude that the system is shift invariant. So, I cannot take an example and say well here you are there is this input I observe the output I shift the input by 1 step forward I find the output is also shifted by 1 step forward and therefore I could conclude the system is shift invariant no. If you wish to prove the system is shift invariant you must do it independent of which input you have given and which shift you have made you must prove it for all shifts and for all inputs. But if you wish to disprove a property it is very easy pick one counter example you see it is like trying to prove or disprove the statement that all residents of a particular colony are liars. Suppose you go to the colony and you wish to prove that all residents of that colony are liars and you go to the house of one of those residents and you detect that resident telling lies always. So, being a liar that is not enough to conclude that all the residents that colony are liars because you have found one liar in the colony does not mean everybody in the colony is a liar. But if you wish to disprove that statement go to the house of one of the residents in the colony and observe him telling the truth at least sometimes and then if by liar you mean somebody who tells lies all the time and if you detect this person telling the truth sometimes there you are you have disproved that statement. So, this is a general principle that we must remember in signals and systems to disprove a statement a counter example is adequate but to prove a statement an example is inadequate anyway coming back to this particular example. So, here you notice that we showed with a counter example that the system is not shift invariant and therefore we can make a conclusion about this system. The system y of t is equal to t times x of t is not shift invariant but it is additive and homogeneous and of course now I am going to leave one exercise for you and before we conclude that session I shall put down the exercise for you. Exercise is analyze y of t is equal to xt plus 5 for shift invariant. Is it shift invariant? And this is a little exercise for you to work out before we come to the next session. Thank you.