 Welcome to this 22nd session where we continue our discussion on discrete systems and we continue from where we stopped in the previous session on discussing one of the properties. Now in the context of discrete systems, we need to build the same properties as we had for continuous variable systems namely homogeneity, shift invariance, causality, stability and of course you could talk about memory. The properties are essentially the same, they have the same broad connotation but there are minor differences in the way we define the properties and those need to be therefore written down explicitly. So we shall take it upon ourselves in this session to put down explicitly all these properties making those little changes that are required to bring it to the context of discrete variable systems. So let us begin by looking at homogeneity as a property. Now recall that homogeneity in the context of continuous variable systems involves scaling the input by a constant. If you are to, we take a discrete system, let us call the discrete system S, we give to it the input x of n. Now please note that x of n is a whole sequence, it is a discrete signal as you might call it, a whole sequence producing the output y n, again a sequence. Now it is a sequence mapped to a sequence. I must emphasize this point again, of course in continuous variable system we encountered this point. A system is a mapping from signals to signals, it is not a point to point mapping, if it is a point to point mapping it becomes a system without memory. The moment a system has memory there is no point to point mapping there. It is a whole signal mapped to a whole signal and it is a mapping from all input signals to all corresponding output signals. The same is true of discrete systems, it is a mapping from the whole sequence to the whole sequence here. I am emphasizing this because when we talk about homogeneity I am talking about scaling the sequence not individual points though they mean essentially the same thing they are different in context or in connotation. So let us now scale this sequence X or the input by a constant alpha. The question is what happens here and that is what determines whether the system is homogeneous or not based on what happens for all possible X and for all possible alpha. Now remember that we have also allowed complex alpha with good purpose. Well in case alpha times Xn also produces alpha times Yn and this happens for all possible alpha this is very important for all possible alpha and for all possible Xn then we say the system is homogeneous. Another way of saying it is the system obeys the property of scaling or it is a system that obeys the property of homogeneity whatever you like to say they all mean the same thing simple enough. Now how do we understand this let us again go back to the tax context population and tax context. So for whatever reason suppose you had double the number of people for example in that state the tax collection could double that is what homogeneity says. In fact double is just one example of an alpha any multiple multiply the population by any factor and the tax gets multiplied by the same factor. Now please note if you multiply the input by alpha homogeneity insists that the output be multiplied by alpha and no other constant that is important alright. Now let us take the next problem we can go a little faster here because we have defined these properties in the context of continuous variable systems and we do not need to go through all the explanation again. However we should be careful to bring out the salient points. So next property is shift invariance. Now here we have to be a little careful because when we talk about shifting the input there are only certain possible shifts namely discrete shifts. So unlike the continuous variable case where you could shift by any tau, tau being any real number here you can only shift by integers that is the little point of difference. So suppose you have this discrete system s and you give to it the input x of n and observe the output y of n and then you shift the input by shift in the input what we mean is we give it x of n minus d where d is any integer this is important it has to be an integer. We ask what is the output in that case and if it is true that s when given x of n minus d produces y of n minus d for every possible x and every possible d this is very important then we say the system is shift invariance. So what is important here is that this should happen for every possible x and for every possible d it is not enough that it happens for some values of the shift and not for others. Now again let us go back to the tax example what shift invariance essentially says is that the taxation rules remain as they are in time what was the taxation rule now remains so next month remains so next year remains so after 10 years. Now of course you will appreciate from this example that shift invariance is a big demand on a system to insist that rule of taxation remain the same over many years is demanding a lot from the government. So therefore in practice too in real life shift invariance systems are going to be rare in the true sense and if you ask me no system really is shift invariant forever and ever that means for all possible d yes in a reasonably big range of d you could have the system behaving in a shift invariant manner but beyond a certain point that shift invariance could be lost. Now the same as true of homogeneous if I scale the population by 100 if we use the same taxation rules of course the tax collected would also scale by 100 but then in a certain sense realistic systems behave differently when you have a larger population all of which pays tax very honestly perhaps the taxation rules could be made a little more liberal and therefore the system would lose homogeneity that liberalization is introduced this is what you would mean in this context by losing homogeneity. So if you kept the same taxation rule no matter what how large the population is then homogeneity is obeyed but if you try and base your taxation rules on the size of the population then homogeneity could be destroyed. Now you could similarly construct other example but what I am trying to bring out is that although we define idealistic properties here we also bring down to reality how far it is reasonable to assume these properties to hold it and I would like you to think more on this after you listen to this session and discuss among yourself too and come back with your own observations on to what extent these properties can be assumed of systems taking different examples and when it becomes unreasonable to demand that they hold forever. We shall look at some more properties in the next session.