 I strongly recommend that the class should review and revise the concepts of partial fraction expansion. In fact, let me write that down as a task to be done. It is a very important task that we should do when dealing with z transforms, revise the principles of partial fraction decomposition of rational functions. I would just like to point out some important things that need to be seen in this revision of partial fraction principles. One of the things is that when you have repeated poles, you must not forget that you cannot obtain the numerator simply by that principle of multiplication by that term and then you know setting z equal to that pole, you cannot do it when there are repeated poles at that point. You need to use slightly more you know intelligent strategies there. So you know I recommend since you know I mean one expects that partial fraction decomposition is done in basic engineering mathematics, I recommend that you review those ideas. Right? It is important. Secondly one must also know how to deal, how to obtain partial fraction expansions with you know by after looking at the degree of the numerator and denominators. You see if in the variable with respect to which you are making the partial fraction expansion, the degree of the numerator is more than the degree of the denominator, you must first extract a series, one must not forget that. Let me illustrate what I mean, you see I am saying suppose you had 1 minus let us take the very simple case of 1 minus half z inverse by 1 minus 1 third z inverse. Now this is not in the standard partial fraction expansion form. In the partial fraction expansion form it should be written down as you know a polynomial or a series in z or z inverse here of finite length plus some constant divided by 1 minus 1 third z inverse and we can obtain this series by a process of long division here. What do you mean by long division? Let us illustrate it. So long division is you know what we have conventionally studied even with integers, but here it is applied on the rationals. So 1 minus half z inverse long divided by 1 minus 1 third z inverse. So of course it goes once and you remove a 1 minus 1 third z inverse leaving you with 1 third minus half z inverse on top. This is the remainder so to speak and quotient. Now you know here the long division actually I intentionally did what is actually incorrect here. Incorrect in sense is not useful, it is not incorrect. But you know the long division here needs to be done on z inverse as the lead. So if we do the z that long, so you know this quotient and remainder would not help us here. So in fact we should do the long division the other way. We should write this as minus half z inverse plus 1 long divided by minus 1 third z inverse plus 1 and you need to multiply minus 1 third by 3 by 2 that gives you minus half z inverse plus 3 by 2 and when you subtract you get a minus half there. So therefore 1 minus 1 third z inverse 1 minus half z inverse by 1 minus 1 third z inverse is 3 by 2. This is the finite series plus minus half divided by 1 minus half z inverse. So this is now the remainder and this is the quotient. I intentionally illustrated both forms of long division here to distinguish between the one that is useful here and the one that is not. When you are trying to express all your partial fraction terms in terms of z inverse then the long division should be done by the lead being on z inverse. So you treat the you know it is a polynomial in z inverse. So you must put the highest power of z inverse first in the long division process. So you know I showed you both those approaches just to distinguish between which one should be used and should not be used in this specific case. Of course it depends on you know you can make an expansion with the rational function being treated as in z inverse or being treated as in z. Either way you would finally arrive at the same sequence. The sequence cannot change but you would arrive at it by slightly different processes just a matter of you know slight adjustment of the process and you know there is no serious difference between the two approaches. Anyway so much so to illustrate that in general I mean you could generalize this. So you know whenever you have a rational z transform any rational z transform can be decomposed in this manner. A finite series in z or z inverse plus a sum of terms sum of what are called proper fractions you know proper fractions. I will use the term proper fraction and proper fractions in a more generalized sense here. What is a proper fraction you know as we understand in the context of integers for example 9 by 7 is treated as an improper fraction because the numerator is greater than the denominator. Now here the greater or lesser is in the sense of degree. So what we are saying is we call a term a proper fraction if the degree of the denominator in the variable with respect to which you are expressing the series is greater strictly greater than the degree of the numerator and improper otherwise even if they are equal as we saw in the example a minute ago it is treated as an improper fraction. So let us write down proper fraction means numerator degree is strictly less than the denominator degree in the variable of consideration. Let us take a couple of examples just to drive the point home. Let us take 1 minus 2 z inverse by 1 minus half z inverse is not a proper fraction 1 by 1 minus half z inverse is a proper fraction. 1 minus 2 z inverse by 1 minus half z inverse the whole squared is a proper fraction we now need to qualify that proper fraction see all these are proper fractions in fact this would be a proper fraction this would also be a proper fraction. If I multiply this by 1 minus 1 third z inverse it would still remain a proper fraction but we do not want something like this you see so we need to qualify a little more you know let us go back to that slide where we said sum of proper fraction we do not want the sum of proper fraction just not any proper fraction here we want proper fractions with poles all at one place sum of proper fractions with unilocated poles. Now unilocated poles does not mean there is just one pole there could be multiple poles but all at the same place. So let us take examples of both 1 minus 2 z inverse by 1 minus half z inverse the whole squared is acceptable as a proper fraction with unilocated poles on the other hand 1 minus 2 z inverse divided by 1 minus half z inverse times 1 minus 3 z inverse if you like is not it is not acceptable it is a proper fraction but it is not a proper fraction with a unilocated pole we want proper fractions with a unilocated pole and we know why we know how to invert such terms we have seen that with a combination of the geometric progression and the property of differentiation we are in a very convenient position to invert such terms.