 A warm welcome to the 22nd session of the 4th module in signals and systems. We have been looking at the rational Z-transform and we go further with that objective here. In fact, we now establish what a rational Z-transform means from a system perspective and then we also put down certain general rules for inversion of a rational Z-transform. So, of course, when you talk about the systems context in the Z-transform, we have a context very similar to the Laplace transform. Let us write down clearly. So, you have this discrete system, discrete linear shift invariant system. It has an impulse response h of n and we hope that this has a Z-transform. Z-transform should be an expression and a region of convergence. Now, if the input to this discrete system x of n also has a Z-transform and so does the output y of n, then we know that y of z is x of z times h of z by the convolution property and therefore, h of z which is called the system function here like the case of the Laplace transform tells you what is the ratio. The system function, this ratio of the output z-transform to the input z-transform importantly is independent the ratio of the input z-transform for LSI systems. That is very important. Now, you see what is the role of again the factors in the denominator and the factors in the numerator. We want to again give them names and in fact, we will choose to give them the same names. So, for rational Z-transform denominator equal to 0 gives what are called the poles. So, you know the term poles and similarly for the numerator equal to 0, we get the 0s. So, you know the word poles and 0s has the same connotation. See the pole comes because you can think of the magnitude of that Z-transform as sitting like a tent on the z plane and where there is a pole that tent is held up, it goes towards infinity. Where there is a 0, it is nailed down to the ground and a tent stands by virtue of the poles that hold it up and the nails that hold it down. So, poles and 0s are very important in a Z-transform, but there is actually one thing much simpler here and that is that it is not so difficult to deal with the numerator being greater in degree than the denominator and so on. So, let us put down first a process for inversion of a Laplace transform of a Z-transform in general. The general process, now here you know when you have a rational Z-transform, you can either think of it as rational in Z or rational in Z inverse, both of them are equivalent. So, for example, you could think of 1 by 1 minus half Z inverse also as by multiplying both numerator and denominator by Z, Z by Z minus half. So, it is rational in Z and also rational in Z inverse, they are equivalent. Now, what we can do is to take one of them with which we are more comfortable. So, let us take the rational in Z inverse interpretation. That means, we will always write down the particular Z-transform as a rational function in Z inverse. So, for example, even if you had something like this, you had something like say Z squared minus Z divided by 1 minus half Z inverse whatever it is, you would force it to become a rational function of Z inverse. So, that can be done very easily. You can take Z squared common from here in the numerator and write 1 minus Z inverse divided by 1 minus half Z inverse. So, this can be written as Z squared times rational function in Z inverse. Now, you know this as you can see you will need to pull out a power of Z. So, it could be a positive or negative power of Z. Now, all that this does, this power of Z is to cause a shift. So, for example, Z squared essentially means shift backwards by two samples. So, we can operate this last after inverting the rest of the Z-transform. We can operate that power of Z. So, let us assume without any loss of generality that we indeed have a rational function of Z inverse. Now, the first step is to check the numerator degree and denominator degree. So, if the numerator degree is greater than or equal to the denominator degree in Z inverse of course, we first carry out a long division. So, for example, let us take the factor that we had you know in the previous example that we were considering here we had this factor. So, let us consider this factor 1 minus Z inverse by 1 minus half Z inverse and let us do the long division there. We would need to multiply by 2. So, I have and thereby we get 1 minus Z inverse by 1 minus half Z inverse is essentially 2 plus minus 1 by 1 minus half Z inverse. So, essentially after long division you would get a quotient and remainder. It gives you a quotient which would essentially be a finite series in Z inverse and it would leave you with a remainder which we keep in the numerator. Now, unlike the Laplace transform where we had to do some work to understand what happens when you have polynomials in S, here dealing with a finite series in Z inverse is no problem. In fact, if you go back to this particular example, if you get a finite series coming out from here of course, the only finite series here is a constant 2. You can combine that finite series with this factor this power of Z. So, in general, so suppose for example, you have a finite series that looks like this say a 0 plus a 1 Z inverse plus and so on and then you had an initial factor of Z to the power D. So, I am saying essentially that this initial power of Z is this and the finite series is this, then you can combine them. So, you can write a 0 Z to the power D plus a 1 Z to the power D minus 1 plus and so on and inverting this is very easy. We simply get a 0 delta n plus t and so on. So, you get a train of impulses, discreet impulses, a finite length sequence and if we invoke the linearity of the Z transform, then one can easily see that you can treat that finite series separately, dispense with it, get that finite length sequence and now deal with the remainder divided by the denominator separately. Now, as far as the remainder divided by the denominator is concerned, we have already seen how to deal with it. We can deal with it by using partial fraction expansion and then that factor of Z to the power of D that hanging power of Z can be taken care of finally, by making a shift. So, we are now well equipped to deal with inverting a rational Z transform. So, if I have a rational system in the Z domain, I know how to find the impulse response. If I have a sequence with a rational Z transform, I know how to find the sequence. The only thing is of course, I have to be able to factorize the denominator into its separate distinct poles and each pole would give us one kind of term. What kind of term? Let us also make that clear. You see every time you differentiate with respect to Z, you are effectively creating a multiplication by n. Of course, give or take some shift and so on. So, what it means is that if I have a factor of say 1 minus alpha Z inverse, let us write that down. If we have a pole at Z equal to alpha, repeated m fold that means, I have a factor. There are things in the numerator and there is a factor of 1 minus alpha Z inverse to the power m in the denominator. Then what is the contribution of this pole in terms of the inverse Z transform? Essentially, this contributes a poly x term like before and this poly x term is a function of n. It would look something like a polynomial in n of degree capital M minus 1 multiplied by alpha raised to the power of n. So, here the only change is you have an alpha raised to the power of n and you would need to associate it either with u of n or u of minus n or whatever you know appropriately shifted. So, you would need to. So, you know you could say u of n or u of minus n or appropriately shifted unit steps. Now, the choice between u of n and u of minus n comes from whether the sequence is right sided or left sided. And the reasoning is exactly and there are so many parallels between the Laplace transform and the Z transform. The same reasoning you know if it is to the exterior of a circle, you have one situation, if it is to the interior of a circle, you have another situation. And therefore, in the next session we are going to see further about what happens to the properties of a rational system. When we look at it from this point of view, we exploit this left sided or right sided sequence point of view. We shall see more in the next session. Thank you.