 So, in a way we have more or less equipped ourselves very well to deal with systems LSI systems where the z transform of the impulse response is rational and the input also has a rational z transform. Let us take an example. You see let us take an LSI system this I will work straight away in the z domain now with impulse response h n. Now I introduce a very important idea the z transform of h n if it exists is called the system function point you to a couple of important warnings here. A system function is not just an expression therefore a system function is an expression with a region of convergence. This is often forgotten in the literature and it is unjustifiably so because the context makes the region of convergence clear. We shall also see very soon how the context can tell us a lot for example if we know the system is causal. Obviously the impulse response must be either finite length or right side it cannot be left side. So, you know the context does tell us a lot about what to expect of the region of convergence. So, very often that region of convergence is not specified, but then technically a system function is a z transform and therefore has an expression and a region of convergence. Now with that observation let us take an example of an x n being given to the LSI system to produce y n and let us take a specific example of x n and h n. So, we have take x n to have the z transform that is straight away work in the z domain to have the z transform 1 by 1 minus 1 4 z inverse 1 minus 1 3rd z inverse and h n to have the z transform of course here I must specify the region of convergence. So, I have different I have 3 possible regions I shall take mod z greater than 1 3rd as the region of convergence and let h n have the z transform 1 minus half z inverse times 1 minus 1 3rd z inverse again with mod z greater than 1 3rd as the region of convergence greater than half and sorry has to be greater than the greater of the 2. Of course, the z transform of the output is going to be the product the z transform of the input and the impulse response. So, we have y z is 1 by 1 minus 1 4th z inverse 1 minus half z inverse 1 minus 1 3rd z inverse and of course here 1 minus 1 3rd z inverse the whole squared. The notice that you have now a multiple pole at z equal to 1 3rd introduced by virtue of convolution or by virtue of the input being subjected to the impulse response. Now of course, you would carry out a partial fraction expansion but you need to know the region of convergence. So, region of convergence here is the intersection here we could take it to be the intersection the intersection you of course happens to be mod z greater than half. So, it is greater than 1 3rd greater than 1 4th and greater than half as well. So, you can invert this we can invert this is not it that is very easy to do. So, y z can be decomposed it can be decomposed in the form a by 1 minus 1 4th z inverse plus b by 1 minus half z inverse. Now please note plus c z inverse plus d divided by 1 minus 1 3rd z inverse the whole squared and you know how to decompose each of these how you know how to invert each of these terms. But what is to be noted here is that by virtue of the input and the impulse response coming together or by virtue of the input being applied to the system even though the input and the impulse response have simple poles at z equal to 1 3rd the output has a double pole. That means although you had just a 1 3rd to the power n kind of term an exponential with exponential factor 1 3rd present in the input and impulse response separately when they came together they gave rise to a new term of the form n times 1 3rd raised to the power of n you know whatever u n and so on that is besides the point. But an exponential multiplied by a polynomial the new term is created by virtue of the same pole being present in the input and the system this would give us a clue how to deal with situations in which the system and the input have similar kinds of terms the system by system I mean the impulse response of the system when they have similar kind of terms there is what is called resonance. In fact we will look at this in slightly more depth when we talk about linear constant coefficient difference equations systems which resonate with the input produce new terms of this kind. Anyway what we intended to do today was to illustrate how we could deal with rational z transforms in much more depth with much more variety and we also intended to put down some principles of inversion. Let me conclude this lecture with a couple of remarks before we move on to the next lecture which would deal further with systems which have rational system functions. The first point I wanted to mention here was that the word system function and transfer function are interchangeable in discrete time signal processing we do not really distinguish between them. In networks in electrical networks people use driving point function and transfer function there is a slight difference if the input and output are at the same place we call it a driving point function like the impedance. But if there are different places we call it a transfer function the notion of having the input and output different at the same location different inputs and outputs the same location does not happen in discrete time system. So we do not have the notion of a driving point function at all we only have transfer functions or system functions as we understand them. Secondly what we have noticed is that inverting the z transform is an art it is like you know if you remember writing differentiation is a process we know how to differentiate functions I mean find the derivatives of functions. Integration of functions is an art it involves knowing derivatives recognizing derivatives and inverting them the same is true for the z transform obtaining the z transform is often a process you know it could be done by using standard approaches of summation of series but inverting the z transform is an art it involves recognizing forms and inverting them. There is another way to formally invert a z transform if you go back to the interpretation that we gave of the z transform in the beginning namely the discrete time Fourier transform of a suitably exponentially weighted sequence. Now I put it as a challenge before you come out with a formal way to invert the z transform based on that interpretation can you treat the inverse z transform as the inverse discrete time Fourier transform of some properly chosen sequence and then get back the original sequence from that properly chosen sequence that is a challenge with that challenge then we will come to the end of this lecture and we proceed to look further at rational systems in the next lecture. Thank you.