 A warm welcome to the 16th session in the fourth module on signals and systems. We have now looked at the importance of rational systems in the continuous independent variable case. We have recognized what the importance is and let us write that down once again with great clarity. Rational systems, they correspond to linear constant coefficient differential equations L double CDEs as we call them. You will remember again that we talked about this to the end of module 1 and we had emphasized that this is the typical kind of system description that you encounter in nature at least as far as linear approximations are concerned, linear shift invariant approximations are concerned. In the first module, we also saw a great deal of these examples. We saw some RC circuit, we saw the corresponding mechanical analogs and of course, you could build higher order systems that is not an issue. But this class of systems is very fundamental, very important in the whole domain of engineering. So rational systems have a physical importance. Of course, we also have certain mathematical conveniences. For example, now we are very clear how we can invert a rational Laplace transform. And now let us formalize that and introduce something more in the context of rational systems. So you see a rational of course, you know we were talking about rational Laplace transforms and I brought in the idea of a system where that Laplace transform corresponds to the impulse response of a linear shift invariant system. So in general, we are talking about rational Laplace transforms. So rational systems are linear shift invariant systems with a rational system function. So there are two or three things that need to be emphasized here. The moment you say rational system, you automatically mean it is linear and shift invariant and you automatically mean that it is impulse response has a Laplace transform. Let us make that very clear, let us write that term. So rational system is one linear shift invariant with an impulse response which has a Laplace transform. The Laplace transform of the impulse response is rational in the Laplace variable s. So essentially it is of the form numerator series in s divided by denominator series in s and notably both these series are finite series. This is important. The finite part is important. It is not alright to have infinite series. Let me give you a very simple example of a system which is irrational and where you can of course have a very simple infinite series. It is interesting. If you have a simple delay, so let us show that, consider the following linear shift invariant system. Let us call it s. So it is a very simple system description. Y of t is x of t minus tau. Tau is fixed. It is a constant. We can very easily obtain the system description here and the corresponding impulse response. So the impulse response is very simple. Apply an impulse. It would be simply delta t minus tau. So simple. And of course this has a Laplace transform. The system function which is the Laplace transform of the impulse response is of course e is the power minus s tau. It is an irrational system function and therefore this is an irrational system. So what is wrong with that? What is wrong with being irrational? Well it is only rational systems which are truly realizable. This is a very interesting idea. Let me emphasize that it is a very important idea we have said. Only rational systems are realizable at least as of the current date, meaning they can be realized or they can be implemented with finite resource. This is important with finite resources. You see you can always ask to implement a system with an infinite resource but that is not practical. So that is very interesting. We are saying if you have an ideal delay or an ideal shift operator it is not exactly realizable or it is not realizable to an exact form by using a finite resource. You can approximate it. In fact this is one of the difficulties in control systems. So you know there people do bring in irrational systems into control, into control engineering but there are issues with it. You can also realize a delay to within a certain approximation that is you know you can make it behave like a constant delay for a large range of inputs and so on. That is possible. So you can always approximate an irrational system as much as you desire but you can never realize an irrational system exactly with finite resources that is the problem. So there is something bad quote unquote bad about irrational systems in that they are not really realizable and of course if you look at this system function e raised to the power minus s tau it cannot be expressed as a ratio of finite series. However, it can be expanded as an infinite series. We can certainly write a Taylor series expansion for this function and let us do that just for completeness. We can expand. This is the Taylor series. In fact you will recall that the Taylor series of e raised to the power of x is summation l going from 0 to infinity x to the power l by l factorial where we should interpret 0 factorial as 1 and 1 factorial also as 1 and following that l factorial is equal to l times l minus 1 factorial for l greater than 1. So for example 2 factorial would be 2 into 1 and 3 factorial would be 3 into 2 into 1 and so on. You will recall that this factorial has a role to play in permutations. So if you have n objects and you want to permute them the factorial is involved recall that yourself. Anyway here it occurs more as a mathematical part of the Taylor series expansion. But what I am trying to say is that if you look at this this is indeed a fine it is not a finite series but this is an infinite length series and unfortunately it cannot be converted to a ratio of finite series. In contrast you know why I am making a point of this cannot be converted is because we may tend to think that the moment you see an infinite length series there you are born it is not rational that is not true. Let us take a very simple example or a counter example to illustrate my point let us take simply 1 by s minus a. Now we can easily expand this if we take into account the conditions on s. So you know we can always write this if you like as 1 by you see you can write that is 1 plus s by minus a you know and then multiply by minus a again in the denominator. So you could do this and then you could of course expand 1 by 1 minus x in the form summation n going from 0 to infinity x to the power n under conditions under the condition mod x less than 1. So here too if somebody wishes to if he puts the condition mod s by a less than 1 which of course is not very meaningful he could make an expansion it would also be an infinite length expansion but we do not we are not talking about that you see the point is whether you can collapse that infinite series into either a finite series or ratio of 2 finite series that is the point for e raised to the power of x we cannot do it and therefore we say the system is irrational. I just wanted to make it clear what irrational means and why irrational is a problem and again this is to reemphasize why we are so fond of rational systems at least in a first course. So now that we have understood the importance of rational systems we must now see how you characterize rational systems. I said you know we are more generally talking about rational Laplace transform but since we want to associate them with systems we would first like to talk about rational systems and then talk about rational Laplace transform the principles are the same you do not have to really make any new principles when dealing with rational Laplace transforms in general. Anyway let us look at the rational system there is a ratio of 2 finite length series. Let us put the numerator and the denominator separately equal to 0. What does it give us? What is the significance of this? How does it help us invert the Laplace transform? These are questions which will not only help us invert the Laplace transform corresponding to system function or the rational Laplace transform but they will also give us certain fundamental insights into the system. We shall take this up in detail in the next session. Thank you.