 A warm welcome to the 33rd session of fourth module on signals and systems, the second module in this course. Now, we have got a reasonable amount of maturity in both the Laplace and the Z transforms. We have also seen at least the beginning of how we could translate Laplace transforms into system realizations. We need to do a little bit to understand how this can be done for the Z transform and let us now complete that exercise. So, let us consider a discrete system function. So, here we have an example of a discrete system function. H of Z is 1 plus Z inverse divided by 1 minus half Z inverse mod Z greater than half. Of course, the discrete system function is Y of Z by X of Z and we can cross multiply and we can invert the Z transform, thereby giving us we can rearrange. Now, this is the central idea in all this. This is an equation involving the two sequences X of n and Y of n. In fact, it is what is called a linear constant coefficient difference equation as opposed to differential and interestingly, this also abbreviates to L double CDE. So, we have L double CDEs for rational continuous independent variable systems and L double CDEs for rational discrete independent variable systems. The only difference is in the continuous independent variable system, the D stands for differential derivatives and in the discrete independent variable D stands for difference. So, go back here. This difference essentially means shifted input and output. So, you have the input and it is once and more delayed versions, the output and it is once or more delayed versions. Now, you know we are only taking delayed versions of the input and the output and there is a good reason. If we want causal processing, we can only allow delays and not advances. So, you know we cannot for example, do X of n plus 1 or something like that, not allowed. If you want causality, but X of minus X of n minus 1 is allowed and X of n minus m is allowed for positive integer m. Similarly, Y of n minus m is allowed again for positive integer m, any m and in fact, we can realize this n minus m business by using one step delays at a time. So, therefore, if we go back to the system function H of z is 1 plus z inverse divided by 1 minus half z inverse with mod z greater than half. We can see it is both causal and stable and we can realize it by expanding the z transform. So, we have Y of z is half z inverse Y of z plus X of z plus z inverse X of z. And now, just as we drew a signal flow graph for the continuous independent variable case, we can draw a signal flow graph for the discrete independent variable case. So, you have an X of z here, you bring it here as it is, use a multiplier of z inverse. What is the physical meaning of a multiplier of z inverse? Remember, the nodes here are the z transforms of corresponding sequences and therefore, multiplication by z inverse amounts to delaying by one step. Now, you keep this as it is delayed by one step and then add there. So, here we have generated this part of the term, I am marking it as 1 and I am saying 1 has been generated here. And now, let me assume that there is an output generated somewhere Y of z, let it have come from another node, this is also Y z. Now, delay this by one step that takes care of this and now, multiplied by half and add and this brings you to Y z once again. So, as you can see, the rules for the signal flow graph are the same here, only the components are different, the interpretation of the nodes is different. The nodes are interpreted as z transforms of the corresponding sequences. The nodes are of course, different sequences at different points and then you have multipliers, these directed edges which transport, they are all transport units with a multiplier embedded. The multiplier is in the z domains, you are multiplying the z transform. It could just be a constant multiplier, which case you are multiplying by a constant or it could be a multiplier dependent on z. When you multiply by z inverse, you are delaying by one step. So, when you have multiple directed edges feeding a node, all that is transported on these edges is deposited at the receiving node. And from the receiving node, if there are multiple edges going out, all of them carry what is already received at that node. It is not distributed uniformly or something of the kind, each of them carries what is there. Each node is very generous to its outgoing edges. It allows all the outgoing edges to carry everything that it has, but what comes to the node is a sum of what is transported on the incoming edges. Now, go back to the signal flow graph. So, here notice that again you have a loop here and the loop indicates feedback. You fed back the output to the input indirectly and you know feedback is a part of recursion as you see it here too. The output depends on the output and the input, that is why you need feedback. Now, what we need to do is to look at the general kind of output which a rational system is likely to give. And that is true both for continuous independent variable rational systems and for discrete independent variable rational systems. We shall give an indication. Let us now come to a general situation. You have a rational system, you have an input and an output. Now, for the moment let us assume the input is also rational. Let us take that situation. That means it has a Laplace transform which is rational. Of course, Laplace transform if we are talking about continuous independent variable or Z transform if we are talking about discrete independent variable. So, what would the output look like? So, you know what I am saying now is true for the Laplace transform and the Z transform. Let me emphasize that. So, output transform is equal to input transform multiplied by system function whether it is the Laplace transform or Z transform. Both of these are rational. Hence, this is also rational. So, what is the situation now? You have a rational input, a rational system function and a rational output. And we can invert the output in the respective transform domain whether Laplace transform or Z transform. The essential principles of inversion are the same. We decompose into partial fractions by identifying the distinct poles and the partial fractions have terms corresponding to each pole. You look at the region of convergence. Look at each term with the corresponding pole. Identify its inverse based on the relative location pole whether way the region of convergence. All this holds for both. We do not need to keep saying it. We have said it often enough in the past. So, each term can be inverted. Now, what kind of terms would you have in the output? So, to answer that question we need to segregate the poles. You see essentially the terms depend on the poles. So, let us ask. Output poles come from input poles which are not system poles. System poles which are not input poles. Now, sometimes the poles can coincide. So, coincident system and input poles and we shall give the corresponding terms a name. So, there are corresponding response terms from the partial fraction expansion. These poles give what are called the forced response of the system. These give what are called the natural response of the system and these give what are called the resonant response of the system. So, when you have a rational input feeding a rational system, you get three parts in the response. One of those parts could be absent for specific situations, but there are three parts in general. A forced response, a natural response and a resonant response. The forced response comes essentially as a reaction to the input because the poles the same as the input poles. What changes is the relative contribution of those poles, but not the nature of the expression that is being fed. The natural response essentially resembles the impulse response of the system or it takes terms from the impulse response of the system. Again the coefficients get played around with because of the partial fraction expansion and finally, the resonant response is something new that emerges. In fact, many of us might have heard about the story of the soldiers walking on a bridge in rhythm. Now, often a bridge can be thought of as approximated by a rational system with certain poles and woe be gone, some of those poles might land up on the imaginary axis in the S plane. Now, if those poles are simple, then the impulse response just has sinusoidal terms or a constant term, but if by chance the marching of the soldiers creates an input which can be approximated as having a pole at the same place, then the pole which was simple now becomes double or triple and then instead of just having a sinusoid as a part of the response, there is a resonant response where you have a sinusoid multiplied by time or multiplied by a polynomial in time and there you have a problem. If you multiply the sinusoid by a polynomial in time, you are talking about ever increasing oscillations and that is why it is said that when soldiers march across a bridge, they are advised to break their rhythm because that resonant response might come into play and do a lot of damage to the bridge on account of ever increasing oscillations. So, resonant responses also have a physical interpretation just as the forced response does and the natural response does. In the subject of control systems or in a subject of digital signal processing, one would investigate these responses in great depth. These responses have a lot of physical significance. They determine what you expect when you feed certain kinds of inputs to certain kinds of systems. I have just given you an example. So, what we have done is to lay the foundation for being able to deal with all such situations, at least a reasonably good class of such situations. We have been able to capture the essence of the transform method generalized also to unstable systems as you can see in the Laplace domain and in the Z domain respectively for continuous independent variable and discrete independent variable systems. All throughout, what was the aim of this module? This module was intended to generalize the results of module 2 and also to carry forth certain ideas that we have learnt in module 1 and also module 3. So, in some sense, this module brought into a new domain, the transform domain what we learnt in the first three modules. This transform domain is more general. All that we need to do is replace S by j omega and you go back to the Fourier domain in module 2, replace Z by erase the power j omega and you go back to the frequency domain as you know in module 2 and 3. Let us write that point down. So, S equal to j omega takes you from the Laplace transform to the Fourier transform. Z equal to erase the power j omega takes you from the Z transform to the discrete time Fourier transform, but this is provided the imaginary axis respectively the unit circle is in the region of convergence. So, there is nothing very complicated now. We have seen the relationships all fitting in and falling all into place. If the Laplace transform admits also particularizing to the Fourier transform, we do it this way. If the Z transform admits particularizing to the discrete time Fourier transform, we do it this way. If we cannot, then we have to make do with the Laplace transform inherently or the Z transform inherently and that can also help us analyze systems which are not stable or where the frequency response is not defined. There are many intricacies and many small notions at points where we have alluded to them, we might not have gone into all the depth that is needed. That requires a deeper understanding and a deeper study. We have gone reasonably deep into the Laplace and the Z transforms in this module, but we have not tried to cover all angles that is rather difficult in one module like this. And in fact, it is not even advisable because it can become confusing. There are places where you must have felt there is a bit of a mystery, there is a bit of unclear discussion that is meant to inspire you to look deeper. The subject is very deep indeed. These techniques that you have learned in all these four modules and particularly in this fourth module are used in many branches of engineering not just electrical engineering or mechanical engineering or systems engineering or control. They are very useful ideas, they are used in many disciplines. And it is very nice that you have come all the way here up to this stage in the fourth module being patient enough to learn these ideas. I have no doubt that these ideas will stand you in very good stead in your understanding of signals and systems in the future. And I encourage you to reflect on them again and again and to look beyond what we have done in this module and in the modules that preceded this one. I do hope we shall have a chance to meet again in further courses and learn things in greater depth. With that then, I bring you to the end of this fourth module in terms of the basic sessions and we show all the very best. Thank you very much.