 A warm welcome to the 32nd session of the fourth module in signals and systems. We have been doing so many of the formal constructions and formal derivations in the previous few sessions. Formalism after a point becomes dull and dry. It is important that we start relating things to systems now and getting a feel of how we can actually understand system construction at least for rational systems, if not irrational. So what we wish to do now is to introduce how one can draw a block diagram of a system and in fact we also wish to connect it to the system description. So once I have a system function and we will begin first with the continuous independent variable case, we would like to draw a block diagram for that system with basic components. So let us get down to that job. The rational systems and we begin first with the continuous independent variable case. Rational systems correspond to linear constant coefficient differential equations. Let us take what we call the rational system of the following form. In fact let us take an example. Let us assume we have a rational system with system function s plus 1 by s plus 2 with the region of convergence given by real part of s greater than minus 2. Now how do we write this in the form of a linear constant coefficient differential equation? So let us write down what the system function means first. y of s by x of s is s plus 1 by s plus 2 and we can cross multiply and we can invert the Laplace transform. Now this is indeed a linear constant coefficient differential equation. Remember that is what we were looking for when we said that there was a correspondence. Now this is alright in terms of writing a differential equation. But in practice suppose I want to draw a diagram which realizes this, so I wanted a system built out of basic components. What basic component should I use? Let us look at the equation clearly. What basic components are needed? You need an adder or more than one adder and of course you can have two input adders and use many of them. You need a differentiator. You need it here and you need it here. You need a constant multiplier like this. So essentially it is three kinds of components, two input adders, differentiators and constant multipliers which should enable you to realize this constant coefficient differential equation. The question is should we be using differentiators or should we be using integrators? We will talk about that after a while. But right now suppose I want to use differentiators. Let us see how I can draw this system in the form of a block diagram or more appropriately we will call it a signal flow graph. So we are now going to talk about representing signals rather systems and signals not just signals with what are called signal flow graphs. A signal flow graph is a collection of nodes and directed edges. In our context the nodes would represent the Laplace transform or the Z transform of corresponding signals. The directed edges are like transportation systems. You could think of them like that or transport units with a processor embedded. And here these multipliers are in the Laplace or Z domain. So what we are saying is as follows, let us take this very same system that we had. We have s plus 2 times y of s is s plus 1 times x of s and we can rewrite this 2 times y of s is s minus s times y of s plus s times x of s plus x of s. So now we can conceive of a signal flow graph. It is best understood by example. You have nodes corresponding to y of s and x of s. Now you use a transport unit with just a unity multiplier. So if there is a unity multiplier it means you are just transporting as is and you are bringing it here. Now what do you need to do? You need to do something similar. So assume that y s is also transported like that. So you see you have an equation which says 2 times y s is something with obtain by combination. So let us take 2 to the other side. We cut off the 2 here and we will divide by 2 everywhere. So of course you need a half excess everywhere. So you might as well multiply by half once and for all here. You can have a multiplier of half. Now what do you need to do? You see y s has been generated. So let us assume y s was generated here and it is brought here. But before generation you differentiated y s. You know you want to take care of this term here. So you differentiated y s that means you multiply it by s. So at this node we have s times y s. Now we need to combine s times y s multiplied by minus half and then half excess. You know half excess has already been created. You need to carry it as it is and you also need to differentiate. So you need to put two edges here. Carry it as it is and differentiate it and then carry. And these need to be added. And now at this node I have generated half excess into s plus 1 and here we have s times y s. You need to multiply this by minus half and you need to add this and this. So I will do that. I will bring another node where I transmit this as it is and I add this. And this is what generates y s. The loop is complete. This is very interesting. What we have done here? Let us go over it again. This is a recursive building of the signal flow graph. By the way this whole thing is a signal flow graph and we will say a little more about the signal flow graph. So as you see a signal flow graph has nodes, it has directed edges. Each of these directed edges should be thought of as transport units. So what do they do? Each directed edge starts from a node and ends at a node. What does it do? It takes what is there at the start node, multiplies it by the multiplier which is on that edge and then dumps it at the end node and take for example this node here. This node has incoming edges, these are incoming edges and these are outgoing edges. Now as far as the incoming edges go, all that the incoming edges bring is deposited at that node and forms the value of that node. The outgoing edges all carry whatever is at that node. So it is not as if you have to distribute it among all the outgoing edges. There is no conservation principle here. So you simply let all the outgoing edges carry whatever that node has. So if you look at it, some nodes have only outgoing edges, these are called source nodes and some nodes have only incoming edges, they are called sink nodes and all other nodes have both outgoing and incoming edges, so they are called intermediate nodes. So remember the value at a particular node is the sum of all that is deposited by the transporting directed edges coming into that node. What goes out on each of the outgoing edges is simply what is on that node, all of them carry the same thing. So this is how we can now understand the signal flow graph. This is a signal flow graph and this is a very common way of representing a block diagrammatic representation of rational systems at least. It could also correspond to irrational systems, we are not getting into that right now. So you see you get what is happening here, you see. You have multiplied x by half, so it is taken care of this factor of half here in both these places. You have essentially multiplied that by s plus 1 and that brings you here. Then that has taken care of this term and then you have ys, assume it is being generated here. You multiply it by s, so it takes care of this and then you multiply it by minus half and then that is added to this. So you have this node and finally that node is the same as ys, so you bring it here. Now you notice something very interesting, there is a loop here. That means you have a set of directed edges which start at a given node and reach back the same node if you go around them. Start from this node, you come here and then you come here and then you come here and you reach back here. So this is an example of a signal flow graph which has feedback, there is a feeding back of the output towards the input side and you are then combining some function of the input with it and generating the output. Recursion requires feedback, so if you are expressing the output in terms of itself and some function of the input, loops are bound to be there. Of course, at this point we are not getting into all the details of how you analyze such a signal flow graph in its own right, there is a whole science behind that and maybe in some subsequent course we can get into it. But right now an awareness of the signal flow graph is important. Now a remark about the way we have constructed the signal flow graph fundamentally. We have constructed the signal flow graph essentially by using multipliers in the Laplace domain and we use differentiators. When you say multiply by s, you are talking about differentiation. Is differentiation a good idea in continuous independent variable systems? In general, no, differentiation tends and in fact let us understand that. If you look at the frequency response of a differentiator, put s equal to j omega and you get just j omega. So what is its magnitude? Notionally, you know of course there is a bit of a catch, it is not really correct to talk about the frequency response of a differentiator because it is an unstable system. But anyway we could cheat a little bit and talk about it and say that it applies in a certain set of finite frequency. So finite frequency is okay. This is the magnitude, mod omega, clearly this emphasizes higher frequencies more and more and higher frequencies are where you find noise typically. Many disturbances, many unwanted components are often present at higher frequencies and these are not a part of your design, they come by virtue of practical implementation. So if you put differentiators into the system, unwittingly you would be emphasizing these unwanted components or noise in most situations. In contrast, if you used an integrator in place of a differentiator, let us see what would happen. When you put s equal to j omega, you get 1 by j omega and therefore the magnitude would look something like this. Now clearly this would over emphasize lower frequencies. In fact, over emphasize the zero frequency that means the constant is a terrible thing to give an integrator. But if you do not give constants, then the integrator is okay because high frequency components are suppressed. So very often an integrator would be used to suppress noise. And therefore it is more traditional to use integrators in realizing system, rational system. So let us see how we could realize the same system by using integrators. I shall show you a few steps in that direction and then leave you to complete the exercise. So let us take the same system that we have. So we will rewrite this in terms of integrators. So we will multiply and divide by s inverse. S inverse is an integrator and now we have a neat signal flow graph. We can write y of s by x of s. So we can now build a signal flow graph corresponding to this. Let us do it right here and take care of terms one by one. So I have a node x of s. I need to multiply it by s inverse and also keep it as it is. Transmit it as it is. Multiply by s inverse. This is the integrator and then add. So I have generated this term here. Now again same way I will keep y of s at this point, use an integrator on it. So I have gone up to here. Multiply by minus 2. So I have taken care of this and now I need to add these two and this should generate y of s. In fact if I really wish to I can pull out y of s explicitly. So this is the signal flow graph with integrators instead of differentiators for the same system. So as you can see there is more than one way of drawing a signal flow graph for a given system and different structure. These are different structures for the same system. The defining equation is the same, the system function is the same but you have different structures. So you have given a feel of how you can have structures for systems. Now which structure is better, which structure is worse is a deep subject by itself. I have given a hint here very often we might prefer integrators instead of differentiators in continuous variable systems. However that is also not universally true. There could be some situations where you want a differentiator for one reason or the other or you may want a combination of differentiators and integrators. And many structures are possible as you can see. Now I am going to conclude this session by just indicating what we would do when the independent variable is discrete. When the independent variable is discrete all that we need to do is to build a similar signal flow graph but we will use constant multipliers, delays, unit delays. So you need a Z inverse term there, a unit delay and of course two input adders as before. I am not going into the construction of those because one would require a whole course on discrete time processing where we would understand such constructions and look at them in depth. At this point I only wish to indicate that the principles of signal flow graphs can be carried through the discrete independent variable as well. We will see a little more about rational systems in the next session. Thank you.