 A warm welcome to the 24th session of the fourth module of signals and systems. We have now looked at Laplace and Z-transforms together and we have in fact focused on rational transforms all the while. Just for this session, we will not restrict ourselves only to rational transforms although whatever we discuss will also apply to rational transforms. So let us ask a very important question. You see, what we have noted is that if you look at a linear shift invariant system and if it is a continuous independent variable system, you can ask whether its impulse response has a Laplace transform. If it is a discrete independent variable system, you can ask whether its impulse response has a Z-transform. If the answer is yes to these questions respectively, we say the system has a system function and the system function is in fact the Laplace transform of the impulse response or the Z-transform of the impulse response as the case may be. The system function is as complete representation of the system as was the impulse response. So whatever we could get from the impulse response, we should be able to get from the system function. Now one important question, you see, the moment you say linear shift invariant, you have already talked about additivity, homogeneity, shift invariance, three properties have already been covered. Now you need to ask about the other two properties. So let us put down the question that we want to address. If a system and of course it has to be a linear shift invariant system, it cannot be otherwise, has a system function. That means if it is a continuous independent variable system, then you are talking about the Laplace transform of the impulse response. If discrete independent variable, now either way the system function gives us complete information about the system. So therefore, it should also be able to tell you about two important questions related to the system which we will now ask. Since complete information is available, we should know about causality and stability of the system. Now what we are going to do in this session is to address this question. How can we determine causality by looking at the system function? Let us now set out to answer that question. Now what does causality mean? In fact let us take the two in parallel, they will be very closely related. Let us take the continuous independent variable system first. The causality is equivalent to the following. The impulse response h of t is essentially 0 for all negative t and therefore the system function as we would know it is an integral from 0 to infinity of h t raised to the power minus s t dt. Now you see what are the kinds of regions of convergence that you have in Laplace transform? They are essentially vertical there between vertical lines, they are bounded by vertical lines in the S plane. So you see what is we have to capture this idea of h t being non-zero only in the positive side or the non-negative side of the time axis. And where will this positive or negative side affect the convergence? You see we have to think about it. What after all, what is it that determines convergence or non-convergence or what do we have to look at in terms of the variable s only the real part of s? So we have to worry about what has happened to the real part of s and how the fact that h t is 0 for all t less than 0 can be reflected with some values of the real part of s definitely coming into the region of convergence that is the thing. So you know in the whole idea here is can we by looking at certain contours in the S plane and checking their inclusion or otherwise in the regions of convergence come to a definite conclusion about causality. So in fact one thing is very clear let us now look at that clear. Let us consider the real part of s to be equal to sigma and let us consider sigma greater than 0 in particular. So you have integral 0 to infinity h t e raised to the power minus s t so minus sigma plus j omega t if you like d t and what is very clear is that since this you see now let us look at the modulus of the integrand is mod the modulus of this quantity only this part and we can simplify that since the modulus of e raised to the power minus j omega t is equal to 1 only this mod h t raised to the power minus sigma t figures and now let us consider sigma tending to infinity you see with sigma tending to plus infinity mod h t this quantity definitely tends to 0 for all t positive and that is where the t positive is important you know the fact that you are considering only positive t becomes important when you look at sigma tending to plus infinity and you can see that if you look at this expression this has no choice but to converge guaranteed to converge as you go to the right extreme you know this is the right extreme so to speak of the Laplace plane. So therefore what is our conclusion? So causal resize continuous independent variable system with system function h s with the region of convergence r h definitely has the real part of s tending to plus infinity included in the region of convergence. Let us sketch it to understand what we are trying to say we are saying if this is the s plane then what we call the right extreme contour you know this is sigma and this is omega. So right extreme contour or real part of s tending to plus infinity is definitely in the ROC. Now it is a natural question to ask what is going to happen to discrete independent variable systems and there we would have to invoke the Z transform. So one way to understand is what is the correspondence for the right extreme contour? Where would the right extreme contour map to or go to you know I mean what does it correspond to in the Z plane? Let us first reason this out independent for a discrete independent variable system. So we have the system function if it exists either the form summation h n Z to the power minus n, n running only from 0 to infinity since the impulse response is 0 for all n less than 0. Now in the discrete domain or in the Z transform what kinds of regions of convergence do we have essentially we have regions between two circles centered at the origin. So we need to worry only about the radii of the circle just like in the case of the continuous independent variable system we needed only to worry about the real part of the s variable. Here we need to worry only about the modulus of the Z variable. So let us take a Q from the context of the continuous case. What is that value? What is that contour in the Z plane which will become critical when you consider only non-negative values of n? You see we want to do something which works only for non-negative values n equal to 0 onwards n equal to 0 to infinity. The moment there is a negative value you should have trouble. So what contour in some sense captures all those non-negative values. Again as I said if we take a Q from the continuous case let us consider more than tending to infinity. You see we have two extreme contours here the innermost here it is in and out not left and right. So innermost and outermost let us consider the outermost contour. Let us look at the sum and let us look at the modulus of the sum and now for all n greater than 0 mod Z to the power of n or rather minus n tends to 0 as mod Z tends to infinity that is very simple to see. So it is very clear that mod Z tending to infinity this is a contour. This contour must be in the region of convergence and we can show that graphically what we are saying is if you have the z plane like this take the outermost rather the you know the most outward contour this must be included. In fact it is these two extreme contours real part of S tending to plus infinity the right most contour in the S plane and mod Z tending to infinity the outermost contour in the z plane which tell you about causality of systems. And in fact for rational systems this is absolutely necessary looking at the outermost contour or the right most contour respectively for the z plane and for the S plane tells you about the causality of the corresponding system. If that contour is in the ROC the system is causing if that contour is not in the ROC the system is not causing there could be trouble. Now there is an issue here you see in fact I have been saying all the time that you have to be careful about these extreme contours you know we keep saying they have essentially the same ROC. But you have to worry about extremes that is where this becomes important sometimes the presence of one nonzero sample on the negative side of n has to force this mod z tending to infinity being excluded in the ROC you know it is interesting. So everything else can be there except for mod z tending to infinity let us take an example see that h n be the following I am using what is called the arrow notation now you have a sequence of 5 samples beginning with n equal to 0 and you can find its system functions. So this is the impulse response you can find the system function it is very easy to see that mod z tending to infinity is definitely in the ROC as mod z tends to infinity the system function which we could call h of z tends essentially to 1 because all these terms vanish. Now let us push it backwards by one step in fact I will show the working right here all that you need to do is simply replace this 0 by minus 1 and now we can write down the new system function. So you have to put z to the power minus of minus 1 that is z to the power 1 into 1 plus minus 1 into z to the power 0 and so on. And now you can see what happens as mod z tends to infinity all the green terms vanish these disappear this one persists but this is just minus 1 but this one diverges and therefore the system function diverges. So it is interesting initially you had a causal system no problem the outermost contour was included then you shifted it backwards by one step suddenly that mod z tending to infinity was not included anymore. So we understand the importance of the outermost contour in the z plane and the extreme right contour in the s plane and we have understood how causality can be determined thank you.