 A warm welcome to the 11th session of module 4 of signals and systems. We have now established a very important way to describe linear shift invariant systems where the impulse response has the Laplace transform if it is on the continuous independent variable or a Z transform if it is on the discrete independent variable. In fact, we have seen that we can characterize such a linear shift invariant system by one single function of the complex variable S or Z that is meter then trying to characterize it by a whole function described as in terms of time or as a function of t that is more difficult to describe. This is nice and compact. So, let us now review that idea once before we proceed further to see further ramifications of that. So, linear shift invariant system where the impulse response has a Laplace transform if on the continuous independent variable and has a Z transform if on the discrete independent variable is said to have a system function. Now, the system function is essentially the Laplace transform or the Z transform of the impulse response and we are now going to write down both of these statements using what is called multistatement notation. So, I am going to introduce now an example of multistatement notation or multistatement format. I give an example of a multistatement here. The system function of a linear shift invariant system is the Laplace transform respectively Z transform. So, we use the words respectively of the impulse response given that the system uses the continuous independent variable and here we say respectively discrete independent variable. So, you know here when you read this statement you must read it with the respectively is carefully obeyed. You are essentially making two statements. You are saying the system function of a linear shift invariant system is the Laplace transform of the impulse response given that the system uses the continuous independent variable. You are also saying in the same breadth that the system function of a linear shift invariant system is the Z transform of the impulse response given that the system uses the discrete independent variable. This is called the multistatement. When there are multiple statements which are very similar to one another and you would like to show them juxtaposed placed next to one another so that you can compare and contrast when we use this format. And it is very useful in many contexts in definitions in signals and systems and in signal processing in general. In fact, in a lot of contexts in mathematics in general or even in the sciences or engineering. When you have similar ideas you know you are trying to define them. Anyway this was just a question of format. Now there is another way to think of the system function of course the more fundamental way is to think of it as the Laplace transform of Z transform of the impulse response. But you can also think of it as a ratio of two Laplace transforms of Z transform. So of course one thing we have to be very clear. The system function for a continuous independent variable system respectively discrete independent variable system. Note we are going to make a multistatement again. Is a Laplace transform? I must emphasize this. It is a Laplace transform respectively a Z transform and therefore has both an expression and a region of convergence. In fact, we can see an expression in S in Z at the respective places. The point is it is not just an expression and in a way this is different from the frequency response for that reason. In module 2 we had described several systems by their frequency response and the frequency response was of course a function of the angular frequency or of the cycles per second frequency. But then there was no notion of region of convergence associated there. This notion of region of convergence becomes important because you are possibly dealing with system which are unstable which could be noncausal system or both. And you have to worry about which of the interpretations of the expression you should use. I do not need to give you examples. We have already seen examples before of cases where the same expression with a different region of convergence would give you different functions underlying and that would be true for the impulse response also. We do not need to repeat that again and again. So, you see it is very important to keep in mind that the region of convergence is there associated with the system function whether or not you state it explicitly. In some cases the region of convergence would be obvious from the context. We can understand that later. For example, if you have e raised the power 2t ut it is an example of a causal system. And if you have the e raised the power of 2t going in the other direction it becomes a noncausal system. So, if you are told the system is causal in addition that there is only one possible region of convergence given that the system function is 1 by s minus 2. So, in that sense the context in many cases might make the region of convergence clear, but it is there. In many cases the context may make it clear what the region of convergence is there, but then the region of convergence is there underlying the system function. And we must keep in mind that the two together constitute a system function so much so. As I was saying we can also interpret the system function as a ratio of two transforms. So, let us write down that definition as well. The system function is also ratio. It is a ratio of the output Laplace transform to the input Laplace transform for continuous variable systems. And of the output z transform to the input z transform for discrete variable systems. Of course, this is all provided these transforms exist. So, that is important. You know you can talk about a linear shift invariant system which has a system function, but if the input does not have a Laplace transform and or the output does not have a Laplace transform then using this definition is not meaningful. So, in that sense looking at the system directly is more fundamentally and in case the input does not have a Laplace transform, but you have a system function in the system then you would have to invert the system function. That means you would have to find the impulse response from the system function and then proceed by convolving of course. So, in fact what we have now seen is that the Laplace transform is a beautiful tool for convolution as well and so is the z transform in the discrete case. So, what is one neat way to convolve? Let us write it down which could also be for the purpose of finding the output of a linear shift invariant system given the input and the impulse response or it could be in general. A neat way to convolve let us write that down, multiply the Laplace transforms respectively z transform of the signals to be convolved in the continuous respectively discrete independent variable then take the inverse transform keeping the ROC in mind do not forget that. So, you see essentially the Laplace transform and the z transform allow you to convolve conveniently if that inversion of the Laplace and the z transform is convenient otherwise of course you have saved nothing at all. In fact you have done more work than necessary because you have gone to a different domain you have worked hard in that domain and you are working hard to get out of that domain. So, it is you see it makes sense to do this if it is easy to go to that domain it is easy to come out of that domain for particular functions if it is not then it is better to convolve in the natural domain directly. Now what is the typical kind of system function which is amenable to convolution by this approach? In fact, we can ask a broader question what is the very typical kind of system function that we encounter in natural systems that we use all the time in engineering? Well we can ask even a more fundamental question what is the typical impulse response that we encounter in many natural systems let us make a note of that. So, the typical impulse response encountered essentially it is a linear combination of what are called poly x terms. What are poly x terms? Poly x terms poly x stands for polynomial multiplied by exponential. So, poly x as an example we can have something like 3 plus 2t into e raised to power minus 2tut and a linear combination of such terms with different exponential parameters. So, this is the poly part and this is the x part exponential part and this is what is called the exponential parameter. So, it is these kinds of impulse responses that we often encounter. So, can we make a job of inversion easier? We shall see in the next session. Thank you.