 A warm welcome to the fourth session of the fourth module in signals and systems. Now, in the last two sessions, we have introduced two new transforms, one which generalizes the discrete case and one which generalizes the continuous case. Let us review a few ideas, because we must get the comparative picture clear. Incidentally, in my exposition of this subject, I am going to take these two transforms in parallel. I am going to deal with them together, looking at them one after the other. That is because there is a lot of similarity, but there are also important differences and you must understand both the similarities and the differences. So, let us begin. So, the plus transform, so the plus transform for signal x t is given by the integral over all time, x t multiplied by e raised to the power minus s t dt, where s is a complex constant and we saw that s could be written as sigma plus j omega. So, s is a complex constant expanded as sigma plus j omega and then we could interpret this. Of course, this must converge and the convergence depends on sigma. We can also view this as we saw to be the Fourier transform, the Fourier transform of x t into e raised to the power minus sigma t for appropriately chosen sigma. Now, we also specified that a Laplace transform is always an expression and a region of convergence. Now, you know it is very important. We illustrated the sampling in the previous discussion. There is an expression and there is a region of convergence. The region of convergence is the one that we might think can be ignored, but no, that is very important because if you do not specify the region of convergence, you could be dealing with the wrong signal. So, let us take an example. We will just repeat the inference that we drew in the previous session to make this clear. We saw. Now, you know we are going to bring in some notation here. When we write like this and we will do it several times in future, script L, it means take the Laplace transform. So, when we take the Laplace transform of e raised to the power of t, u t it gives us 1 by s minus 1 with the region of convergence. The real part of s is greater than 1 and we shall often further abbreviate this by writing r e in bracket s is greater than 1. This is a short form for real part of s greater than 1. Of course, you could also say sigma is greater than it all mean the same thing and we also saw minimally modifying what we did in the previous discussion that minus e raised to the power of t u minus t when Laplace transformed gave us 1 by s minus 1 once again, but with the region of convergence real part of s is less than 1. So, the region of convergence matters. You had the same expression 1 by s minus 1, but with the two different regions of convergence you are essentially talking about two different signals. Of course, I do agree that the two signals are in some sense derivatives of the same wave form they are really derived from one exponential, but there is one right sided exponential and one left sided exponential. One exponential whose domain is on the right side of 0 and the other whose domain is on the left side of 0 the fundamentally different signals. In fact, what this also brings to notice let us sketch both these sequences together now on the same graph we will get something interesting there. So, this is the time axis this is the point t equal to 0 we begin from 1 and when you exponentially grow you get e raised to the power of t u t and then you could exponentially decay, but on the left side and you get e raised to the power of t u minus t and of course, you can visualize how e raised to the power t u minus t multiplied by minus 1 looks that would look like this. So, these are exponentially decaying, but left towards and this is exponentially growing right towards. Now, you will also notice that this would anyway have a Fourier transform in fact, the Fourier transform is the same as its Laplace transform putting sigma equal to 0 or putting real part of s equal to 0 and in fact, you know this is also reminiscent of what we have done just previously right now. In this case for this region of convergence here real part of s equal to 0 is included, but here it is not that also tells us something about what we just said that this does not a priori have a Fourier transform. You need first to constrain it or hold it down however, in this case no such holding down is required it would have a Fourier transform as we have said. So this also brings us to the observation that the Laplace transform is indeed a very good generalization of the Fourier transform because for those signals that have a Fourier transform they also have a Laplace transform, but there are many other signals which would not a priori have a Fourier transform which now do have a Laplace transform after holding them down. And what are we really saying in terms of which signals are amenable at least the Laplace transform if not to the Fourier transform let us capture that idea now. So, we are saying a signal which can be made absolutely integrable after multiplication by a suitable decaying exponential is amenable to Laplace transformation. So, you see it is important you essentially have to have this complex not complex actually a real an exponentially decaying function you just need erase the power minus sigma t it is a real function a decaying exponential. So, you need you should be able to multiply it by a suitable decaying exponentials where it becomes absolutely integrable if that happens that function has a Laplace transform or at least we are assured that there is a convergence in the Laplace integral of course, we are talking about quote unquote reasonable functions that we often encounter in nature we most of time encounter in nature not some pathological cases where you know you have essentially the function not being integrable. So, let us not get into those pathological case, but otherwise you know for all the reasonable function that we encounter in nature wherever the function might not itself be absolutely integrable, but on the multiplication by a decaying exponential makes it absolutely integrable in such a case a Laplace transform is possible. So, you know it does not have to be just a simple exponential like you had here it could be something on one region something on another something on another. So, you know you could for example visualize the Laplace transform something like this and I give this to you as an exercise. So, you could think of a signal which begins at t equal to 0 with the value of 1 it continues to have the value of 1 and darkening it all this while up to t equal to 1 and then grows exponentially from that point onwards. First write the expression for this wave form and then find the integral interesting. Now, let us make a similar summary for the Z-transform so, the Z-transform the Z-transform of a sequence x of n is summation n going from minus to plus infinity x n into Z raise the power of minus n it is often denoted as capital X of Z and this is also an expression or a function of Z together with the region of convergence. Once again we must emphasize the region of convergence matters the same expression with different regions of convergence could correspond will correspond in fact to different sequences. We have seen examples. Now, we shall deal more elaborately with these transforms in the subsequent sessions. Thank you.