 A warm welcome to the first session of the fourth module in the course signals and systems. You will remember that in the second module, we had gone from what we call the natural domain to what we call the frequency domain. So, we said that there are some very great conveniences in thinking of synusoids as a basis for expressing and representing signals. And in fact, also looking at what systems do to signals, particularly linear shift invariant systems. And in fact, based on that, we built ideas in module 3, where we could bring continuous and discrete systems together, we could bring equivalences between them. Now, unfortunately, one of the things we kept saying again and again and again, when we talked about the Fourier transform and we talked about the spectrum is the signal should have a spectrum, we keep saying that, is not it? We also kept giving examples of signals that do not have a spectrum. For example, in exponentially growing waveform or for that matter, even the unit step technically, it does not have a Fourier transform in the conventional sense. You need to bring in impulses and things like that, you see. So, all the while, we must have been wondering, can we generalize? That is one of the strategies that we use. Can we generalize the whole idea of alternate domains? And now that we have worked with continuous time and discrete time or continuous independent variable and discrete independent variable. It is an important question, whether we can generalize what we gain from the Fourier domain analysis to a more general domain. So, for example, can we do something to be able to deal with unstable systems as well? I mean, we do not have to worry if the system is unstable. We can still use some kind of a transform domain approach, that is what I mean. We would like that to happen and we would like this to happen both with continuous time systems and with discrete time or continuous independent variable systems and with discrete independent variable systems. So, how do we go about it? Well, that is the whole theme of module 4. So, let me put it down to explain to you in a few sentences. What do we want to do in module 4? We want to generalize the Fourier transform. So, what we want to do is to accommodate systems whose impulse responses do not have a Fourier transform. An example from continuous independent variable or continuous time. Suppose you had this impulse response, h t equal to e raised to the power of t u t, u t is a standard unit step. So, how would it look? On the t axis from t equal to 0, you would have an exponentially growing wave form. This is how h t looks. Now, clearly this has no Fourier transform. In fact, you cannot make any tweaking on the Fourier transform to accommodate this and let this be the impulse response of obviously, an unstable system. Let us take a discrete time example. H n is 2 raised to the power of n u n. So, obviously, mod h n summed over all n is essentially mod n going from 0 to infinity 2 raised to the power of n and this is divergent. So, system is unstable. Both of these are examples of unstable systems and obviously, there is no frequency response. So, then what do we do? If there is no frequency response, are we incapable of dealing with the system at all using a transform domain or do we need to generalize? Can we build a more general transform? That is the first part. Now, the second part is to generalize the properties of the Fourier transform to this more general context. And what properties are we talking about? For example, convolution, shift, differentiation, modulation and so on. What we are saying is, can I in some sense use all the conveniences that I got from the Fourier transform, but then extend the domain of application to these unstable and troublemaking systems? Well, you know, I think we should we have now seen that there are good unstable systems also. So, these systems which are not otherwise amenable to Fourier analysts, that is the way we should think about them. Unstable and stable is after all just a property. So, the third thing that we would like to do is to characterize systems more generally. That means, be able to deal with more general classes of inputs. You see, if you looked at it carefully, there is a certain restrictiveness in the Fourier transform. By and large, the Fourier transform is directly or indirectly dealing with sinusoidal inputs. You know, we want to deal with more general inputs. For example, exponential inputs or exponential inputs multiplied by some sinusoid and by the way, exponentials multiplied by sinusoid are common in nature. So, this desire to deal with the more general class of inputs, again using the conveniences, that is what we are trying to evolve. So, in all of module 4, what we will notice is that we will have many of these objectives attained at once, once we set up a more general transform framework. So, let us now talk a little bit about what we are going to do in this more general transform framework. What will this more general transformation framework be? You see, we would first like to begin with the philosophy of this generalization. And the philosophy is very simple. Capture the source of instability first. Capture or ensnare and then we could simply apply the Fourier transform on the thus modified sigma. Let us take an example of continuous time. Let us take that troublemaking input that we had, h t is equal to e raised to the power of t u t. Now, let us multiply h t, you see, where was the trouble in h t? The trouble came due to exponential growths. Now, capturing or ensnaring means multiply by a decaying exponential, stronger than this growing one. In other words, what you could do is to multiply h t by e raised to the power minus sigma t, where sigma is greater than 1. For example, if sigma is equal to 2, what will happen? You will get h t e raised to the power minus 2 t, which is e raised to the power of t u t into e raised to the power minus 2 t, which is now e raised to the power minus t u t and this becomes absolutely integrable. Now, if you do this, first capture or ensnare with this multiplying function, which then brings it in control and then use the Fourier transform. Very simple. The idea is very simple. Put the trouble making function in a cage, so to speak and then put all your processing on top of that, in the cage and then after doing all your processing, which could be in the Fourier domain, then release the function from the cage. Now, this whole idea of first capturing and then releasing is the essence of the generalization that we are going to make. We could make a very similar generalization for discrete time signals or sequences, essentially capture with an exponentially decaying sequence stronger than the growing one and then do all the Fourier processing that you want and then release it. So, in fact, we are going to give formal names to these two kinds of processing that we do for continuous time and for discrete time. For continuous time or continuous independent variable, we are going to call this whole business the Laplace transform after the scientist Laplace, who came up with this idea and for a discrete independent variable context, we are going to call it the Z transform. We will build more ideas about these two transforms in subsequent sessions. Thank you.