 A warm welcome to the 31st session of the third module on signals and systems. We are now embarking upon a different kind of venture altogether, namely we have now the tools to deal with discrete time systems in place of continuous time systems. So, you know there is an underlying continuous time system that we want to realize and we can do it all on a computing device. We have laid the foundations for it in the last couple of sessions, but now let me put all our ideas down very clearly that we see the picture in its complete form and structure. So, what we are talking about is general discrete time systems in place of continuous time systems. Of course, when we want to do something like this is going to come at a cost, you cannot expect that every continuous time system can be replaced by an equivalent discrete time system. First, as far as this module goes we are talking about continuous time signals which have a Fourier transform and we are talking about continuous time systems which have a frequency response. So, let us put that context down clearly. So, this is very important we are talking about signals and systems where Fourier analysis is meaningful. Now, if not it is not that we cannot deal with that situation, but we will have to wait until module 4 to be able to do it. So, right now let us confine ourselves to this context. Now, having said this it is not all after all you could not expect to have been able to replace all continuous time systems even with a Fourier transform or Fourier processing possible with discrete time processing. There is something more and that has been captured in the last few sessions. We are talking about band limited. So, in other words we are talking about a continuous time signal x of t whose Fourier transform capital X of omega looks like this. It is non-zero only in a limited range around the zero frequency. So, it could be from minus capital omega m to plus capital omega m. I mean we can choose capital omega m suitably for this to be true provided the signal is band limited. Now let us get some notation clear. The moment it is band limited we have now all the tools to process it with only its samples. Again there we have a lot of choice. We could either do ideal sampling or close to ideal sampling or we can do flat top sampling. We could naturally sample by multiplying by a train of pulses which are not quite impulses or there are so many things we can do and all of them ultimately can be brought down to a repetition of the spectrum around every multiple of the sampling frequency. Now, what I am going to do right now is to assume that we are doing almost ideal sampling. So, then I do not have to worry about the fact that these copies might get diminished in amplitude as you go away from zero frequency and so on. I neglect those effects. I am doing close to ideal sampling also I am assuming that the sampling frequency is reasonably more than the maximum that the signal allows that is you know I mean the maximum the minimum sampling frequency or the maximum sampling interval that the signal allows that is what I mean. So, we have seen the Nyquist theorem it says that if you have a maximum frequency component angular frequency component of capital omega m in the signal whereby you would have a maximum hertz component of omega m divided by 2 phi in the signal your sampling rate needs to be more than twice that more than twice remember the more than is required because you want to have realistic filters for reconstruction. So, let us put all these things down clearly we have a continuous time band limited signal now omega m can be taken as 2 pi times f m we can reconstruct the signal from its samples in principle perfectly at least in principle now we have given names to these processes so essentially we have called this process analog to digital conversion and we have called this process digital to analog conversion and we have assumed that the sampling is almost ideal. So, if we simply do this sample and reconstruct nothing much is going to be achieved. So, the beauty of this whole thing is being able to do something between that analog to digital converter and the digital to analog converter then there is some beauty in this and that leads us to a whole new subject a whole new subject and we will now name that subject and we shall give a flavor of that subject in the next couple of sessions. So, the whole subject of discrete time signal processing is a consequence of being able to do work between these two important blocks and what work do we do we could either do the work of a non-linear system or we could do the work of a linear system or we could be even more specific and do the work of a linear shift invariant system with a frequency response and then we realize that if we are going to work upon this continuous time signal with a linear shift invariant system which has a frequency response it is equivalent to restricting ourselves to that band the band between minus omega m and plus omega m and seeing what the system does in that band. So, let us make that clear let us draw the situation to make it very clear what we are saying is this suppose you had the underlying continuous time linear shift invariant system we pass the underlying continuous time signal through this system. So, we have this system let us call it script s with small h t as the impulse response capital H omega as the frequency response we have given to it the input continuous time system continuous time band limited signal and we produce the output continuous time of course band limited signal the input continuous time band limited signal x of t has a Fourier transform given by capital X omega and the output continuous time band limited signal y of t has the Fourier transform capital Y of omega. Obviously, if we invoke the relation between the input and the output in the Fourier domain we immediately have Y of omega is equal to X of omega times H of omega we know that we have done enough of this when we studied the Fourier transform. So, one immediate consequence of this relationship is as follows let us draw it and illustrate what we are saying capital X of omega ranges between minus f m and plus f m on the f axis on the Hertz axis or cycles per second cycles per time axis or you could also mark the corresponding omega. So, I will do both this is the cycles per time axis I am marking in green the corresponding omega and in general omega is 2 pi f let us always remember that. So, what I am saying is that capital X whether you think of it as a function of capital omega or of f is present only here it is 0 outside and we are saying that capital Y of omega is capital X of omega times capital H of omega. So, in this capital Y of omega if either of these terms is 0 if either capital X is 0 or capital H is 0 Y is 0. So, it is very clear that Y has no choice but we present only here as well and further it is also equally clear and let me mark this in blue that we need only consider H in this region. So, we need only consider H in this region we need only consider X in this region and we need only consider Y in this region. So, all our considerations are in that limited range minus capital omega m to plus capital omega m. So, in fact, now we can band limit everything because the input signal is band limited we can band limit everything we can band limit the input we can band limit the frequency response we can band limit the output and we can work with this band limited situation by sampling. Let us see more of this in the next session. Thank you.