 a warm welcome to the 32nd session of the third module in signals and systems. We are now embarking on the general paradigm of discrete time signal processing as an alternative to continuous time signal processing. Let us put down the structure clearly. What we are saying is a discrete time signal processing system does the following. It takes a continuous time band limited signal, band limited to angular frequency capital omega m. This is sampled. So, then unlock the digital converter sampled with a sampling rate according to the Nyquist principle. So, suppose this continuous time band limited signal is called small x of t, then what we generate here is a sequence of samples or an input sequence of samples. And we will put square brackets here, x square bracket n T s where T s is 1 by f s. And instead of keeping on writing x square bracket n T s we will just write x square bracket n. So, essentially we are saying T s is taken as the unit time and that means f s now becomes the unit frequency. So, this process is called normalization. This process of calling this the unit time and this the unit frequency is called normalization. Normalization to go from continuous time to discrete time. Now, we have this discrete sequence here. So, we put it into a discrete system and the discrete system has an impulse response as usual. Of course, here we are talking about a linear shift invariant discrete system and it generates the output sequence. y n is related to x n through convolution with h n. And you know the expression for convolution or of course, you could interchange the role of h and x, you already know all those properties of convolution. Now, the issue is what is this discrete time system in relation to the equivalent continuous time system that we want to operate? What is this system? What is this discrete time system which will give you the same effect if you reconstruct the output sequence into a continuous time signal and you want the effect to be what you would have if you were to do the processing in continuous time. We had begun the discussion to answer this question towards the end of the previous session, but we had not completed the answer. Let us do so now. So, what we are saying is you had this original paradigm. You wanted to process it with a continuous time system and of course, it is linear shift invariant. It has a frequency response. Its frequency response is capital H of omega and you wanted to produce a continuous time output signal. Now, of course, the output signal has to be band limited and we also saw that we are quite safe if we just restrict this 2 minus 2 plus omega and that is what will give us the discrete system. So, all that we need to do to get the discrete system is take the underlying continuous system that you wanted to operate, restrict it to minus capital omega m to plus capital omega m and then treat that as the frequency response of the discrete time system. So, simple. So, let us put it down clearly. We are saying that in this system that we had here essentially the Fourier, the discrete time Fourier transform capital H of omega of the sequence h n, well actually we should be writing capital H of the small omega because we want to normalize. So, we should strike out that capital omega and write small omega that is the first change we need to make and the discrete time Fourier transform of this is essentially the continuous time Fourier transform capital H omega restricted and then normalized. Now, let us draw that properly. So, what I am saying is the following. You had this original frequency response capital H of omega as a function of the angular frequency omega. Now, it may extend all over capital omega. However, we restrict it, we chop it down. So, you know if you had something like this as capital H omega, you cut it down and now we have this whatever it be. Now, this omega m maps to pi or rather not omega m, but you know that in principle we have sampled at omega s on the angular frequency axis. So, let us draw omega s on the same axis. You know that omega s by 2 was more than omega m and therefore, we should in fact broaden our perspective a little. Although this particular signal is band limited to capital omega m, we can admit signals which are band limited to less than capital omega s pi 2. So, instead of chopping it down to this region, we could actually bring it all the way up to minus omega s by 2 to plus omega s by 2. We can retain that why throw it away. So, we keep up to there and now all this which I am marking in green is what we have kept. So, this thing and therefore, now it is not omega m which maps to pi, but omega s by 2 which maps to pi a slight change and therefore, minus omega s by 2 maps to minus pi here. So, let us redraw this since we have made a few changes. We retain capital H omega in this region and remove the rest and reduce the rest to 0. Omega s by 2 maps to pi small omega equal to pi and minus capital omega s by 2 maps to small omega equal to minus pi on account of normalization. Here we are talking about the discrete time Fourier transform or the DTFT as we have called it and all other frequencies proportionately mapped and that gives us capital H of small omega. The frequency response of the discrete time system. Now, we are all equipped. We can now take as we did the input stream of samples or input sequence x of n. Process it with this H of omega or the corresponding impulse response small h of n which is the inverse discrete time Fourier transform. That produces an output y of n. Now, subject this to digital analog conversion or reconstruction and there we get the desired output y. That is very interesting. What we were trying to do in continuous time? We can now do with an equivalent discrete time system. And why is this a good idea? All this discrete processing the system with frequency response capital H of small omega in discrete time can be realized on a computer. Changing the system is essentially changing a computer program not changing components. Not only that you can make upgrade the system you can simplify the system all that is essentially changing a computer program or a microprocessor program. It does not require to reconfigure components to buy new components. This was the whole power of sampling and reconstruction not just to sample reconstructor to transmit, but being able to do processing in a totally different way. We will see more of this in the next session. Thank you.