 A warm welcome to the final session, the 34th session of the third module in signals and systems. We looked at an example of constructing an equivalent discrete time processing system for an RC circuit in the previous session. Let us recall where we were. We said that we could think of the RC circuit that we constructed if we chose its half power frequency to be 3 kilohertz equivalent to 3 kilohertz. See I said equivalent please remember essentially capital omega is 1 by 1000 is an equivalence 2 pi times. Whatever equivalent to 3 kilohertz that means that we saw that 9 kilohertz would be bringing down the magnitude to 10 percent, the power to 10 percent. And then it depends on how we wish we could of course take the exact chopped system the RC response itself between minus pi and pi treat that as the frequency response of the discrete time system and go about everything else or we could do something simpler. We could choose an ideal low pass filter which behaves just very similar to what this RC circuit does. And when we say very similar we have to think of a cutoff for the ideal low pass filter. So the cutoff could be anywhere between this 3 kilohertz and 6 and 9 kilohertz extremes depending on where we are satisfied. So 3 kilohertz is the half power point 9 kilohertz is the 0.1 power point 10 percent power point 10 percent is perhaps too low half power is perhaps too high. So we could choose something in between we could agree that 6 kilohertz is the corresponding cutoff of the ideal low pass filter. And the ideal low pass filter if it has a cutoff at 6 kilohertz is equivalent to a discrete time system with the following frequency response. This is the h of small omega that we are talking about 1 between minus pi by 2 and plus pi by 2 and 0 between minus pi and minus pi by 2 and plus pi by 2 and plus pi. Of course, h omega is the discrete time Fourier transform of the corresponding impulse response. So, let us write down that expression too if the corresponding impulse response is small h of n then this is the discrete time Fourier transform of the impulse response. And clearly this is periodic with period 2 pi all that we have to do is to note that this is 1 for all integer n and therefore, the periodicity. Now what is the mechanism for constructing the inverse? So suppose I know the ideal frequency response, how would I get the ideal impulse response? Well it is very simple you could think of all our time and frequency as based on the normalized scale now. So, you are essentially thinking of capital H of omega as the Fourier transform of a continuous time impulse response which has been sampled with a unit sample interval you are saying essentially and therefore, the corresponding continuous time impulse response is the inverse Fourier transform which is and now we can get the discrete sequence of the impulse response of the discrete system by replacing t by integer values here. So, simple that means we are essentially saying the impulse response of the discrete time system is given by what we call the inverse discrete time Fourier transform of capital H omega or inverse DTFT for short and we can evaluate this very easily for this particular case. In fact, I leave it to you as an exercise and just write down the integral for you and you can evaluate it. Now, I recommend that you evaluate separately for n equal to 0 and n not equal to 0 and show that this is the answer. So, I have just shown you one example but what it leads to is a very rich set of possibilities. You could construct an equivalent discrete time processing system for a wide range of continuous time processing systems. Here we over simplified the discrete time system. The RC circuit was far from this response it had a variation as a function of omega and if you want it to be more accurate you could have a second or better approximation to this RC circuit may be not as general as the RC circuit because that is exponential you could of course, invert that to it is not impossible. So, you know I am trying to explain it is not impossible for you to find the exact inverse. In fact, let us let me just indicate how you could find the exact inverse. So, capital H of omega for the RC circuit is like this and you made a replacement omega capital omega is small omega by T s. So, therefore, capital H of small omega is this and we restricted this and therefore, the exact H n would be the inverse GTFT of this expression. And how would that inverse GTFT look? Let me just write it down for the fun of it if you like. Now, as I said it is not impossible to evaluate this integral you could do it if you really desire, but I strongly recommend you compare those of you who are keen to those of you who are also capable of doing so calculate this exact inverse GTFT and compare it with what you get with the approximate system that we have taken a low pass filter with cutoff pi by 2. It would be interesting to see what you notice. Anyway, that is something that I leave you to do. But now I am going to say a few things in conclusion of this module. Where are we in this module? What have we done in this module? We have essentially made a movement and a marriage between continuous independent variable and discrete independent variable. Now, continuous independent variable is more real life it occurs at least seems to occur more frequently in real life and therefore continuous independent variable systems make a lot of sense if you want to understand real life situations properly. Discrete independent variable occurs sometimes in real life. If you recall in the first module we had more difficulty in giving meaningful examples of discrete time systems inherently discrete time systems because of course construct artificial discrete time systems. So, discrete time systems are perhaps not so, now at least not as common as continuous independent variable systems in real life. There are a few good examples. Economic systems are good examples. Financial systems are good examples. Systems that operate at certain points of the clock are examples. But in general it is continuous independent variable that dominates seemingly in the world around us. But continuous independent variable is much more difficult to understand much more difficult to process. The convenience of processing discrete independent variable systems comes from the fact that all this discrete time convolution, discrete time operation can be done by storing the input samples in a computer, processing them on a computer with a discrete time system which is implemented in the form of a small program. You see for sequences the moment you are doing discrete time you can program it. You can say you can have a loop that says for every instance of n do a certain amount of computations, do a certain amount of work on the input xn and output the sequence y of n. So, I am saying let me draw the figure here. A discrete time system can be implemented with computing devices. In fact, modern computing devices if you like input sequence can be stored in memory as a stream of samples. The discrete time system can be a computer program. This would generate the output sequence at a different set of memory locations. So, we have all this, we have this beautiful situation where once you have got the input sequence you can write a computer program to process it and get the output sequence. You can come here with analog to digital conversion, ADC is analog to digital converter interface. So, come here with an ADC interface, go back with a DAC interface, go back with a digital to analog conversion interface. So, interface a modern computing device with an analog digital conversion and a digital to analog conversion and you can do what you like inside that computer program much more versatile, much more flexible, much more robust because you do not expect the computer to go bad as quickly as do components at times. Anyway, you know sometimes it is easier to fix a computer than to fix individual components at least in the modern world because you can always replace that computer and it may be worth doing that because you know you could do so many different things with the same computer or maybe if you are not willing to invest in a computer invest in a microprocessor. The same microprocessor could be doing so many different things. There is a reason why we want to go digital, it is not without meaning. So, this module 3 has given you those tools, how to go digital from analog, how to go discrete time or discrete independent variable from continuous independent variable and it makes a lot of sense in the world as far as processing power goals. So, while continuous independent variable is the reality, discrete independent variable is the convenience and module 3 has brought them together in an effective manner. Thank you so much and we will meet again in module 4 where we shall now make a generalization of the paradigms for transforms. Thank you so much.