 A warm welcome to the 27th session of the third module of signals and systems. We have now embarked upon a very different kind of signal and system paradigm where we use discrete time systems to do what you would like in other continuous time system to do. And I think it is appropriate that we give some concrete example. Let us know that. Now, let us take the example of wanting to process a speech signal. Now, of course, you will recall we have talked about this before. Speech is typically band limited to about 4 kilohertz. So, if we look at the terms of the spectrum or the Fourier transform and if I am talking about the cycles per second frequency which I am denoting by f, we are talking about an occupancy from 0 to 4000 and minus 4000 to 0 on the other side, on the negative side. Now, we have been talking about sampling and reconstruction. So, we can of course, put down a requirement on sampling. We need to sample the speech signal at greater than 2 into 4000 or 8000 when I say hertz here. So, we have the right samples per time per unit time. So, for example, remember you know we wanted a margin. So, let us choose a sampling rate of say 10 kilohertz, whereupon what we expect is that if we have the speech spectrum, let us take what we call a prototype speech spectrum or let us just draw some spectrum for the speech signal. So, you know instead of keeping on writing 4000, I will divide f by 1000. So, I will write 4 and minus 4 here and have some spectrum. I will just show it by something like this. Now, you know I must caution you that when we make diagrams like this, it is essentially for conceptual understand. We must not take the shape of the spectrum too seriously. You see it is just to explain what is going on and in fact, when I say a prototype spectrum, I am talking about simple shapes, where I can ensure that every point in the relevant part of the spectrum has a different magnitude and I do not bother about the phase. Anyway, coming back to this, if we sample this, we get a spectrum that looks like this. I compressed the frequency axis a little bit and so on. This is the original spectrum here and these are the aliases. Now, what are we doing when we do discrete time processing? We are not in a hurry to reconstruct and that is the whole theory we are now going to develop. We are not going to be in a hurry to reconstruct. We would like to do something to these samples inherently. So, I have this spectrum of the sample signal, I have a stream of samples coming. I operate a discrete system on them. Now, let me recall how I understood discrete systems. How do I describe a discrete system? Let us write that down. The discrete system takes the sample signal as input and produces a concurrent output train. What is the input sequence here? The sequence of speed samples. It produces an output sequence. And this would again be a sequence of desired speed samples or processed speed samples. And having generated these processed speed samples, you can now subject it to a digital to analog converter. What is a digital to analog converter? It is essentially a reconstructor. Now, that reconstructor of course needs to have a few analog components which we have not discussed in detail. That is also true of the digitizer. So, you have these analog samples which have come from speech. How could they have come? For example, you could have been recording the speech. Today, of course, we know about digital recording media. That is not a problem. I mean, you could have a digital recording device and then automatically whatever is being spoken, speech is thought of as analog. And then you are essentially using a device which automatically converts it first into a train of samples and then each sample generates a stream of bits. Now, you know, since we do not assume a knowledge of electronic systems for this course, I am not going into the electronic circuit here which could do analog digital conversion or digital to analog conversion. But for those of you who are interested and who have a background in electronics on electronic circuits, those of you who have a hobby are interested are strongly encouraged to go and look up these terms. Analog digital conversion in a book on digital electronics or in a book on electronic systems. They are not difficult to obtain. Look up analog digital conversion and digital to analog conversion. What I will do right now is just to give you a quick feel of the idea behind analog digital conversion. So, you know, there are different kinds of analog digital converters. One of them is called a successive approximation converter. So, I will just give you the principle of it. You know, how you go from the continuous time sample that you had that is the analog value to the corresponding stream of bits which represents it in the digital memory. Let us see you have an analog sample which has come. And let us assume that you have a device which can make an analog comparison. On the other side, we have one memory or one discrete time stream of bits. We convert this into an analog value by using a system of resistors and voltage sources. What is called a voltage combiner and that gives you an approximated analog value. These are the two inputs to the analog comparator. The analog comparator generates a difference or it says more or less generates a binary output more or less which then tunes the discrete stream of bits. So, you know, let me explain what are we really doing here? For example, suppose this discrete stream of bits had 8 bits in it. You know, I think many of you would understand what I call a binary representation of a number. So, for example, in a binary representation of a number, you give places each of which holds either 1 or 0. Just as when we write a number in the decimal system, we give places and in each place we write one of the digits from 0 to 9. We, of course, also have what is called the fraction point or decimal point and that concept would be there in any representation, binary representation or decimal representation. And the idea is you look at the most significant place. You know, let me take an example here. So, what we are saying is, for example, suppose your number is 14 or 14, let us say little, you know, let us put a fractional part there as well. So, let us say 14.5. So, you can write this down in binary. In binary it will be written as 2 raised to the 3, that is 8 plus 2 raised to the 2, that is 4, 8 and 4 makes 12 plus 2 raised to the 1, that is 14 for you and then you have 2 raised to the minus 1 plus half. Essentially, you have written it to the base 2. Now, how would you write this actually in terms of 1s and 0s? You would write places which denote powers of 2 with a binary point or fractional point. This point would correspond, for example, to 2 raised to the power minus 1. This is 2 raised to the 0, 2 raised to the 1, 2 raised to the 2 and you have 2 raised to the 3 here. And you can now fill in the values. In each place, you need to fill either 1 or 0. That is easy to do. So, all that you need to do is to put down, let me do that for you. Put down a 1 here because there is a 2 raised to the 3, put down a 1 here, put down a 1 here but there is no 2 raised to 0. So, you put a 0 there and you put a 1 here. Then, of course, you can put 0 subsequently. This is the binary representation. Now, this is what is called the most significant bit and this is called the least significant of the bits. By the way, bit means binary digit or the number filled in the binary place. So, the philosophy is you start by questioning what is the most significant bit? Is it a 1 or a 0? Now, if it is a 1, the number has to be more than 2 raised to the 3. So, you put down a 2 raised to the power of 3, that is 8 and compare it against the number. If you find that the number is more, then you should retain, retain 1 there. In contrast, suppose for example, we had the number 7 and you made 2 raised to the power of 3 place equal to 1. You would get the number 8 on making it 1 which is more. So, you know the number actually is less. So, if you find that is what I meant you know in the previous drawing when I said more or less that is what I was talking about this thing you know this more or less. If you find the analog sample is more, then you keep a 1 in that place and if you find the analog sample is less, then you make it 0. And you keep doing this bit by bit starting from the most significant bit and moving towards the least significant bit. This is one essential idea that can be used for analog digital conversion. I just gave you this idea although I have not shown the full detailed circuit, if some of you are interested you might want to look up how all these things are done. But no, we should have a conceptual idea and we are talking about all this analog digital conversion, we must have some concrete idea what is happening. We will see more about this system in the next session. Thank you.