 A warm welcome to the 28th session of the third module of signals and systems. In this session, we will continue with the discussion which we had commenced upon in the previous session, namely to think of a practical system involving speech processing, which took speech as a continuous time input, generated speech samples at the rate of 10 kilo samples per second or 10,000 samples per second, processed those speech samples in a particular manner that we desire, generated a stream of output samples, if we call it an output sequence and then reconstructed speech. So, you know I told you that the process of going from continuous time speech to the input sequence is simple, use a speech recording device, it would essentially give you a stream of analog samples, we saw one philosophy to move from analog samples to a discrete representation or a digital representation. So, in fact, I gave you what was the concept behind one of the analog digital converters that are commonly used and then we assume that this stream of bits, the digital word corresponding to that sample is stored at the appropriate location in memory. So, you have the sequence of samples there, now you could subject that sequence of samples to some discrete time processing, it would generate an output stream and output sequence and each output sequence point, each sample of the output sequence would again be a digital word. So, in fact, one can reconstruct the analog value from the digital word using again a combination of resistances and voltage sources, what is called a voltage combiner, you know. So, essentially you could have some fixed voltage sources which are in relative values of powers of 2. So, you know I will tell you the principle, so the actual circuit is a different issue, so what would be the principle? The principle is use a number of resistors and this is not too difficult to understand, you know we can also understand it with the electricity that we do in high school. So, you have a system here where you give inputs on one side of the resistance which are either 5 volts or 0 volts in magnitude, they are all direct DC inputs, you join all the other output points of the resistor together and then connect it to one resistor across which we look at the net voltage. So, this point is connected to 0 volts always, now you know you could take these resistance values as I have written here in powers of 2. So, for example, you know let me if you had just 4 of them, you could make the resistance values sum r 0, 2 times r 0, 4 times r 0 and 8 times r 0 and this is r L here. Just to take an example, these r 0 values could be in kilo ohms, so r 0 could be 1 kilo ohm for example, so that would make 1 kilo ohm, 2 kilo ohms, 4 kilo ohms and 8 kilo ohms and r L could be may be say 5 kilo ohms or 10 anything I mean depends on what kind of currents we are willing to bear. Now, let us write the Kirchhoff's current law at this point and let us call these voltages here, so they are either 5 volts or 0. Let us call this voltage say v 0, v 1, v 2 and v 3 and the net voltage is v net. Let us write down the Kirchhoff's current law it says that the net voltage as seen here would obey that the sum of currents coming in here must be 0. What does that say? That tells us v 3 minus v L or v net rather by 8 r 0 that is the current you know. So, let me keep showing you one current after another. So, in fact, let us write down the you know current names as well that will make it easier. Let us call this current I 3 similarly I 2 and let us call this current I L. Let us write an expression for all of them one after the other. So, you have this is I 3 here similarly I 2, I 1 and I 0 and I L is of course, v net by R L and from Kirchhoff's current law I L must be the sum of all the others. All the others come in and I L goes out. So, that leaves us with v net by R L is v 3 minus v net by 8 r 0 and plus and plus and so on and all that we need to do now is to rearrange a few terms. Let us do that let us rearrange the terms. Now, let us give this whole you know let us simplify let us call this whole thing here this big expression let us call it G L 1. G stands for conductance reciprocal of resistance and let us denote 1 by R 0 as G 0. So, what are we saying? We are saying v net is equal to G 0 by G L 1 into 1 by 8 times v 3 and this can continue. So, you notice that v net has a combination of 4 terms it is proportional to a sum of 1 by 8 times v 3 plus 1 by 4 times v 2 plus 1 by 2 times v 1 plus v 0. So, v 0 is being given the most significance v 1 is being given half that significance v 2 1 4th and v 3 1 8th. So, you could think of v 0 as being the most significant bit here and v 1 v 2 and v 3 the in decreasing order of significance. You know scientist I wanted to put a concrete example of what is what principle could be used a simple structure. Now, if you have not quite understood suppose you are not familiar with even these basic electrical circuits do not worry too much about it. But I thought many people might be familiar with simple resistance calculations and therefore, I put this example down. So, you get a concrete feel of how we can do digital to land lockman version anyway you know it is important that we get a link to the physical reality what are we doing we are doing all these things abstractly what are we doing in the physical domain what are we doing actually on the ground it is a good idea sometimes to be aware of that anyway. So, now, you have a mechanism of course, when you will get a voltage which is proportional to the word that you have created and that voltage can then of course, be converted back to speech we know how to do that by a player you could use some kind of a device that converts voltage you know that that is not a play essentially a speaker I mean in a way after all something that converts a voltage to speech is a part of so many of the audio devices that we have at home we do not have too much of difficulty in visualizing those parts, but let us not go into all the electronic circuitry. Now, what is important here is that the discrete system that we have in between can do a lot of things in a very simple way and let me start with a very simple example. Let us take a very simple discrete system which would do something interesting and before we do that we must now build some conventions in our expression for the discrete system. So, let us agree on a few things for the discrete system. So, let us look at the discrete processing system and let us build a few conventions you know we need to put down certain things that will make our system universal otherwise each time we have a different sampling rate we have to start writing different symbols. So, let us agree on a few things for all the discrete system that we are going to build let us agree that we shall use the unit time equal to the sample interval. So, that means between whatever be that sample interval it could be 1 millisecond it could be 0.1 millisecond we will agree that we treat that as our unit all along the discrete processing. As a consequence the sampling frequency becomes the reciprocal of this on the frequency axis we think of frequency equal to 1 as the sampling frequency the unit of frequency also becomes 1 now and having agreed to this the maximum frequency component that can be present in the signal before sampling is half now. In fact, it should be less than half to be very precise and why do we want this we want this to avoid a legacy to obey the Nyquist principle otherwise those aliases the copies would start polluting the original spectrum. Now, this is in what is called so this is called the normalized cycles per second frequency or it is the actual cycles per second frequency divided by the actual sampling frequency. So, it is called normalized normalized the sampling frequency, but then you can also have a normalized angular frequency and that normalized angular frequencies 2 pi times the normalized cycles per second frequency. So, therefore, the maximum normalized angular frequency component which we can have in the signal should be less than 2 pi into half which is pi. Now, let us get the picture clear let us draw a picture to understand what we are saying. We are saying that you had this original signal spectrum in the case of speech it went say from minus 4 to 4 and you have sampled it at 10 kilohertz. So, you got this spectrum and so on. Now, what we are saying is that the normalized frequency this is of course the actual cycles per second frequency in kilohertz and let me write the corresponding normalized angular frequency in red. So, at 5 we would have pi at 4 therefore, it becomes 0.8 pi this corresponds to 2 pi and so on. Now, we will have to see more of this in the next session. Thank you.