 A warm welcome to the 29th session of the third module of signals and systems. In the last couple of sessions, we have been building ideas about how we could build a speech system, a discrete time processing system for speech. And we were at the following point, let me review the ideas here. We said this is how we had the spectral situation. We had the original spectrum here. We sampled the 10 kilohertz and drawing the frequency axis in the normal kilohertz framework right now. Now on sampling, we got this. This was the original spectrum and these are the carbon copies of aliases. Now, we were writing down the normalized frequencies. So, of course, here this would be 6, this would be 14, this would be 20 minus again, you know, so 20 minus 4, so 16 and 24 and so on. And we have 5 in between, 5 has a significance here and minus 5. So, let me write down the normalized angle of frequencies again. 5 would correspond to half into 2 pi, which is pi and therefore, 4 would correspond to 0.8 pi and similarly, the others. 10 would correspond to 2 pi, the sampling rate and of course, you could put down all the other values. I leave it to you to do that. Now, this normalized angular frequency is often denoted by small omega and we shall use that convention. You see, let us do a very simple discrete time operation now. Let us give it the input X of n and it produces the output Y of n. Let us call the discrete system script S. In fact, let it be a discrete linear shift invariant system. Let the description of the system be that Y of n is half of X of n plus X of n minus 1. So, essentially it takes the current and just the past sample, sample one step before and it averages them. Let us see what the system would do to sinusoids of different frequencies. Now, there again, you know, let us review our discussions in module 2. We have agreed that instead of dealing with sinusoids directly, it is a good idea to deal with the corresponding complex exponential, the rotating complex number. And here, you notice that there is a certain indistinguishability. We have been talking about this all the while in sampling. You also agree that our unit time now, in just the previous session, we agreed on some normalizations, on some conveniences that we would use to describe this discrete time system. Our unit of time now, once you have come to the discrete domain, the unit of time is your sampling interval. The unit of frequency is the sampling frequency. Of course, the sampling frequency written in normalized angular frequency is 2 pi, then in normalized cycles per second frequency, it is 1. So, look at this system here. We have y of n is half x of n plus x of n minus 1 and n refers to the sample number. So, we agreed to look at what happens when we give sinusoids, you know, when we want to understand what the system does. The first thing we can ask, given our experience in module 2, is what happens when you feed a sinusoid to it. Now, in speech that makes sense, there is a physical meaning to the sinusoidal frequency. Typically, lower frequencies, you know, if you look at male voices and female voices, typically in speech, the chances are that female voices have higher frequencies on the whole and male voices typically have lower frequencies. So, between the 0 and 4 kilohertz band, the upper range of the band is largely occupied by female voices and the lower parts of the band are likely to be occupied by male voices. So, suppose you have a mixture of male and female voices, you have several people speaking and some of them are male and some of them are female. The likelihood is that much of the signal coming from the male speech is likely to occur in the band, in the first half band and much of the content of female speech is likely to occupy the second half band. The first half band meaning between 0 and 2 kilohertz and the second half band being beyond 2 kilohertz. Now, what does the system do? So, let us understand this first in terms of the actual frequency and then normalized frequency. So, how would you get this spectrum in the first place? The sample signal spectrum. So, we are talking about X of T, continuous T as the speed signal, at 1 by T s, where of course, 1 by T s is 10 kilohertz or 10 kilo samples per second, that is more appropriate. And therefore, what you really have is the samples given by X evaluated n T s, where n is all the integers. So, after sampling, what is the signal formally? It is X T multiplied by the train of uniform train of impulses, which can be written in the following way. And what we are calling X of n is really just this, this is X of square bracket n. Now, let us take its Fourier transform. Now, we know how this impulse operates and let us hope this Fourier transform converges. It would be expected for speech, is not it? We can interchange the orders. We can write this as summation n going from minus to plus infinity. We can take the dependence on n outside and put the impulse inside. And now, this is very easy. We can simply lift the value at the point where the impulse occurs. So, that leaves us with, this is the Fourier transform of the sample signal. And this is what you saw several drawings ago. This is what you saw. What we just calculated here is this really. This is that Fourier transform of the sample signal that we have. So, now, let us talk in terms of the normalizations that we have done. X of n T s is essentially X square bracket n and e raised to the power minus j omega n T s can be rewritten. What is this quantity omega by 1 by T s? It is 2 pi times the actual cycles per second frequency divided by the actual sampling frequency. And this is essentially the normalized angular frequency small omega. So, therefore, we have very simple interpretation for this quantity. This is e raised to the power minus j small omega n simple. And therefore, this quantity that we obtained just previously here summation n going from minus to plus infinity X of n T s e raised to the power minus j omega n T s can now be rewritten. Rewritten as summation n from minus to plus infinity X n e raised to the power minus j small omega n. This is essentially what is called the discrete time Fourier transform. The discrete time Fourier transform of the sequence X of n. Now, you know, we have an interpretation for this. We can denote it by capital X of omega. That is a common convention. And the interpretation that we had, you know, recall the ideas in the second module. It is the projection or the component of X n along e raised to the power j omega n. So, it is how much of e raised to the power j omega n there is in X n by finding the dot product, the inner product. Remember, this is the inner product that we have here. We will see more about this in the next session where we will also now continue to analyze what the system would do based on this background. Thank you.