[0:19]Okay. So in the last class, what we have discussed uh is how from uh last few class probably I should say. How from Fourier series forwards Fourier transform? So that means uh we have a energy signal, from there, means we have a power signal, from there we get Fourier series, that is something we have already explored. And then if uh instead of uh energy means power signal, we have a non-periodic signal that is energy signal, which is time bounded. For that how do we actually evaluate uh the frequency components? So what we have seen that uh whenever means we have put a trick over there. So instead of uh means uh doing the analysis completely uh in a new fashion, we have actually borrowed the idea from Fourier series. So what we did, we have taken a time period which is bigger than the existence of the signal. So that means if the signal is defined from, let's say, A to B, we have taken the time period as a time period which includes that entire thing. And then we said that immediately if I start repeating it up to time plus infinity and minus infinity, so immediately it becomes a uh again uh power signal. Okay, uh and it's a periodic signal. So we can get Fourier series accordingly. And then we started stretching this T to infinity. So immediately we could see that we were getting a Fourier transform case. So we have given interpretation of Fourier transform. What exactly it means. So if uh for a signal GT, we get a Fourier transform, which is GF. What's the meaning of that? We have said that uh at any frequency uh F, the value of GF is not actually the spectrum component, as it was for a discrete Fourier series case. So we have told that that generally gives me zero. But if I just multiply GF with some Del F. Okay. Then within this Del F, how much frequency component is there that can be actually evaluated? So that is why GF is called as spectral density, because GF Del F is almost similar to that DN. And then GF Del F divided by Del F is actually giving me GF. Okay, so that means DN by Del F. So this is actually the spectrum divided by the frequency part, so it becomes spectral density. So this is something we have already discussed. Okay, so we have given some interpretation about that GF what does that mean? Now what we wish to do, uh is now from here to measurement. We have already defined that energy or power is a kind of measurement for us. Okay, that's almost like in vector also, we get the distance of a vector, which is measurement. We have shown that similarly if we just integrate either G squared T or mod GT squared, we get energy. Okay, and correspondingly if it is a power signal, then we can calculate evaluate power just dividing by time and stretching the time up to infinity. Okay, so we'll now try to get into the uh energy or power of a particular signal. So our target is similar to what we have done. We have also evaluated the power of a periodic signal, right? That's something we have already talked about and we have given the equivalence uh means power calculation theorem, which is called Parseval's power theorem. Okay, so similar theorem will be trying to derive for a time limited signal or a energy signal. So let's try to see if I have a signal GT, which is time bounded. Okay, so the signal can be anything. I'm just taking this signal. It might be of any nature, defined between some finite amount of time. Okay. So if I just take that GT, I wish to now evaluate the energy of this signal. Okay. So you might be saying, okay, it's it's very easy. We already have defined this particular thing. So, it's all about integrating mod GT square DT, right? Or if it is a real signal, then it's just integrating G square T DT, right? So this is our means this happens when the signal is real. This happens when the signal is complex. Right? Or we can write that mod GT square as GT G star T DT. Right? So what we'll do? We'll probably take the more generic definition, which is the complex signal definition, because complex signal already uh takes the real part, right? Uh if it is real, then G star T will be GT, and immediately we get G square T, right? So for that case, I wish to evaluate this.
[6:03]You might be asking, so this is my energy. You might be asking this is by definition. Already we have defined this. We have also from the definition of Fourier series and Fourier transform, we have already defined that this should be the energy, right? So this is something was already derived. So what we are targeting now? Like what we have done in Parseval's theorem uh for Fourier series, we wish to also define energy from the frequency spectrum. This is what we are seeing from the time. Okay. So GT has equivalent Fourier transform. Let's say that is GF. Because GT is a band time limited signal. So it must have a corresponding Fourier transform, which is well defined. Okay, as long as GT has some uh property that we have discussed already. So GF is already defined. So I want to see in the frequency domain, can we evaluate the power or energy, similarly? For this signal because it's a time limited, so it should be a energy signal. So we will be interested in energy.
[7:22]So let's try to see if we can define that thing.
[7:28]So what we can do is something like this.
[7:37]We have already talked about inverse Fourier transform. Okay. So in that inverse Fourier transform, we have said GT is represented as long as GF is known. It's minus infinity to plus infinity GF to the power plus J 2 pi F T D F, right? So that's the theorem we have proven already, that's the inverse Fourier transform. So, this G star T will be replacing by this. Okay. So it will be one integration over F G F. Of course, because we have to take complex conjugate, because this is GT, we need to get G star T. So G star T will be G star F, and it will be there will be a minus sign over here.
[8:29]So I can replace it by that. So it will be G star F, into the power minus J 2 pi F T D F, right? No problem in that. Whatever we have derived so far, we are just using those things. So uh, because we want to take it to frequency domain, that is why we are applying this inverse Fourier transform. So the time domain signal, I'm actually representing it through inverse Fourier transform to frequency domain signal. But so far, I have only succeeded in replacing one of them. Of course, the DT should be still there. So this is where we have replaced G star T. Okay.
[9:14]We have to see what happens to this GT. Now, let's see this integration limit. Okay. So they are not dependent on each other, so I can always flip these two integration. So I can take the frequency integration outside. So whatever terms which involve only frequency, I can take that outside. So G square F will be sorry G star F will be there. Inside the time integration should be there, where GT is already there, E to the power minus J 2 pi F T. This is also a thing which is dependent on time, so I cannot take that outside that integration. So DT integration is being done first and then D F. Right? No problem. I have just rearranged it and I'm allowed to do that rearranging because the limits are not dependent on each other. Okay. And uh whichever functions are dependent on both F and T, I have not taken them out. So I have done all the things correctly. So now let's see what happens over here. Can you identify this thing? So it's it's it is actually the Fourier transform of GT. We have already defined that. Okay. So and we have derived that. So this is Fourier transform of GT. Therefore this must be GF. So that's GF DF. So basically what has happened, we have started the definition of energy from time domain, and eventually we could get into frequency domain of the same energy. So what happens in frequency domain? It is nothing but mod GF square DF. Right? So basically in frequency domain, now you can see also what's happening. My GF might be a complex term. We have seen that already. It might have a phase component, it might have a amplitude component. What here we are doing because we are measuring a real quantity EG. So basically we are taking the formula says that I must take the modulus of GF, which is real again, and square of that I integrate over the entire frequency domain. Whatever I get back, that's actually energy. So in the frequency domain, whatever pattern I'm getting, this mod GFT square if I plot, okay, so suppose this is frequency, and I'm plotting this, I have a corresponding GT. Let's say that GT looks like this. And I'm plotting mod GF square, which will be eventually looking like this. Okay. So you will later on see if GT is this box function, it becomes sink square. Okay. So sin x by x square of that. Okay, so it looks like this. So if my GT is this, I plot. This is actually mod GF square.
[12:47]This I call it as energy spectral density. Or ESD. Why it is called so?
[12:58]Because first of all, if you integrate it from minus infinity to plus infinity, that means all the frequency component, you are actually trying to evaluate something, which is mod GF square. I still don't know what that is. Okay. So if I integrate that, I get the overall energy. Good. And if I suppose try to see what are the frequency component it has or a measurable quantity of some frequency component it has. If I try to see that, what I'll be doing? I'll be passing this through a band pass filter, very narrow band pass filter of, let's say, a band pass filter, centered around F C, and with this one as Del F. So the filter if you correctly plot that, it will, the transfer function will look like this. So it will have in the plus F C, there will be a flat band where it will pass every frequency. And minus F C also because of the symmetric nature in Fourier transform. So it will have something. Okay.
[14:26]So if I pass this, what will happen? I will have a multiplication of these two. Right? So immediately what I'll get only at that frequency, this mod GF square will exist, because it will just pass, and in all other frequency it will be zero.
[14:49]And then if I just try to see that, suppose the band is over here, let's say. So what I'll get, I'll just get the same, almost similar pattern over here. And if I just now integrate, what I should expect now? That in those frequency band how much power it has.
[15:19]Okay. So in that Del F, because the overall power is if I integrate it fully, now I'm passing it through filter. Filter will be passing only on that band whatever it has, exactly as it is. So if I now after passing through filter if I now wish to calculate the energy, that must be those frequency component energy because those are only getting passed through this filter. So that frequency component energy is becoming just the integration of this mod GF square over that frequency band.
[16:11]So this is the reason why this is called energy spectral density, because mod GF square gives me an idea that at any frequency I choose per unit band, how much energy is being there in that signal.
[16:30]So it actually characterizes the overall signal, because we have told that this energy is a measurement. So it characterizes the overall signal. It just says every frequency term, how much energy or means spectral density. I should not say spectrum. It should be spectral density. How much spectral density it has at that targeted frequency. Okay, because it characterizes that. So that is another way of characterizing the whole signal.
[17:22]Okay. So that's the importance or significance of this. So we have now got some measurement or measurable quantity in the frequency domain as well. Because we have now defined two uh separate domain for representing a signal. One is the time domain, another one is the frequency domain. Now we have got a frequency domain representation of measurable quantity, which is energy. Okay. Or energy spectral density, that's something we have got.
[18:00]Now we'll introduce another thing.
[18:14]Okay. So that is that if I have a signal GT, Let's say this is GT.
[18:31]How do I get my energy spectral density? The technique is very simple. First, do our Fourier transform. Okay. So whenever I do a Fourier transform, I'll be getting corresponding mod GF square, right? Means first I'll be getting GF, and then from that GF I can calculate the mod GF square. So that becomes my energy spectral density. This is one way of evaluating energy spectral density. But I would say most of the time the helpful one is not this one. Okay. So we'll try to see the what is the other method to actually evaluate energy spectral density. So for that, we'll try to define a particular term, which is called time auto correlation function.
[19:31]What is time auto correlation function? So suppose I have a time bounded signal GT. It might be like this. It might be anything, but bounded in time. Okay. So this GT, if I multiply with a either advanced or delayed version of that same signal. So it might be T plus tau. Let's say it's advanced by tau amount of time. And if I integrate this over the entire time. Okay. So I take a signal, so it might be this signal. And then probably this means actually it will be delayed. Okay. So it will be uh shifted backwards. So if I just shift it by this amount. So this is actually tau. And then I'll multiply these two. So this is GT and this is GT plus tau. Okay. If I multiply these two signal and start integrating it, we are saying this is we are just defining it. Okay, right now probably you won't appreciate why we are defining this, but later on you'll see that this has a big implication in communication or signal processing. So we are defining this as phi G tau. Of course, we are integrating it over T. So the tau will remain, so it's a function of tau for different values of tau, this integration will be different. You can see already. If I I put a tau which is sufficiently large than this particular uh duration of this signal, then it will become zero. Whereas here it is not zero. It has a overlapping part. Okay, so it depends on. Of course, this will be dependent on tau. Whatever tau I choose, so it is a function of tau. Okay, so this is our time auto correlation function. First of all, let's see some property of it. We'll we'll later on link this to something of our interest and then you'll appreciate why we are defining this. But let let us first try to characterize this particular signal. So first thing we wish to prove is that it is a even symmetric function. So let's say my definition was this.
[22:04]Okay. Now let us substitute T plus tau as Z. Okay. So what will happen to T? That should be Z minus tau. Right? And DT will be tau is a constant, so DT will be for this integration tau is a constant. So DT will be D Z. So this phi G tau should be minus infinity to plus infinity G T. Now we have to do the substitution. So T will become G Z minus tau and T plus tau will become Z and this become D Z. Right? So again, Z is just a dummy variable. I can replace that as T. So it will become minus infinity to plus infinity GT G T minus tau DT. Just think about this and this definition. What is happening? If this is phi G tau. Okay. This is actually phi G minus tau by this definition. Because if tau is replaced by minus tau, this automatically becomes. So if I just replace tau by minus tau, this becomes phi G minus tau, which is exactly equivalent to this particular thing. But this is equal to this. Therefore this must be equal to this. So I can see my phi G tau is a even symmetric function, because the negative of means tau if I replace by minus tau, I get the same value. So it's a even symmetric function. Right? Okay. Next, the most important thing that I will be doing. So now what I wish to do is I wish to take the Fourier transform of this one. It will be very clear after sometime that why I'm doing this. Okay. So right now, just doing it. So I wish to take the Fourier transform of this particular signal. Right? Phi G tau. Of course, when I am saying that I am taking a Fourier transform, my Fourier transform is actually transforming it from tau domain to some other domain. Okay. Let's call that as F domain. So it's no longer the T domain, because here the means independent variable is tau. So Fourier transform should be on tau. So let us put that Fourier transform. Fourier transform means I have to put the function. So which is nothing but minus infinity to plus infinity GT GT plus tau DT, right? So this is my whole signal that has to be Fourier transform. Okay. Into E to the power minus J 2 pi F T. Sorry F tau. Because the variable is now tau, D tau. Integration minus infinity plus infinity.
[25:35]So that is the Fourier transform. Right? So let's now try to see if we can evaluate this. Again, I'll do a means change of this integration, because they are not dependent on each other. The limits are not dependent. So I can I can keep the T integration out and put the tau integration inside. So whichever is free of tau, that will be going out. So GT goes out. So I have minus infinity to plus infinity G T plus tau, E to the power minus J 2 pi F tau, D tau, right? So this is something which is there inside. Okay. Now I'll just do again same trick. T plus tau I'll replace as some Y. Let's say. Okay. So immediately what do we get? Minus infinity to plus infinity GT. Now, minus infinity to plus infinity this is becoming my Y. So G Y. Okay. Now, tau is a variable. So tau must be and for this integration T is a constant, because this integration is over tau. Okay. So T is actually a constant for that integration. So I should not bother about T. So tau becomes Y minus T, right? So this becomes E to the power minus J 2 pi F Y into E to the power plus J 2 pi F T, right? I have just replaced tau by Y minus T. So Y that minus is there, and minus T becomes plus. Okay. And DT. Right? So this is that integration. Now let's see, for this particular integration inside, this is a constant term. Okay. Because it has nothing dependent on tau. So I can take that out. So GT goes out. So I have minus infinity to plus infinity GT plus tau, E to the power minus J 2 pi F tau, D tau. Right? So this is something which is there inside. Okay. Now I'll just do again same trick. T plus tau I'll replace as some Y. Let's say. Okay. So immediately what do we get? Minus infinity to plus infinity GT. Now minus infinity to plus infinity. This is becoming my Y. So GY. Okay. Now, tau is a variable. So tau must be and for this integration T is a constant. Because this integration is over tau. Okay. So T is actually a constant for that integration. So I should not bother about T. So tau becomes Y minus T, right? So this becomes E to the power minus J 2 pi F Y into E to the power plus J 2 pi F T, right? I have just replaced tau by Y minus T. So Y that minus is there, and minus T becomes plus. Okay. And DT. Right? So this is that integration. Now let's see, for this particular integration inside, this is a constant term. Okay. Because it has nothing dependent on tau. So I can take that out. So GT goes out. So I have minus infinity to plus infinity G T plus tau, E to the power minus J 2 pi F tau, D tau. Right? So this is something which is there inside. Okay. Now I'll just do again same trick. T plus tau I'll replace as some Y. Let's say. Okay. So immediately what do we get? Minus infinity to plus infinity GT. Now minus infinity to plus infinity. This is becoming my Y. So GY. Okay. Now, tau is a variable. So tau must be and for this integration T is a constant. Because this integration is over tau. Okay. So T is actually a constant for that integration. So I should not bother about T. So tau becomes Y minus T, right? So this becomes E to the power minus J 2 pi F Y into E to the power plus J 2 pi F T, right?
