 Hi, I'm Zor. Welcome to InDesign Education. I would like to spend some time analyzing mathematics behind frequency modulation. So basically I call this lecture frequency modulation equation. It's a mathematical expression of how exactly the frequency modulated signal looks like. This lecture is part of the course called Physics 14. It's presented on Unisor.com. I suggest you to watch this lecture from this website. You have to go to Unisor.com. Then there is a course, Physics 14, the part of the course which this lecture is part of. It's called Waves and among the waves there is a topic radio. So this lecture is part of that topic radio. Okay, now why? Because every lecture has notes, very detailed notes on the website. So if you find it, let's say on YouTube, you will not be able to read it. And notes are written as a textbook. On the same website you will have Math 14's prerequisite course, which contains all the math which I am using today and in many other lectures. Math is an absolute necessity for people who study physics. Now there are many problems for those people who would like to take exams. They can actually take exams as many times as they want. Now there is a certain educational functionality in the website, which means if you are studying under somebody's supervision, then that somebody can enroll you in certain courses, take a look at your exams, etc. And finally the website is totally free. There are no ads, no strings attached. You don't even have to sign in if you don't want to this type of supervised studying. Okay, back to frequency modulation. Well, first of all, what do we try to accomplish by frequency modulation? Just as a reminder, there is an amplitude modulation. So if you have a high frequency carrier signal and some kind of a sound wave, whatever sound wave is, and sound wave is basically the changes of the air pressure and it has much lower waving than the high frequency signals. Then we are changing the amplitude of the high frequency in sync with the way how the sound wave signal goes, how the air pressure goes, which means the result would be higher amplitude of oscillation in this case than lower here, high here again, lower here again. So this new signal is an amplitude modulated signal which represents this particular curve of the sound wave, of the air pressure basically. Okay, so that's amplitude and it's kind of easily understood. Frequency modulation is a little bit more obscure. So again, let me tell what we want to do. For instance, this is a curve which represents the air pressure as a function of time. Now, how our high frequency modulation should look like if you would like to modulate frequency, not the amplitude. So the amplitude should remain exactly the same in the high frequency oscillation. But whenever there is a higher pressure, frequency must be greater, which means our curve should go tightly in this case. Now here, we have a lower air pressure and our oscillation should go with the same amplitude but a little bit less frequently. Then again, we have high frequency here and lower frequency here. So again, if you can look at this as a spring actually, now the spring will be tighter here and it will be not as tight in cases of the low pressure. Again, tighter here and the greater the intensity of the sound, the greater the air pressure, the tighter should be these loops of the spring. So that's how it goes. Now that's the purpose. And today, I would like to express this particular curve, how it looks mathematically based on how this curve looks. The modulating signal is this one. This is a function of time. So the function of time is a modulating signal. This is basically air pressure as a function of time at some particular point in space, let's say where our microphone is, because the microphone is converging this air pressure into electric signal, let's say current in some circuit. And that current should somehow overlap with this current, which represents the current in the main carrier LC circuit. Remember, LC is inductor capacitor circuit which produces oscillating current. We spoke about this. So this is the main carrier circuit and it has certain main base frequency, depending on L and C. Omega 0 is equal to 1 over square root of L, C. So that's all we know. So there is something like a main frequency. The main LC circuit of a transmitter is supposed to run on. Now, this modulating signal should change the main frequency. I mean, without the modulating signal, our frequency, whatever we are transmitting, should look uniformly, with the same angular frequency, this is angular frequency, all the time as a function of time. So it basically would be... So this would be the main signal. So it's a sinusoidal signal, constant amplitude, constant frequency, without any modulation. My purpose is... Now, and this is function of time, obviously. This is unmodulating signals, unmodulating oscillations of this circuit. Now, my purpose right now to come up with another formula, which basically gives me modulating oscillation, modulated oscillations of this main circuit. So the frequency should be changed based on modulating signal. So how this particular new formula, mathematical formula would look like, which represents the function of current in the LC circuit based on time and modulating signal. So, again, all I know is that with a higher modulating signal, I have to have higher frequency. The question is, how can I represent it in formula? So that's what we're going to talk about today. Now, before everything, I would like to step aside a little bit and talk about frequency. I know some people have a little problem with understanding this concept as far as the frequency is concerned, because there is some differentiation, integration involved, etc. So what is frequency? Now, we used to think about frequency as either just regular frequency or angular frequency. Now, regular frequency f is number of cycles per time. So if you have something which is just oscillating, which we are saying like five oscillations per second, it means that every second there are five full circles of oscillation. Every two seconds, we have 10. Every 20 seconds, we have 100. Now, that's regular frequency. Now, when we are talking about frequency, we very often express our function as this type of equation. And whenever we are talking about full cycle, so the frequency is how many cycles per, how many seconds or how many cycles per second actually, we are talking about a period of the function cosine. So the full period and the function looks like this. So full period is, let's say from here to here or from here to here, whatever, doesn't really matter. So how many full periods per second this particular function makes? That's the regular frequency. Now, we know that this function has period 2 pi. Now, every period of the function cosine has length 2 pi. Now, what's the period of this function? Well, the period of this function is 2 pi divided by omega 0. That's the period. Now, this 2 pi comes out as kind of an inconvenience. And what people actually did, they were considering this type of function as a representation of certain mechanical movement. Now, what's the mechanical movement? If you have a circle and you imagine your point here, and let's say this circle has a radius A. So this point is at x-coordinate A. And then it starts moving, uniformly moving. Now, what is its x-coordinate? Well, if the radius is A, the x-coordinate is A times cosine of this angle. This angle, considered as a function of time, is basically the argument to cosine. And if our movement is uniform, now, uniform movement means that phi of t, the angle of rotation is equal to some kind of a constant by time. This is a uniform rotation. Now, in this particular case, for the uniform rotation, we see that our x-coordinate is exactly what we actually have to have. So let's talk about oscillations as a result, basically of some kind of a point, uniformly rotating, as I was just describing, on a radius A with a frequency, with a speed. Actually, now it's angular speed. You see, there is a very direct analogy between the distance and speed in kinematics and this angle of rotation and so-called angular frequency. This coefficient is called angular frequency, basically. In this electronic, radial, et cetera, kind of a physics type of things. It's a direct analogy. So this is the distance. This is a uniform constant speed. This is time. So speed times time would equal to distance. In this particular case, the distance is measured as an angle in the region. That's why whenever we have a full circle, we have two pi regions, right? A radian, two pi regions. That's why the period of function, cosine in this particular case, is equal to two pi. Now, if the point is rotating faster, this is faster. The period will be proportionally smaller, right? So if the regular cosine of T has the period two pi, then if I multiply the speed of rotation by factor omega zero, my period will be reduced, obviously. So whenever we are talking about this equation as representing, let's say, radio waves, we definitely can model it as a rotation of a point around a circle of radius A with a speed. I'm using the word speed. Actually, it's angular speed. Omega zero, radians per second. So there is a direct analogy between kinematics, distance and speed terminology, and this radio electronics when we are talking about angular frequency and amplitude. Now, if there is a direct analogy, let's just push this analogy a little bit further. So, we obviously know this. Now, this is good because for uniform rotation it works and it fits perfectly. Now, we know, again, back to kinematics, that distance can actually be the result of multiplication of speed by time. For uniform movement along the straight line with a constant speed. But, again, a little further in studying the mechanics, we were actually studying something like accelerated movement or, in theory, any kind of a movement with variable speed. And what is speed if you consider that you know the distance as a function of time? Well, let's go back to our mechanics, actually. And the first derivative of the distance by time is actually a speed as a function of time. This is a momentary speed, speed at any particular moment of time t. Why is it derivative? Because, obviously, if you take some kind of a moment of time t and then the next moment of time t plus a little increment, you can consider that during this very little time interval the speed is not actually changing. And your speed can be considered during the whole interval as being equal to v of t, the speed at this particular moment because these are very, very close to each other moments. And now, at the same time, we know that function of time, function distance of time at this particular moment t plus delta t has value t plus delta t. And the previous moment, distance was like this. So this is increment of the distance. And if you divide it by time, which passed from one moment to another, you will have an average speed during this very small amount of time. And whenever we will reduce the amount of time, this time increment, we will make these two events very, very close to each other. So this will be smaller and smaller and this will be smaller and smaller. And in the limit, we will have something which is called the first derivative. And that's why this is goes to v of t as delta t goes to zero. So that's why this formula, which implies that if our rotation is not uniform but has a changing frequency, we have absolutely analogous relationship between angle of rotation and instantaneous, first derivative of angle of rotation is instantaneous angular speed. Exactly the same. So for uniform movement, yes, if this is a function, the first derivative of this function is omega zero. This is a simple linear function and that's uniform movement. But if the movement is not uniform, if our point is moving faster at one particular time and then slower and then faster and then slower, etc., then the instantaneous angular frequency is actually instantaneous angular speed. It's just a tradition that radio engineers are calling it frequency. It's basically angular speed. Angular speed depends on angular rotation and angle of rotation in this particular fashion. In exactly the same fashion as distance and speed are related to each other. Because angular rotation is actually angular distance and the angular speed is basically how fast we are moving. This is very important to understand. Now, and this is a very simple concept which is actually no different than kinematics concept of distance and speed in case of non-uniform movement. So this is the same. This and this are the same. Okay, great. Now, from the terminological standpoint, angle of rotation is called a phase. And again, it should not be understood in any different way than just an angle of rotation. So what's the phase of some particular oscillation? Well, if oscillation is uniform, then it's basically an angle by which the point managed to turn during certain time t. So in this particular case, if the point is here, then this is an angle. So phase is a function of time which represents basically the angle by which the point managed to turn during the time t. So phase is a function of time. Phase is angle of rotation by the time from zero to some time t. That's what phase actually is. And again, it should not be understood in any other sense, but just the angular distance by which our point managed to turn. And as usually, x-coordinate of the point, we are considering as our main variable which we are studying. We are always studying the x-coordinate of the point. And the point is rotating around this particular radius a in this particular case. Now, if we know this, we know from angle of rotation, we know how to determine the instantaneous speed of rotation or angular frequency if you would like. So angular frequency and angular frequency, this is our omega t. So these are basically radio-electronic kind of terminology. The angle of rotation and angular speed are more mechanical, so to speak, language. But they mean exactly the same. And the relationship should be exactly the same. The same relationship as between distance and speed. We have between phase and angular frequency. Now, there is a small misunderstanding because sometimes the word frequency is used in angular sense and sometimes in the sense of how many periods per second. Now, number of periods per second, but if the period is actually 2 pi divided by omega, then the frequency is equal to how many periods per seconds each period is this. So frequency is equal to basically omega divided by 2 pi, which is basically 1 over 2. So this is, sometimes people are mixing this frequency with this frequency. Now, this frequency is number of periods per second. This is number of radians per second. And since each period is 2 pi radians, that's why we have this relationship. And sometimes in formulas you have omega, but sometimes instead of omega you have 2 pi f, 2 pi f instead of omega. Formules are exactly the same. It's just what exactly we are talking about frequency, frequency in number of periods per seconds or number of radians per seconds. It's just two different scales, but they mean absolutely the same. And the difference is in this multiplier 2 pi. Now, I will use omega because it's just simpler. You don't have to write 2 pi. Okay, so. And there is one more very important piece about frequency, angular frequency. When I'm saying frequency, it means angular frequency from this. And from very simple mathematics. You know if there is a function and there is a first derivative of this function. Well, from function we can get first derivative. Can we get, from first derivative, can we get the main function? Well, very simple. Well, in pure mathematics it's supposed to be minus infinity instead of zero. In physics, we don't deal with minus infinity, we are talking about movement. So the amount, the angle of rotation by the time t is basically integral. Integral is a reverse of differentiation. So this is differentiation, this is integration. And that would give me angle of rotation from moment of time t0 to moment of time t. Well, I shouldn't use the letter t in both cases because that's kind of confusing. Let's just use Greek letter tau. It's just a variable of integration, doesn't really matter. So this is an elementary mass, which you kind of must know. And that's why there is a mass for genes, course, prerequisite course on the same website. So how to do differentiation and how to do integration should be of no problem. In elementary cases, I'm not talking about some high complexity things. So, so far this is sufficient for us to talk about frequency modulation. And here's how. Now, whenever we are talking about frequency modulation, we are talking about not a constant frequency. It's not a mega zero, but it will be frequency which depends on time. That's very important because sometimes when my modulation, modulating signal is stronger, I have to have higher frequency. Whenever it's weaker, I have to have a lower frequency. So basically what I would like to talk about is a modulated, modulated frequency. This modulated frequency obviously should depend on main frequency of our circuit, which produces main base frequency on which basically our transmitter is transmitting signal, plus something which depends on the strength of the modulating signal. So if my modulating signal is some kind of a function of t, I have to add this component. Usually it's added with some kind of a multiplier lambda called modulating index. This is a contribution. This index actually kind of controls the contribution of the input signal to frequency. And basically that's it. I mean that's how our frequency modulation is done. We are adding to the main frequency on which this particular transmitter is tuned in. We are adding something which depends on the modulating signal. So this is the sound, let's say. Now, it has certain limits of strength. Now, our widths of the band around this particular frequency should be obviously restricted because there are some other transmitters which would like to also transmit some information. So for each transmitting radio station we give the certain band of frequencies. And we were talking about this. For FM radio it's from 88 megahertz to 108 megahertz, divided into 100 pieces. And within each 100 we will have some kind of central frequency and frequency around it to compensate this particular thing. So we should use the modulating index to narrow down this particular thing. So what happens now is basically pure mathematics which you definitely are familiar from whatever I was talking before. If you know this, how can you recreate our main equation? Well, again, let's go back to main equation. We started with this unmodulating signal. Now we would like to have the modulating signal. Now, our current in the LC circuit is a cosine. Now, instead of this, I will use phi of t. In uniform case, angular speed times t is angle of rotation. In a non-uniform, we have a different speed, but we can recreate this thing very simply. Modulated angle of rotation as a function of t is integral of 1 of t, omega modulated as a function of tau. Let's use a different letter, d tau. This is a derivative. This is the angle of rotation, angular distance. So this is angular speed, angular distance, and the distance can be derived from speed using simple integration. So let's use this here. And what do we see now? We see the phi modulated at t is equal to integral from 0 to t. Omega 0 plus lambda modulating signal of tau. Sorry about that again. d tau equals... Well, if you multiply this by this, you will have one component, which is this, plus another component, which is lambda goes outside m tau d tau, which is equal to... Now, this is a simple integral, which is omega times t. And this is as it is. I don't want to do anything about this, from which I... So the current modulated as a function of t is equal to a cosine modulated f, which is omega t plus lambda integral from 0 to t, m of tau d tau. That's the final formula of modulated signal. And what I did in the notes for this lecture, I had an example. I had a function m of tau as some of... I don't remember the numbers, but it's something like, you know, 3 times cosine 2t plus cosine t or something like this. I took this as m of t as a combination of two different oscillations. Maybe there are two musical instruments. One is playing one note, another is playing another note. One is, let's say, louder, another is softer. This is producing higher pitch and this is lower pitch. So it's like an orchestra from two different instruments. Whatever it is, it doesn't really matter. I could take any function here. It doesn't really matter whether it's trigonometric or not trigonometric. It's just usually when we are talking about sound, we have a combination of many different harmonic oscillations, and not so harmonic, but in any case some kind of oscillations. So that's why I took this as m of tau. This is m. And then I basically integrate it. I mean, it's easy to integrate cosine. You know, the derivative of sine is a cosine, integral of cosine is a sine, and the derivative of sine is cosine. So it's very easy to integrate. And I come up with a formula and draw the graph. So the graph of m of t. So I draw this graph and this graph. And it appears actually exactly, if this is the sound, my frequency was much more intense here and much less intense here. Then more intense here and less intense. And the amplitude was the same. So I had these two graphs presented in notes for this lecture. And you can take a look at this. Well, that's it. My purpose was to come up with this formula and draw a particular example how exactly this frequency modulation actually looks like. Now, in my example, as a base frequency, I did not use a very high frequency because otherwise my graph would not be really easily decipherable when it's too high frequency. You don't really see the difference in frequencies if it's really too high. So I had exactly the number where you see that sometimes it's rarer and sometimes it's more frequent waves of the resulting graph. In theory, this frequency is significantly greater and it allows to... You see, this frequency is from 88 MHz. So it's 88 MHz of oscillations per second. That's high. Okay. So it allows to accommodate any kind of a sound, even high-fi. So little deviations maybe from some smoothness which basically represents some particularity of the sound can be accommodated using a very high frequency as long as we basically know how to produce this signal and how to receive it and convert back frequency oscillation, frequency modulation into some kind of a sound on the receiver side. And basically I don't know all the details but as far as I know this frequency modulated signal after it's received is converted somehow into amplitude changes because eventually we have to really produce the sound through some kind of a speaker and that's why we need the amplitude. So there is some kind of a conversion but that's beyond the scope of this particular course. It's more for radio engineers. So if you will be a specialist in this particular area you will know exactly how it's done, how to transmit this signal and how to receive it on the receiver side. Okay, that's it. I suggest you to read the notes for this lecture including the graph which is presented there. Actually click on the right button on the graph and open it up in a new tab on your browser so it will be bigger and much more decipherable. Well, that's it. Thank you very much and good luck.