So let's talk a little bit more about the idea of the spectrum of a waveform and how that's related to the tone of the signal. So the spectrum is just a display of the frequency content of a signal. Now, if we plot the spectrum, it's plotted, typically, as the frequency axis long the x direction. And the amplitude of each frequency component of that waveform along the y-axis. So here's a typical waveform spectrum. So it has a fundamental frequency at f0. And then one sees a whole series of overtones in this spectrum. And the If one likes you can also draw a, envelope over the, series of overtones and the shape of that envelope is often called the format of that spectrum. Now If, here's an important definition, if the overtones are integer multiples of the fundamental frequency. So let's say this is at 100 hertz. If these were 200, 300, 400, so on. So if they're at just integer multiples of that fundamental, then you call the overtones harmonics. They're harmonically related to the fundamental frequency. So, the, component at 2f0 would be called the second harmonic. The one at 5f0, the fifth harmonic, and so on. Now, most of the musical sounds that, one hears, are, do contain harmonics. in fact, there is some, interesting physics; for example, in, physics of a piano, the, overtones of piano strings are not quite harmonically related to their fundamental, which has to do with the stiffness of the strings. there are other types of instruments, like a gong. drums most percussion instruments where they, the overtone series is not harmonically related to the fundamental. There are those like the timpani where the shape of the timpani actually forces the overtones to be at or near harmonics of the fundamental. So timpani has much more of a pitch than most percussion instruments, but that's a whole other topic a very interesting one. Um, [COUGH] so that's the, this is just the basic idea of what a frequency spectrum of a waveform looks like, and some of the terminology that we are going to be using. So let's go on and talk about the idea of Fourier series. We saw first, simple examples of that with the square wave and the triangle wave recipes that we demonstrated earlier with the MATLAB simulation. But, it's possible using a Fourier series to build, any periodic waveform, with a period of one over the fundamental frequency from sine and cosine functions. Now, this periodic waveform restriction is not as strong as it, it seems. if I have any sound, as long as it has a finite duration. So let's say I have one minute worth of sound clip. Well, I can take and mathematically, I can repeat that one minute. Sound clip after I can sort of stich it on to itself and I can make that periodic. So any waveform that is finite in length, in time duration can be made to satisfy the periodic waveform conditioned for Fourier series. So Fourier series are, are, are very powerful and apply basically to, to virtually any waveform. Now, [COUGH], here's the, the scary mathematical formula for a Fourier series. So, here's the waveform I'm trying to build x of time, will say, and there's some constant part that I'll call a0, so this could just be sort of the, the DC-level of that signal. And then we're able to then we have two series of cosines and sines and each cosine is at n times the fundamental. And now this summation here runs from n equals one all the way out to infinity and the fundamental frequency f0 is the one over the periodicity of the signal that we're trying to represent. And so I have n times f0. So the first tu, term is just at one times f0. And I have a cosine and a sine component at that frequency. And then a1 and b1 are the amplitudes of the cosine, wave, and sine wave at the frequency. And then the next term in this summation is when n equals 2, and so at 2 times f0, I have cosine and sine components, and I have a2 and b2. Now, these a's and b's. Those are basically the recipe for building the complex waveform, so you can kind of think of it as you know, the recipe for a cake. how much of of the fundamental frequency do I need? how much of the second harmonic? How much of the third harmonic? And these are the amounts of all of those ingredients that go into making this complex waveform. Now this is great if you want to just go about playing around trying different sets of a's and b's to see what you get. that's fine, but what you really need to be able to do is you want to start with some function y of t, that some complex function of time. And I want to compute these Fourier Coefficients, these a's and b's. And so it's kind of like unbaking the cake, you have to someone hands you the entire cake or the entire complex, waveform and you have to figure out how much of each frequency is in that mixture. And the way you do that is with these sort of scary looking, expressions that are integrals and I'm going to explain what that is. And so the a n's and the b n's are the integral over one cycle of this original periodic waveform times the cosine at whatever the frequency, the harmonic frequency corresponding to n. [COUGH] so I'm going to show some, draw some pictures and show what this means but I multiplied the original signal itself by a cosine function at nf0. And after I've multiplied them, I add them up over the entire cycle to see how much, total, area I get, from this, so-called integral. Now, the b's are just calculated, with the same input signal, but sines. And so, I have to compute of these integrals for every value of n. And so this is the an-s are the cosines, and those apply for n, the greater than or equal to zero, the first one is the the DC part. And then because the zero frequency, cosine of zero, is unity and this, the cosine, the a0 cosine gives you the DC part. And then the sines are over here and those are good for n equals one to infinity. Because sine at 0 sine of 0 is 0 and so there is no, sine does not contribute to the DC offset part. So I know this is, a little bit daunting to see this big long formula. But this is just the general form of those formulas that I showed for the, the square wave and the triangle wave. And the interesting thing, now, is how we can take a complex waveform apart, or analyze it, do Fourier analysis. To figure out the a's and b's that are required to construct that waveform.