OK, come on, come on. So we would think about-- in the case of the Fourier, we think about integrating over the period sifts out a component. Use OCW to guide your own life-long learning, or to teach others. So let me draw two orthogonal directions. It's a little hard to see. Cosines, the complete ones, the complex coefficients. You see the ripples moving over there, but their height doesn't change. OK? And every time we do it, we see, you understand the decay rate now? Well, sines are odd functions. We're going to be multiplying Fourier series. No enrollment or registration. And we'd like to understand that better. Because all those series are series of orthogonal functions. Speech synthesis and recognition technology uses frequency analysis to accurately reconstruct vowels. With taking derivatives. What about b_3? It's because the resulting waveform no longer has an integer number of periods in the interval capital T equals 3. It's not as bad as usual. You integrate over an integer number of periods, you get 0. So you get two different spectra, depending on the filter shape. Yeah? Odd means that S(-x) is -S(x). OK, so the point is that we've already covered, even though we've only done a little bit of work in lecture, we've already covered all of the theory. Is the cos of 2 pi t orthogonal to the sine of 2 pi t over the interval capital T equals 3? But even more, professional singers often have a lot of trouble with the enormous stress that happens on this structure with repeated use and repeated overuse. This is a Fourier decomposition of that periodic waveform. You don't need to memorize that. This S(x) is, let's see. Loo. When does it work? What would be the formula for c_k? So the inner product depends on the period, because the inner product has something to do with integrator sum. Here-- how many periods? So what's up? Gibbs. What do we have to know how to do and what should we understand? Now let's see what these numbers are. I'll talk more about the MATLAB this afternoon in the review session right here. So linear equations. It's so easy, it jumps at you. And double it. I can see, what's my formula, what should c_k be if I know the d_k? You see the pattern. Because what happens is when you want to make a high sound, you tense the structure. And it should have period 2pi. But you have to take into consideration this was made with X-rays. Because there's just one formula. And so here's a table showing measured formant frequencies, F1, F2, and F3, for whatever, six different sounds for three different categories of speakers. Massachusetts Institute of Technology. Don't forget that it's four on the right-hand side and not one, so if you get an answer near 1/4 at the center of the circle, that's the reason. And then comes the b_3 guy, would be b_3 sin(3x) sin(2x). And then it goes back down. So I have net minus minus one, I get a two. That's the pattern. This is if u itself has coefficient c_k, then -u'' has these coefficients. Two. - Bat, bait, bet, beet, bit, bite, bought, boat, but, boot. And the sine vectors are an orthogonal basis. The frequency response depends on how I've put my tongue in my mouth and how I've opened my lips and stuff like that. I don't care. Made for sharing. If you were to integrate this over the period t equals 3, if I didn't multiply them if I just did that, if I just thought about that integral, I wouldn't get 0, right? So that's one good reason to look at the complex form. I'm going to focus today on things that we call voiced-- in a voiced sound. But that's ultimately the source of speech. AUDIENCE: Is there a way to think about orthogonality using the Fourier [INAUDIBLE].