Wednesday, March 05, 2008

The soundless cacophony

Now for the Big Idea, what's big about it, and its limitations. Although it had some precursors, the general method was defined and proved by the French mathematician Fourier in the early nineteenth century. His name is pronounced "FOO-ree-ay."

NOTE: I use the italic f to represent both frequency and a function of time. Frequency is just f; the function is f(t).

Phase, frequency, amplitude: the spinning circles. If you want a geometric picture of many fundamental harmonic motions oscillating at once, think of many circles, each one having a size indicating the oscillation's amplitude and turning around steadily at its distinctive frequency. If each circle is marked on its edge to indicate the starting point, the phase of each circle's oscillation is just how much that circle's mark is offset from some fixed reference angle at the start time.

Periodicity over a finite interval of time. Let's think of some sort of motion or signal that repeats over a time interval or fundamental period T. That is, we're dealing with a function of time f(t) that is periodic: f(t+T) = f(t). By recursion, this also mean that f(t+2T) = f(t+T) = f(t), etc. The function f(t) can have up to a countable infinity of discontinuities, but it must be finite everywhere.*

Fourier's theorem says that is possible to represent such a function as an infinite series of sines and cosines, each vibrating at one harmonic of the fundamental frequency f = 1/T. This overtone series fn = n/T, with associated angular frequencies ωn = 2πn/T, is familiar from music. The second harmonic, for example (n=2), is an octave above the fundamental frequency.

Each sine and cosine is weighted by their respective amplitudes (An, Bn). The combined amplitude, related to the energy or power stored in that harmonic, is usually stated as Cn2 = An2 + Bn2.

As we'll usually do, we've ignored phase. It's taken care of by using the sine and cosine together. The Fourier series, if we don't care about phase, is then just characterized by all the Cn and the ωn.**

Fourier's series method allows us to analyze or decompose any such motion satisfying the restrictions into an infinite series of vibrating modes. For example, each musical instruments has a different profile of amplitudes Cn - its Fourier spectrum - giving each instrument (piano, guitar, violin, and so on) a distinctive timbre or sound-color. The same note played on each has the same fundamental pitch and the same overtone frequencies. But the strength of those overtones, relative to the fundamental, is different for each instrument.

Note the ratio of any two frequencies in the overtone series is a rational number: m/n, and any pair of frequencies is commensurate. We know that the motion has to repeat after waiting a time T anyway, but it's reassuring to see this come out of the analysis automatically.

Periodicity over an infinite period of time. Can this Fourier trick be extended to an infinite time interval, T → ∞? Yes, with some subtleties. If T → ∞, the fundamental frequency f → 0, and the countable infinity of discrete harmonics merges into a continuum of possible frequencies. The profile of amplitudes Cn now becomes a continuous function of f or ω, the Fourier transform F(ω) or F(f). It shows how much vibration at each frequency contributes to the whole motion being analyzed. The square of the amplitude is often related to the energy or power stored in each vibrating mode.

It might seem strange to say that a function f(t) defined over an infinite time interval should be periodic, but that's what the continuous version of Fourier's method (the Fourier integral or transform) requires. That is, f(-∞) = f(+∞). In practice, this is ensured by assuming that the function "turns on" slowly in the far past (t → -∞) and "turns off" slowly in the far future (t → +∞). Then the function is trivially periodic, being zero at the beginning and end. If the function approaches a non-zero constant in the far past and far future instead, that constant can just subtracted off of f(t) before it's Fourier-analyzed.

Long time intervals contribute mainly to low-frequency amplitudes, or the low-frequency part of the Fourier spectrum. The condition of periodicity at the infinite past and future then requires that the amplitude spectrum cut off (drop to nothing) below some non-zero frequency.

Fourier lives inside your car, your cellphone, your radio - and a lot else besides. Fourier analysis is fundamental to all branches of physics and engineering. It's how signals are transmitted on phones, on radio and television, and on computers. Engineers design filters to block out signal components at certain frequencies, based on Fourier analysis of the overall motion or signal. It's essential to clear communication on your cell phone, for example, where the frequencies are audio range (100s or 1000s of Hertz); and to automatic controllers, where the frequencies are much lower (tenths to 10s of Hertz).

The continuum of frequencies: Waiting for the grand recurrence. This world of continuous Fourier analysis is infinitely-pitched world with a continuum of frequencies. Buried in the continuum of frequencies is a countable infinity of frequency sets, each with a countable infinity of fundamentals and overtones. But these are rational number "stars" embedded in the continuously infinite background of irrationals - incommensurate frequencies that prevent any system from exactly repeating itself. The continuous babble of pitches is ultimately the result of irrational numbers, numbers that aren't ratios of integers - which is to say, most numbers.†

Only if the Fourier transform's start and endpoint conditions are satisfied, does the system repeat, after waiting a "long" time - and this implies that the Fourier spectrum of amplitudes is cut off at some low but non-zero frequency. The overall repetition of the system's motion requires that its Fourier spectrum not extend to arbitrarily low frequencies.

Searching for chaos. For millennia, this "atonal" babble of incommensurate frequencies was thought to be what we today call "chaos." †† It certainly seems "chaotic" in the colloquial sense. But what we call "chaos" today, first envisioned at in the late 1800s and fully (re)discovered in the 1960s with the advent of computers, is something else yet more subtle. It has "no tone" and no periodicity at all - it's the note that never vibrates. We'll meet this unheard tone next.
* A finite discontinuity means that the function is defined at the time of discontinuous "jump," but not the function's derivatives. That is, it's not differentiable at those times.

** You might wonder: can an infinite series of combined oscillations like that converge to a finite number? Indeed, it does, under certain restrictions, which hold automatically for the Fourier series if the original function f(t) is finite everywhere. The overtone amplitudes Cn have to fall rapidly enough for higher n. In practice, the first few overtones are often the only ones strong enough to matter for practical purposes.

† Because they can be put into one-to-one correspondence with the integers, the rational numbers are countably infinite. This countable infinity is "smaller" than the continuous infinity of points on a line, even a line of finite length. Using one-to-one correspondence, it's perfectly possible, contrary to an old philosophical prejudice, to work with actual infinities.

The catch is that certain laws of logic don't hold with actual infinities, in particular the law of identity. For an actual infinity, A is not A, parts are not smaller than wholes, and wholes are not bigger than parts.

†† Schoenberg, the first composer of full-blown atonal music, wanted it to be called "pantonal" instead, which is indeed a far better description, both of his mature music and of the Fourier frequency continuum.

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