Monday, April 30, 2007

It can drive you nuts, and it can help you hear

kw: book reviews, nonfiction, acoustics, signal processing

NOISE by Bart Kosko is the most intensely mathematical book I've yet seen at the local library, outside the Reference section. The author is a professor of Electrical Engineering, a popular author, and judging from the diversity of his formal education, a polymath. Judging from the text, he is also an autodidact in a number of areas.

I'll begin with a quibble: Dr. Kosko has the teacher's habit of repetition—we all know our students don't get it the first time—, but takes it to an extent I found tiresome. There is a little English bird named the great tit (Parus major) that is an easy subject, so it is studied extensively. A number of studies have shown that it sings on a higher pitch in the presence of industrial noise. According to the index to the book, this bird appears three times. The indexer missed the other four times the story is repeated. Dr K? You had me by the second time. Your meme has now successfully reproduced!

Noise has a Jekyll-and-Hyde quality. It helps and it hurts. I have tintinnitus (I think tinnitus is the more modern term) in both ears, probably due to mowing too many lawns without hearing protection. I've never used firearms or listened to loud music; my brother, once a rock drummer, got his hearing problems the musical way. I found out something about tinnitus, when I had physical therapy for a sore neck. Certain head positions caused the ringing to get much louder. I went to an audiologist. He said the cause was that, moving the head that way triggered the "too loud" muscle that closes or narrows the ear canal in response to loud sounds (drummer's earache, my brother called it, when it went on too long and got sore). Tightening that muscle increases the resonance inside the ear, making the tinnitus louder. Tinnitus due to hair cell damage is an actual sound produced in the cochlea; a sensitive microphone can record it.

When I nap, as distinct from sleeping at night, I usually find that turning on the radio so I can barely hear it helps me nod off. It isn't quite white noise, but it masks a lot of other sounds, and its familiarity seems to help. Some noise is helpful. Indeed, noise of my choosing may be sufficient to let me rest when a neighbor is using a lawn mower or stereo up too loud. The author goes into legal implications of noise and noise ordinances, showing his familiarity with law (as it turns out, that's one of his degrees).

Low levels of noise also help detecting certain signals. This works only in nonlinear systems, but I have yet to find a truly linear system in nature. I've also occasionally heard a third tone when a friend and I whistle two loud, different tones. The third tone is always lower, the subtraction of the two higher frequencies. Theoretically, a fourth tone, the sum frequency, is also present but I haven't heard it. This phenomenon arises from nonlinearity, which is slightly present in the air transmitting the sounds, but is much greater at the air-eardrum interface, and perhaps equally so in the drum-stirrup-anvil-cochlea chain.

Anyway, this nonlinearity means a faint sound could be more discernible in the presence of an even fainter hissing noise ("white" noise). I suspect the tinnitus I already have would overcome both signals, so someone else will have to do the experiment.

In the realm of signal processing, myriads of experiments have been performed, as Dr. Kosko writes. He goes quite deeply (for me...and I am a mathematician by trade!) into the math of noise statistics and signal-noise convolutions. One aspect from which I learned much was the diversity of "bell curves". Like most classically-trained statisticians, the only bell curve I knew was the Gaussian normal. Of course, I know quite a variety of bell-like distributions (Weibull, Lognormal, Logistic, and a number of others). However, only the Normal curve has a related central limit theorem. Thanks to this book, I learned that there are a number of central limit theorems, leading to a family of symmetric bell curves, of which the Gaussian is on end-member. There were hints of other families thereof, not as relevant to signal processing.

All of these bell curves apply to the analysis of white noise, which is noise with a flat spectrum. Of course, an entity with a truly flat spectrum from f=0 to infinity is impossible, because the energy required is infinite for any nonzero signal level. In real systems, a signal can have a flat spectrum over a range of interest, but fall off at higher frequencies. Any "near-white" signal will be very jittery, but the shape of the jitteriness can vary. Thus, "well-behaved white noise" with a uniform range of excursions in sound pressure will sound like a steady hiss, "gaussian" noise with a larger number of small excursions and fewer large ones will sound crackly, and more leptocurtic distributions of sound pressure such as "cauchy" noise sound like popcorn over a fainter hiss.

Much more common are pink noise, more rumbly because the sound power at higher frequencies falls off steadily as 1/f. I suspect real "pink" noise is more like lognormal noise, with the lowest frequencies sharply attenuated, but a 1/f response above a low-frequency mode. I learned of brown noise, whose frequency spectrum is essentially the square of pink noise, and black noise, which is cubic or higher. I suspect black noise sounds a lot like an earthquake rumble, or is perhaps felt but barely heard. (NOTE to self: transduce pink, brown, and black signals into WAV files to see how they sound).

Stephen Hawking was once told that each equation in a book cuts the potential audience in half. If this were true, NOISE would have an audience of one or fewer. But the text is actually quite readable, and I can attest that, if one simply skims over the equations and scans the accompanying explanations, sufficient understanding results.

A large part of the book illustrates and explains stochastic resonance, the enhancement of detection that faint noise can confer on small signals. I can add an example. Astronomers have been taking advantage of this for decades. When recording a negative, an astronomer will balance the exposure time so that sky brightness (which is never zero) produces a density of 0.5. This means there is a background speckly gray caused by several percent of the silver grains.

It is known that for the best astro emulsions, it takes ten or more photons to make a grain "convert". If there is a faint nebula in view, so faint that only one or two photons will strike the average grain during the exposure, then statistically a larger proportion of grains will "convert" because of the added contribution of the sky brightness, which is noise to the uninitiated. The nebula would be undetectable in a much darker sky without a much longer exposure (which would then of course be possible).

The author shows how stochastic resonance works for "linear" signals and for images, and how it seems to help our neurons do their job better. He speculates that, though noise probably isn't the cause of life, it probably makes life possible.

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