The Calculus Diaries - Jennifer Ouellette [95]
Digital signal processing (DSP) would not be possible without Fourier analysis. DSPs are microelectronic devices that determine which sound wave is required to cancel noise. A DSP contains a resonator that vibrates in response to specific incoming frequencies. It then re-creates that sound wave—minus the frequency it is trying to cancel—and amplifies it through speakers or headphones. The end result is near silence. Most cell phones, CD players, and hearing aids now contain one or more DSP devices. The Fourier transform is a mathematical resonator, an efficient tool to filter signals.
A DSP first selects a sampling of a given signal measured at regular intervals. Let’s say we have 1,000 such samples of that signal, perhaps the infrasonic murmurings of a breaking ocean wave. We don’t know exactly how many “partial” sine waves make up that complex waveform, but the number of samples gives us an upper limit and a lower limit. Once we have that range of values, we can determine how many sine waves would fit between those two limits. In most cases, we will need as many sine waves as we have samples (i.e., 1,000) to perfectly reconstruct a given signal.
So now we know the frequency of our component sine waves. We also need to know their amplitude, that is, how much of each partial sine wave to mix in as we rebuild our original signal. If a given signal is heavy on the bass, there are more low-frequency sine waves mixed in than high-frequency ones, and vice versa if the signal is shrill with more treble in the mix. How can we figure that out? Why, by taking an integral, of course. We multiply that sine wave with signal samples and add the results together to get the amplitude of the partial sine wave at the frequency of interest. We do this for every possible frequency of sine wave, weed out those that we identify as ingredients, and voilà! We end up with the recipe for rebuilding that signal.
That evening finds me unwinding from the day’s exertions over a refreshing tropical cocktail in the hotel’s outdoor bar. The sun is setting, casting a pinkish orange glow over the water as hotel staff readies for the night’s luau performance. I find myself listening to the rhythmic cycle of waves crashing on the shore and reflecting on the fact that I can hear those waves because my brain is performing Fourier transforms constantly. It’s an essential part of how we hear.
The brain senses the incoming pressure wave and performs a Fourier transform on that signal to identify the frequency and amplitude of the “sound.” The ear measures change in pressure as a function of time. Sometimes it is a single note; sometimes it is several notes together, as in a musical chord; in each case, the brain uses the Fourier transform to determine which components make up the total sound wave. Similarly, every time we gaze at a sunset and identify specific hues—a spectacular orange-red, or a more subtle pinkish glow—our brain has taken a Fourier transform to isolate specific frequencies of light. And when a surfer eyeballs incoming ocean waves, his or her brain is making similar calculations.
The Fourier transform has a personal significance as well. Shortly after becoming engaged, Sean and I drove from a conference in San Francisco to our new home in Los Angeles via the scenic route along the Pacific Coast Highway. At sunset, we stopped briefly to refuel just north of Malibu and found ourselves admiring the brilliant orange, red, and purple hues stretching across the darkening horizon, savoring the peaceful sound of waves lapping against the shore. It was the perfect romantic setting to cap off a long day’s drive. Sean is nothing if not romantic. He is also the quintessential physicist. So he put his arms around me and whispered, “Wouldn’t it be fascinating