White, Pink and Brown Noise: The Same Randomness, Three Different Filters

Same randomness, different filters

All three noise colors covered here start from the same idea — a signal whose value at every instant is unpredictable from the last — and differ only in how their power is distributed across frequency. White noise has equal power per unit frequency (a flat power spectral density); pink noise falls off as 1/f, meaning power halves each time frequency doubles; brown noise (also called red noise, from Brownian motion) falls off faster still, as 1/f-squared. The underlying randomness is philosophically the same in all three; only the spectral shape differs.

White noise: flat by definition

True white noise, the kind produced by an ideal random-number generator feeding a digital-to-analog converter, has exactly equal expected power at every frequency, which is why it sounds harsh and hissy — human hearing is roughly logarithmic in frequency, so a flat spectrum on a linear frequency axis packs disproportionately more audible energy into the high octaves than the low ones, and the ear perceives that imbalance as brightness or hiss.

Pink noise: equal energy per octave

Pink noise's 1/f falloff exactly compensates for that logarithmic hearing: because each successive octave spans twice the frequency range of the one below it, and pink noise's power per hertz drops by half each time frequency doubles, the total power contained in each octave band works out equal. This is why pink noise sounds noticeably more balanced and natural to human ears than white noise, and why audio engineers use it as the standard reference signal for calibrating loudspeakers and room acoustics — a flat-sounding response to pink noise corresponds to a genuinely flat perceptual response.

power spectral density:
white: S(f) = constant                total power per octave rises with frequency
pink:  S(f) proportional to 1/f       total power per octane is constant
brown: S(f) proportional to 1/f^2     total power per octave falls with frequency

Brown noise: the sound of a random walk

Brown noise gets its name not from color perception but from Brownian motion — it is literally what you get by integrating white noise over time, i.e. taking a running sum of independent random steps, exactly the mathematical model of a pollen grain's random walk under molecular bombardment that Einstein analysed in 1905. That integration is what produces the steeper 1/f-squared falloff: integrating in time is equivalent, in the frequency domain, to dividing by frequency, and doing it once already turns white noise's flat spectrum into pink noise's 1/f — do it in the other direction and differentiating pink noise's spectrum recovers white noise, which is the formal relationship linking all three colors as successive integrals or derivatives of each other. Because low frequencies dominate so heavily, brown noise sounds deep, rumbling and muffled, like a distant waterfall or heavy rain, with almost none of white noise's hiss.

Generating each one

White noise needs nothing more than an independent random sample per output tick. Brown noise is a simple running sum (cumulative integral) of white noise, clamped or leaked slightly to prevent the random walk from drifting off to infinity. Pink noise is the awkward middle case: a mathematically exact 1/f filter is not realisable with a finite number of simple filter stages, so practical implementations approximate it, most commonly with the Voss-McCartney algorithm, which sums several independently-updated white-noise generators running at octave-spaced update rates, or with a short cascade of first-order filter sections tuned to approximate the 1/f slope across the audible range.

Where each color actually gets used

White noise calibrates audio equipment and is used in dither for digital audio and in scientific tests of a system's frequency response, since by construction it excites every frequency equally. Pink noise is the industry-standard reference for loudspeaker and room-acoustics calibration precisely because its equal-energy-per-octave shape matches how humans judge tonal balance. Brown noise, along with pink noise, is popular as an ambient masking sound for sleep and concentration, because its low-frequency emphasis is unobtrusive and covers sudden environmental noises without the harsh hiss of white noise.

Frequently asked questions

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