Pink noise is a random signal whose power spectral density decreases as 1/f: S(f) = k/f. In other words, each octave carries the same power. It is also called “1/f noise”.
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How to measure it in class
FizziQ can generate pink noise and then analyze its spectrum with the frequency analyzer (FFT) and the spectrogram, making the comparison with white noise immediate.
Steps:
- Generate white noise and record its averaged spectrum for a few seconds
- Generate pink noise and record its spectrum under the same conditions
- Overlay the two readings: pink noise shows a steady downward slope, white noise a plateau
- Read the pink noise level at 200 Hz, 400 Hz, 800 Hz and 1,600 Hz - each doubling of frequency should lose about 3 dB
- Check that these -3 dB per octave indeed correspond to a 1/f decrease of the spectral density
- Listen to the two noises in succession and describe the difference: white noise hisses, pink noise evokes a waterfall or wind
Scientific activities on this topic
To discover what pink noise sounds like, you can listen to this YouTube video
The following activities help understand what pink noise is and how to use it:
- Analyzing the spectrum of pink noise and comparing it with white noise
- Destructive sound interference - understand how a broadband noise is attenuated by superposing a wave in phase opposition
Possible extensions with FizziQ: measuring the speed of sound by attenuation of non-resonant frequencies in a tube.
Learn more
Why does pink noise sound “balanced” to the ear?
Because the ear does not analyze frequencies in bands of fixed width in hertz, but in proportional bands - widths that grow with frequency, modeled by critical bands or, more simply, by octaves. In white noise, the 5,000-10,000 Hz octave is fifty times wider than the 100-200 Hz octave: it receives fifty times more energy, hence the hissing impression. Pink noise corrects exactly this imbalance by compensating the widening of the octaves with a 1/f decrease. Every octave carries the same weight, and the sound seems uniform, close to a waterfall or wind in the leaves.
This is why pink noise, and not white noise, serves as the reference signal for tuning sound systems and for room acoustics measurements.
Where the -3 dB per octave comes from
The calculation is short and worth doing in class. The power contained between f and 2f for a density S(f) = k/f is the integral of k/f between f and 2f, that is k·ln(2), independent of f. The power per octave is therefore constant - this is the very definition of pink noise. Symmetrically, in a band of fixed width Δf, the power decreases as 1/f. Since doubling the frequency halves the density, the loss is 10 × log(2) ≈ 3 dB per octave, or 10 dB per decade.
A noise found everywhere
1/f noise is not an acoustician’s curiosity. It is observed in a remarkable number of seemingly unrelated systems: the low-frequency noise of transistors and semiconductors, the fluctuations of the Nile’s flow, the variations of the Earth’s rotation period, the light intensity of quasars, the heart rate, the temporal distribution of notes in tonal music. This universality remains partly unexplained: unlike thermal white noise, whose physical origin is well identified, 1/f noise has no single known mechanism. It typically appears in systems featuring a superposition of processes with very varied time scales.
Its mathematical signature is that it has no characteristic scale: stretching the time axis by any factor gives back a statistically similar signal. This is called self-similar behavior, or scale-free noise.
Pink, white, red: a continuous family
The “colors” of noise form a continuum indexed by the exponent α in S(f) ∝ 1/f^α:
- α = 0: white noise, constant density, +3 dB per octave in energy per octave band
- α = 1: pink noise, constant energy per octave, -3 dB per octave in density
- α = 2: red or Brownian noise, -6 dB per octave, very muffled; it is the integral of white noise
- α = -1: blue noise, +3 dB per octave, very hissing
Pink noise therefore occupies exactly the intermediate position between white and red, which its name reflects.
Beware of the display scale
A very frequent misinterpretation in lab work: on an FFT analyzer with a linear frequency axis, pink noise does not look like a straight line but like a curve that plunges very quickly and then flattens out. One then believes to see a defect in the signal. Only on a logarithmic axis does the 1/f law become a straight line with a slope of -3 dB per octave. Checking the type of axis before drawing conclusions is part of reading a spectrum.
Formula
Power spectral density of pink noise:
S(f) = k / f
Power contained in one octave, from f to 2f:
P = ∫ (k/f) df = k × ln(2), independent of f
Density difference between two frequencies:
ΔL = 10 × log(f₁ / f₂)
that is, about -3 dB per octave (f doubled) and -10 dB per decade (f multiplied by 10).
General family of colored noises:
S(f) ∝ 1 / f^α with α = 0 (white), α = 1 (pink), α = 2 (red)
where:
- S(f): power spectral density (W·Hz⁻¹)
- f: frequency (Hz)
- k: constant setting the overall level of the noise
- α: spectral exponent, dimensionless
- ΔL: level difference (dB)
Application examples
- Tuning a sound system. Pink noise is played into the room and the spectrum received at the audience’s position is measured. Since pink noise is balanced per octave, a spectrum measured flat in third-octave analysis means that the room and the system are correctly equalized. Deviations directly indicate the bands to correct.
- Quantified comparison with white noise. Between 500 and 1,000 Hz and then between 1,000 and 2,000 Hz, pink noise delivers the same power in both octaves. White noise delivers twice as much (+3 dB) in the second one, since it is twice as wide in hertz.
- Loudspeaker endurance testing. Standards use pink noise rather than white: white noise would concentrate excessive power in the treble and destroy tweeters without representing realistic musical use. The average spectrum of music is in fact fairly close to pink noise.
- Sound masking and sleep. Noise generators for sleep or tinnitus use pink or Brownian noise, better tolerated over long periods than white noise, which is perceived as harsh.
- Flicker noise in electronics. Below a few kHz, the noise of a transistor is dominated by a 1/f component that overtakes thermal noise. This is what limits very low frequency amplifiers, and why delicate measurements are often shifted to higher frequency by modulation.
- Time series analysis. Detecting an exponent α close to 1 in the fluctuations of a biological or geophysical signal is an indication of complex dynamics, with no preferred time scale.
FAQ
Q: Is pink noise a kind of white noise? A: No. They are two distinct noises. White noise has a constant spectral density - the same power per hertz. Pink noise has a 1/f density - the same power per octave. You go from white to pink by applying a filter with a slope of -3 dB per octave.
Q: Why is pink noise more pleasant to listen to than white noise? A: Because the ear analyzes sound in proportional bands, not in bands of fixed width in hertz. In white noise, the higher octaves, being wider, concentrate far more energy: the sound seems hissing. Pink noise distributes the energy equally between octaves, which makes it uniform and close to natural sounds like rain or wind.
Q: Where does the -3 dB per octave figure come from? A: Moving up an octave doubles the frequency, and therefore halves the spectral density S(f) = k/f. Halving a power corresponds to 10 × log(1/2) ≈ -3 dB. The same decrease gives -10 dB per decade.
Q: My pink noise spectrum does not look like a straight line, is that normal? A: Yes, if the frequency axis is linear - the default setting of most analyzers. The 1/f law only appears as a straight line on a logarithmic scale. Add to that the fluctuations of a random signal, which require averaging over several seconds.
Q: Why is this noise called “pink”? A: By analogy with light. A light spectrum decreasing toward high frequencies is depleted in blue and enriched in red: the result is a pinkish hue. Pink noise sits between white noise (flat spectrum) and red noise (1/f² decrease, even more pronounced).
Related concepts
White Noise - Decibel - Octave - Fourier Transform (FFT) - Timbre - Sound Intensity - Active Noise Reduction