White Noise

White noise is a random signal whose power spectral density is constant: every frequency band of the same width in hertz carries the same power. Its name comes from an analogy with white light, which contains all the wavelengths of the visible spectrum.

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How to measure it in class

FizziQ makes it possible to produce white noise with its sound generator, then to visualize its frequency content with the spectrum analyzer (FFT) and the spectrogram.

Steps:

Generate white noise with FizziQ, or play it from an external source
Open the spectrum analyzer and observe the shape of the spectrum: an approximately horizontal plateau, but highly fluctuating from one instant to the next
Activate averaging, or make a long capture with the spectrogram: the plateau smooths out as you average
Compare this spectrum to that of a pure tone (a single line) then to that of pink noise (a decreasing spectrum)
Listen alternately to white noise and pink noise and describe the difference in timbre
Repeat the analysis by displaying the levels per octave band if the tool allows it: white noise then shows a rising slope of +3 dB per octave

Scientific activities on this topic

White noise: spectral composition - analyze the spectrum of white noise and compare it to that of other sounds
Speed of sound: tube resonance - use white noise as a broadband excitation to reveal the resonance frequencies of a tube

Learn more

In addition to white noise, there are several other types of noise, each with specific spectral and statistical characteristics.

Here are some of the most common types of noise:

Pink noise: Pink noise, also known as “1/f noise”, has a power spectral density that decreases as the inverse of the frequency. The energy per hertz therefore decreases as the frequency increases, but the energy per octave remains constant: this is what makes it perceptually balanced, that is, −3 dB per octave relative to white noise.

Blue noise: Blue noise has a power spectral density that increases proportionally to frequency (+3 dB per octave). It is used in image processing for dithering and in electronics.

Red noise: Red noise, also called “Brownian noise”, has a spectral density that decreases as 1/f², that is, −6 dB per octave. Its decrease is therefore faster than that of pink noise, not slower: it is much duller, dominated by the low frequencies. It corresponds to the integral of white noise and describes the random walk of a Brownian particle, hence its name, which comes from the botanist Robert Brown, not from the color.

Gaussian noise: Gaussian noise, also called “Gaussian white noise”, follows a Gaussian (normal) distribution of amplitudes. It is the type of noise most commonly used in statistics and signal processing.

Impulse noise: Impulse noise is characterized by the occurrence of sudden peaks or impulses in the signal. It can be caused by electromagnetic interference, transmission errors, or other disturbances.

Thermal noise: Thermal noise, also known as “Johnson-Nyquist noise”, is generated by the thermal fluctuations of electric charges in a conductor. It is omnipresent in electronic components and has a Gaussian distribution.

Quantum noise: Quantum noise is associated with quantum fluctuations and is observed in quantum systems, notably in nanoscale electronic devices. It can include “shot noise” and “generation-recombination noise”.

These different types of noise have varied applications depending on their characteristics. For example, white noise is often used to calibrate systems, while pink noise is used to simulate natural acoustic conditions, and Gaussian noise is commonly encountered in everyday random situations.

White and Gaussian: two independent properties

This is a very frequent confusion, even in textbooks. “White” describes the spectrum: the distribution of power according to frequency, in other words the absence of correlation between successive samples. “Gaussian” describes the statistics of the amplitudes: the way the instantaneous values of the signal are distributed. The two notions are independent. There are non-Gaussian white noises: for example a signal that only takes the values +1 and −1 at random, whose spectrum is perfectly flat but whose distribution is binary. And a Gaussian noise can very well be colored, hence not white at all. The usual “Gaussian white noise” simply combines the two properties.

Ideal white noise does not exist

A rigorously white noise would have a constant spectral density over an infinite frequency band, hence an infinite total power. It is a mathematical idealization, convenient but physically impossible. Any real white noise is in fact band-limited: flat between two frequencies, zero beyond. In audio, one restricts oneself to the audible band, 20 Hz - 20 kHz. In digital signal processing, the limit is imposed by Shannon’s theorem: a signal sampled at 44.1 kHz cannot contain anything above 22.05 kHz.

Where do we really encounter it?

White noise is not just a laboratory object. The thermal noise (Johnson-Nyquist) of a resistor is white up to very high frequencies: it is the unavoidable noise floor of all electronics, and its density equals 4k_BTR. The shot noise of an electric current, linked to the discrete nature of the electron’s charge, is also white. These noises set the ultimate sensitivity limit of a sensor: that is why the detectors of telescopes are cooled.

Why excite a system with white noise?

This is the most useful use in lab work. Since white noise contains all frequencies simultaneously, sending it into a system (a tube, a room, a filter) makes it possible to test all its resonance frequencies in a single measurement, instead of patiently sweeping frequency by frequency. The spectrum of the signal collected at the output then directly gives the frequency response of the system: the resonances appear as peaks, the absorbing zones as dips. This is exactly what is exploited in the tube resonance activity.

Formula

Power spectral density of white noise (constant over the whole band):

S(f) = S₀

Power contained in a frequency band [f₁; f₂]:

P = S₀ × (f₂ − f₁)

The power therefore depends only on the width of the band, not on its position. For comparison, in pink noise the density equals S(f) = k/f and the power per octave is constant.

Level per octave band of white noise, from one octave to the next:

ΔL = 10 × log(2) ≈ +3 dB per octave

Thermal noise of a resistor (white noise):

S = 4 k_B T R

where:

S(f): power spectral density (W·Hz⁻¹, or V²·Hz⁻¹ depending on the quantity)
S₀: constant value of this density
f: frequency (Hz)
P: power in the band considered (W)
k_B: Boltzmann constant, 1.38 × 10⁻²³ J·K⁻¹
T: absolute temperature (K)
R: resistance (Ω)

Application examples

Measuring the response of a room. White noise is played and the spectrum is recorded at different points: the dips and bumps reveal the normal modes of the room and the frequencies absorbed by the materials. A single measurement replaces a complete sweep.
A numerical white/pink comparison. Between 100 and 200 Hz (width 100 Hz) and between 1,000 and 1,100 Hz (width 100 Hz), white noise contains the same power. But between the octaves 100-200 Hz and 1,000-2,000 Hz (width 1,000 Hz), it contains ten times more in the second, that is, +10 dB. Pink noise, on the other hand, would contain as much in both octaves.
Sleep and tinnitus. Noise generators designed to mask unwanted sounds more often use pink or Brownian noise than white noise, which is judged too hissing precisely because of its excess energy in the high frequencies.
Electronic noise floor. A 1 kΩ resistor at 300 K produces a noise voltage of about 4 nV per square root of hertz, that is, roughly 0.6 μV rms over the whole audio band. This is what limits the sensitivity of a microphone preamplifier.
Telecommunications. The reference channel for evaluating a transmission system is the AWGN channel (additive white Gaussian noise): it is on this model that the maximum capacity of a link is calculated.
Loudspeaker testing. White noise, rich in high frequencies, is used to detect tweeter defects; but pink noise is preferred for endurance tests, because it does not impose excessive power at high frequencies.

FAQ

Q: What is the real difference between white noise and pink noise? A: It lies in the way the energy is distributed. White noise has a constant power per hertz: all bands of the same width in Hz are equivalent. Pink noise has a constant power per octave: its spectral density decreases as 1/f, that is, −3 dB per octave. Since the ear analyzes by octaves, it is pink noise that sounds balanced; white noise sounds hissing.

Q: The spectrum I obtain is not flat at all, did I fail my measurement? A: Probably not. The instantaneous spectrum of a random noise always fluctuates by several decibels. Flatness is a statistical property, which only appears after averaging over several seconds. Add to that the frequency response of the loudspeaker and the microphone, which is never flat either.

Q: Are white noise and Gaussian noise the same thing? A: No. “White” qualifies the spectrum, “Gaussian” the distribution of amplitudes. They are two independent properties: a noise can be white without being Gaussian, and Gaussian without being white.

Q: Why is this noise called “white”? A: By analogy with white light, which contains all the wavelengths of the visible spectrum. The analogy has its limits: perceived white light does not have a flat spectrum, and the other “colors” of noise (pink, red, blue) were named later by extension of the same image.

Q: Can perfect white noise exist? A: No. A constant spectral density over an infinite band would imply infinite power. Any real white noise is band-limited, in audio to the audible band, and in digital by the sampling frequency via Shannon’s theorem.

Pink Noise - Fourier Transform (FFT) - Complex Sound - Decibel - Resonance Frequency - Timbre - Microphone