Power-law spectra

The power spectrum of pink noise is ${\displaystyle {\frac {1}{f}}}$ only for one-dimensional signals. For two-dimensional signals (e.g., images) the average power spectrum at any orientation falls as ${\displaystyle {\frac {1}{f^{2}}}}$, and in ${\displaystyle d}$ dimensions, it falls as ${\displaystyle {\frac {1}{f^{d}}}}$. In every case, each octave carries an equal amount of noise power.

The average amplitude ${\displaystyle a_{\theta }}$ and power ${\displaystyle p_{\theta }}$ of a pink noise signal at any orientation ${\displaystyle \theta }$, and the total power across all orientations, fall off as some power of the frequency. The following table lists these power-law frequency-dependencies for pink noise signal in different dimensions, and also for general power-law colored noise with power ${\displaystyle \alpha }$ (e.g.: Brown noise has ${\displaystyle \alpha =2}$): ^[7]

Power-law spectra of pink noise

dimensions	avg. amp. ${\displaystyle a_{\theta }(f)}$	avg. power ${\displaystyle p_{\theta }(f)}$	tot. power ${\displaystyle p(f)}$
1	${\displaystyle 1/{\sqrt {f}}}$	${\displaystyle 1/f}$	${\displaystyle 1/f}$
2	${\displaystyle 1/f}$	${\displaystyle 1/f^{2}}$	${\displaystyle 1/f}$
3	${\displaystyle 1/f^{3/2}}$	${\displaystyle 1/f^{3}}$	${\displaystyle 1/f}$
${\displaystyle d}$	${\displaystyle 1/f^{d/2}}$	${\displaystyle 1/f^{d}}$	${\displaystyle 1/f}$
${\displaystyle d}$, power ${\displaystyle \alpha }$	${\displaystyle 1/f^{\alpha d/2}}$	${\displaystyle 1/f^{\alpha d}}$	${\displaystyle 1/f^{1+(\alpha -1)d}}$

Distribution of point values

Consider pink noise of any dimension that is produced by generating a Gaussian white noise signal with mean ${\displaystyle \mu }$ and sd ${\displaystyle \sigma }$, then multiplying its spectrum with a filter (equivalent to spatially filtering it with a filter ${\displaystyle {\boldsymbol {a}}}$). Then the point values of the pink noise signal will also be normally distributed, with mean ${\displaystyle \mu }$ and sd ${\displaystyle \lVert {\boldsymbol {a}}\rVert \sigma }$.^[7]

Autocorrelation

Unlike white noise, which has no correlations across the signal, a pink noise signal is correlated with itself, as follows.

1D signal

The Pearson's correlation coefficient of a one-dimensional pink noise signal (comprising discrete frequencies ${\displaystyle k}$) with itself across a distance ${\displaystyle d}$ in the configuration (space or time) domain is:^[7]

$${\displaystyle r(d)={\frac {\sum _{k}{\frac {\cos {\frac {2\pi kd}{N}}}{k}}}{\sum _{k}{\frac {1}{k}}}}.}$$

If instead of discrete frequencies, the pink noise comprises a superposition of continuous frequencies from ${\displaystyle k_{\textrm {min}}}$ to ${\displaystyle k_{\textrm {max}}}$, the autocorrelation coefficient is:^[7]

$${\displaystyle r(d)={\frac {{\textrm {Ci}}({\frac {2\pi k_{\textrm {max}}d}{N}})-{\textrm {Ci}}({\frac {2\pi k_{\textrm {min}}d}{N}})}{\log {\frac {k_{\textrm {max}}}{k_{\textrm {min}}}}}},}$$

where ${\displaystyle {\textrm {Ci}}(x)}$ is the cosine integral function.

2D signal

The Pearson's autocorrelation coefficient of a two-dimensional pink noise signal comprising discrete frequencies is theoretically approximated as:^[7]

$${\displaystyle r(d)={\frac {\sum _{k}{\frac {J_{0}({\frac {2\pi kd}{N}})}{k}}}{\sum _{k}{\frac {1}{k}}}},}$$

where ${\displaystyle J_{0}}$ is the Bessel function of the first kind.

Precision timekeeping

The ubiquitous 1/f noise poses a "noise floor" to precision timekeeping.^[12] The derivation is based on.^[20]

A clock is most easily tested by comparing it with a far more accurate reference clock. During an interval of time τ, as measured by the reference clock, the clock under test advances by τy, where y is the average (relative) clock frequency over that interval.

Suppose that we have a timekeeping device (it could be anything from quartz oscillators, atomic clocks, and hourglasses^[21]). Let its readout be a real number ${\displaystyle x(t)}$ that changes with the actual time ${\displaystyle t}$. For concreteness, let us consider a quartz oscillator. In a quartz oscillator, ${\displaystyle x(t)}$ is the number of oscillations, and ${\displaystyle {\dot {x}}(t)}$ is the rate of oscillation. The rate of oscillation has a constant component ${\displaystyle {\dot {x}}_{0}}$and a fluctuating component ${\displaystyle {\dot {x}}_{f}}$, so ${\textstyle {\dot {x}}(t)={\dot {x}}_{0}+{\dot {x}}_{f}(t)}$. By selecting the right units for ${\displaystyle x}$, we can have ${\displaystyle {\dot {x}}_{0}=1}$, meaning that on average, one second of clock-time passes for every second of real-time.

The stability of the clock is measured by how many "ticks" it makes over a fixed interval. The more stable the number of ticks, the better the stability of the clock. So, define the average clock frequency over the interval ${\displaystyle [k\tau ,(k+1)\tau ]}$ as

$${\displaystyle y_{k}={\frac {1}{\tau }}\int _{k\tau }^{(k+1)\tau }{\dot {x}}(t)dt={\frac {x((k+1)\tau )-x(k\tau )}{\tau }}}$$

Note that ${\displaystyle y_{k}}$ is unitless: it is the numerical ratio between ticks of the physical clock and ticks of an ideal clock^{[note 1]}. The Allan variance of the clock frequency is half the mean square of change in average clock frequency:

$${\displaystyle \sigma ^{2}(\tau )={\frac {1}{2}}{\overline {(y_{k}-y_{k-1})^{2}}}={\frac {1}{K}}\sum _{k=1}^{K}{\frac {1}{2}}(y_{k}-y_{k-1})^{2}}$$

where ${\displaystyle K}$ is an integer large enough for the averaging to converge to a definite value.

For example, a 2013 atomic clock^[22] achieved ${\displaystyle \sigma (25000{\text{ seconds}})=1.6\times 10^{-18}}$, meaning that if the clock is used to repeatedly measure intervals of 7 hours, the standard deviation of the actually measured time would be around 40 femtoseconds.

Now we have

$${\displaystyle y_{k}-y_{k-1}=\int _{\mathbb {R} }g(k\tau -t){\dot {x}}_{f}(t)dt=(g\ast {\dot {x}}_{f})(k\tau )}$$

where ${\displaystyle g(t)={\frac {-1_{[0,\tau ]}(t)+1_{[-\tau ,0]}(t)}{\tau }}}$ is one packet of a square wave with height ${\displaystyle 1/\tau }$ and wavelength ${\displaystyle 2\tau }$. Let ${\displaystyle h(t)}$ be a packet of a square wave with height 1 and wavelength 2, then ${\displaystyle g(t)=h(t/\tau )/\tau }$, and its Fourier transform satisfies ${\displaystyle {\mathcal {F}}[g](\omega )={\mathcal {F}}[h](\tau \omega )}$.

The Allan variance is then ${\displaystyle \sigma ^{2}(\tau )={\frac {1}{2}}{\overline {(y_{k}-y_{k-1})^{2}}}={\frac {1}{2}}{\overline {(g\ast {\dot {x}}_{f})(k\tau )^{2}}}}$, and the discrete averaging can be approximated by a continuous averaging: ${\displaystyle {\frac {1}{K}}\sum _{k=1}^{K}{\frac {1}{2}}(y_{k}-y_{k-1})^{2}\approx {\frac {1}{K\tau }}\int _{0}^{K\tau }{\frac {1}{2}}(g\ast {\dot {x}}_{f})(t)^{2}dt}$, which is the total power of the signal ${\displaystyle (g\ast {\dot {x}}_{f})}$, or the integral of its power spectrum:

${\displaystyle \sigma ^{2}(1)}$ is approximately the area under the green curve. When ${\displaystyle \tau }$ increases, ${\displaystyle S[g](\omega )}$ shrinks on the x-axis, and the green curve shrinks on the x-axis but expands on the y-axis. When ${\displaystyle S[{\dot {x}}_{f}](\omega )\propto \omega ^{-\alpha }}$, the combined effect of both is that ${\displaystyle \sigma ^{2}(\tau )\propto \tau ^{\alpha -1}}$.

$${\displaystyle \sigma ^{2}(\tau )\approx \int _{0}^{\infty }S[g\ast {\dot {x}}_{f}](\omega )d\omega =\int _{0}^{\infty }S[g](\omega )\cdot S[{\dot {x}}_{f}](\omega )d\omega =\int _{0}^{\infty }S[h](\tau \omega )\cdot S[{\dot {x}}_{f}](\omega )d\omega }$$

In words, the Allan variance is approximately the power of the fluctuation after bandpass filtering at ${\displaystyle \omega \sim 1/\tau }$ with bandwidth ${\displaystyle \Delta \omega \sim 1/\tau }$.

For ${\displaystyle 1/f^{\alpha }}$ fluctuation, we have ${\displaystyle S[{\dot {x}}_{f}](\omega )=C/\omega ^{\alpha }}$ for some constant ${\displaystyle C}$, so ${\displaystyle \sigma ^{2}(\tau )\approx \tau ^{\alpha -1}\sigma ^{2}(1)\propto \tau ^{\alpha -1}}$. In particular, when the fluctuating component ${\displaystyle {\dot {x}}_{f}}$ is a 1/f noise, then ${\displaystyle \sigma ^{2}(\tau )}$ is independent of the averaging time ${\displaystyle \tau }$, meaning that the clock frequency does not become more stable by simply averaging for longer. This contrasts with a white noise fluctuation, in which case ${\displaystyle \sigma ^{2}(\tau )\propto \tau ^{-1}}$, meaning that doubling the averaging time would improve the stability of frequency by ${\displaystyle {\sqrt {2}}}$.^[12]

The cause of the noise floor is often traced to particular electronic components (such as transistors, resistors, and capacitors) within the oscillator feedback.^[23]

Humans

In brains, pink noise has been widely observed across many temporal and physical scales from ion channel gating to EEG and MEG and LFP recordings in humans.^[24] In clinical EEG, deviations from this 1/f pink noise can be used to identify epilepsy, even in the absence of a seizure, or during the interictal state.^[25] Classic models of EEG generators suggested that dendritic inputs in gray matter were principally responsible for generating the 1/f power spectrum observed in EEG/MEG signals. However, recent computational models using cable theory have shown that action potential transduction along white matter tracts in the brain also generates a 1/f spectral density. Therefore, white matter signal transduction may also contribute to pink noise measured in scalp EEG recordings.^[26]

It has also been successfully applied to the modeling of mental states in psychology,^[27] and used to explain stylistic variations in music from different cultures and historic periods.^[28] Richard F. Voss and J. Clarke claim that almost all musical melodies, when each successive note is plotted on a scale of pitches, will tend towards a pink noise spectrum.^[29] Similarly, a generally pink distribution pattern has been observed in film shot length by researcher James E. Cutting of Cornell University, in the study of 150 popular movies released from 1935 to 2005.^[30]

Pink noise has also been found to be endemic in human response. Gilden et al. (1995) found extremely pure examples of this noise in the time series formed upon iterated production of temporal and spatial intervals.^[31] Later, Gilden (1997) and Gilden (2001) found that time series formed from reaction time measurement and from iterated two-alternative forced choice also produced pink noises.^[32][33]

Electronic devices

The principal sources of pink noise in electronic devices are almost invariably the slow fluctuations of properties of the condensed-matter materials of the devices. In many cases the specific sources of the fluctuations are known. These include fluctuating configurations of defects in metals, fluctuating occupancies of traps in semiconductors, and fluctuating domain structures in magnetic materials.^[8][34] The explanation for the approximately pink spectral form turns out to be relatively trivial, usually coming from a distribution of kinetic activation energies of the fluctuating processes.^[35] Since the frequency range of the typical noise experiment (e.g., 1 Hz – 1 kHz) is low compared with typical microscopic "attempt frequencies" (e.g., 10¹⁴ Hz), the exponential factors in the Arrhenius equation for the rates are large. Relatively small spreads in the activation energies appearing in these exponents then result in large spreads of characteristic rates. In the simplest toy case, a flat distribution of activation energies gives exactly a pink spectrum, because ${\displaystyle \textstyle {\frac {d}{df}}\ln f={\frac {1}{f}}.}$

There is no known lower bound to background pink noise in electronics. Measurements made down to 10⁻⁶ Hz (taking several weeks) have not shown a ceasing of pink-noise behaviour.^[36] (Kleinpenning, de Kuijper, 1988)^[37] measured the resistance in a noisy carbon-sheet resistor, and found 1/f noise behavior over the range of ${\displaystyle [10^{-5.5}\mathrm {Hz} ,10^{4}\mathrm {Hz} ]}$, a range of 9.5 decades.

A pioneering researcher in this field was Aldert van der Ziel.^[38]

Flicker noise is commonly used for the reliability characterization of electronic devices.^[39] It is also used for gas detection in chemoresistive sensors ^[40] by dedicated measurement setups.^[41]

In gravitational wave astronomy

Noise curves for a selection of gravitational-wave detectors as a function of frequency

1/f ^α noises with α near 1 are a factor in gravitational-wave astronomy. The noise curve at very low frequencies affects pulsar timing arrays, the European Pulsar Timing Array (EPTA) and the future International Pulsar Timing Array (IPTA); at low frequencies are space-borne detectors, the formerly proposed Laser Interferometer Space Antenna (LISA) and the currently proposed evolved Laser Interferometer Space Antenna (eLISA), and at high frequencies are ground-based detectors, the initial Laser Interferometer Gravitational-Wave Observatory (LIGO) and its advanced configuration (aLIGO). The characteristic strain of potential astrophysical sources are also shown. To be detectable the characteristic strain of a signal must be above the noise curve.^[42]

Climate dynamics

Pink noise on timescales of decades has been found in climate proxy data, which may indicate amplification and coupling of processes in the climate system.^[43][44]

Diffusion processes

Many time-dependent stochastic processes are known to exhibit 1/f ^α noises with α between 0 and 2. In particular Brownian motion has a power spectral density that equals 4D/f ²,^[45] where D is the diffusion coefficient. This type of spectrum is sometimes referred to as Brownian noise. Interestingly, the analysis of individual Brownian motion trajectories also show 1/f ² spectrum, albeit with random amplitudes.^[46] Fractional Brownian motion with Hurst exponent H also show 1/f ^α power spectral density with α=2H+1 for subdiffusive processes (H<0.5) and α=2 for superdiffusive processes (0.5<H<1).^[47]