Chapter 5

The colors of noise

Point a spectrum analyzer at almost anything in a lab — a photodiode, a resistor, a laser's frequency, a magnetometer left running overnight — and the noise floor you see is, over decades of frequency, a straight line on a log–log plot: a power law $S(f) \propto f^{-\alpha}$. The single exponent $\alpha$ is a personality. It fixes what the noise looks like in time, whether averaging longer helps, and whether "the RMS" even means anything. Learn to read slopes and you can diagnose an experiment from one plot.

One exponent, a whole personality

Chapter 4 gave us the power spectral density: a one-sided function $S(f)$, in units of (signal)² per Hz, whose integral over frequency is the variance of the signal. Real measured spectra are rarely exotic. Overwhelmingly often they are described, over the band you can see, by

$$ S(f) \;=\; \frac{h_\alpha}{f^{\alpha}}, $$

with $h_\alpha$ a constant setting the level and $\alpha$ setting the slope. By an old audio-engineering analogy with light, the family members have colors: $\alpha=0$ is white (equal power at all frequencies, like white light), $\alpha=1$ is pink or flicker noise (tilted toward the red/low-frequency end), and $\alpha=2$ is brown or red noise (named less for the color than for Brownian motion — it is a random walk).

Before any formulas, get a feel for what $\alpha$ does. The machine below synthesizes noise with an exact $f^{-\alpha}$ spectrum. Sweep the slider slowly from 0 to 2 and watch the time trace change character: from memoryless fuzz, to wandering-but-tethered, to a drunkard who never comes home. The underlying random numbers are kept fixed while you drag, so the change of character you see is purely the exponent, not sample-to-sample luck.

Notice the two extremes. At $\alpha=0$ every sample is a fresh start: zoom in anywhere and the trace has no memory of what came before. At $\alpha=2$ the trace is nothing but memory: each value is the previous one plus a small kick, so excursions build up and the trace spends long stretches far from zero. Pink noise, in between, is the strangest of the three — it wanders on every timescale at once, a point we will make precise below.

White noise ($\alpha = 0$): no memory

A flat PSD means equal power in every 1-Hz slice of frequency. In the time domain the correlation function is (ideally) a delta function: knowing the value now tells you nothing about the value one sample later. White noise is what chapter 3's independent-samples process looks like through a spectrum analyzer.

Why is white noise everywhere? Because it is what the central limit theorem does in time: any signal that is the sum of very many fast, independent microscopic kicks — thermal agitation of electrons, discrete charges crossing a junction, photons arriving at a photodiode — looks, on any timescale much longer than a single kick, like Gaussian noise with no memory. Three workhorse examples, with numbers worth memorizing:

Check the first one yourself: $4 k_B T R = 4 \times (1.38\times10^{-23}\, \mathrm{J/K}) \times 300\,\mathrm{K} \times 50\,\Omega = 8.3 \times 10^{-19}\ \mathrm{V^2/Hz}$, whose square root is $0.91\ \mathrm{nV}/ \sqrt{\mathrm{Hz}}$. Every low-noise amplifier data sheet is implicitly competing against this number.

There is a neater way to state the thermal result — one that makes the $R$ disappear. $4 k_B T R$ is the resistor's open-circuit voltage noise, and it grows with $R$; but a large $R$ is also a feeble power source. Ask instead for the maximum power the resistor can deliver: connect it to a matched load (another resistor $R$), which sees half the voltage and so receives $V^2/(4R) = k_B T$ watts per hertz. The resistance cancels:

$$ N_0 \;=\; k_B T \ \text{ watts per Hz, independent of } R. $$

Every warm object drives a matched circuit with the same universal noise density. Note what kind of density this is: real watts per hertz — power you could dissipate in a load — not the $\mathrm{V^2/Hz}$ of a recorded waveform. That is exactly why $R$ could cancel. There is also a cute equipartition shortcut to the same number: each degree of freedom holds $\tfrac12 k_B T$ of energy, and a circuit offers about two of them per second per hertz of bandwidth (one per quadrature), so thermal equilibrium hands a matched load $k_B T$ of energy per second — $k_B T$ watts — per hertz.

Shot noise: light arrives in lumps

The middle rows of the table deserve more than a row each, because they are the noise floor of essentially every optical experiment. Current and light are not continuous fluids: they arrive as discrete, independent lumps — electrons crossing a junction, photons landing on a photodiode. Independent arrivals at a mean rate $N$ per second follow Poisson statistics: the counting law for events that are independent and memoryless — each arrival neither knows nor cares when the last one came (the telegraph fluctuators below hop at the same kind of random times). Poisson statistics have one defining property: the variance of a count equals its mean. Count for one second and you get $N \pm \sqrt{N}$. Because each arrival knows nothing about the others, the fluctuation power is spread evenly over all frequencies: shot noise is white. For a current $I$ of charges $e$ the one-sided PSD is $S_I = 2eI$; for a beam of power $P$ made of photons of energy $h\nu$, the fractional power fluctuation $\delta P/P$ has the flat PSD $2h\nu/P$, i.e. an ASD of $\sqrt{2h\nu/P}$ per root hertz.

Make that concrete with a lab workhorse: $P = 1$ mW at 1064 nm. Each photon carries $h\nu = 1.87\times10^{-19}$ J, so the beam is a stream of $R = P/h\nu = 5.4\times10^{15}$ photons per second. Count them for one second: the fractional fluctuation is $1/\sqrt{R} = 1.4\times10^{-8}$ — fourteen parts per billion, for free, from a beam that fits in your hand. As a spectral density, the shot-limited relative intensity noise (RIN) is $2h\nu/P = 3.7\times10^{-16}\ \mathrm{Hz^{-1}}$, which laser engineers quote logarithmically as $-154$ dB/Hz. The photocurrent picture gives the same answer: a photodiode with quantum efficiency $\eta \approx 0.8$ turns the beam into $I = \eta e P/h\nu \approx 0.69$ mA, and $S_I = 2eI$ implies a relative current noise $S_I/I^2 = 2e/I = 2h\nu/(\eta P)$ — the shot noise of the detected photon stream $\eta R$. Shot noise looks identical whether you count photons or electrons; the only cost of imperfect detection is the $1/\eta$ from counting fewer arrivals.

Note the scaling that runs the economics of precision optics: the signal grows like $P$ but the shot-noise ASD only like $\sqrt{P}$, so the signal-to-noise ratio of a shot-limited measurement grows as $\sqrt{P}$. That is why LIGO circulates hundreds of kilowatts in its arms (chapter 9), and why "just turn up the laser" is always the first move — and eventually a losing one: technical intensity noise is fractional, so more power does not dilute it, and at high enough power radiation-pressure back-action starts shaking the very mirrors you are trying to read out.

A warning before you budget for this floor: a real laser only reaches it at high Fourier frequencies $f$ — the axis of the intensity-noise PSD $S_{\delta P/P}(f)$, i.e. how fast the detected power is wiggling. (Nothing to do with the optical frequency $\nu$: a slow, 1 kHz flicker of a 282 THz beam lives at $f = 1$ kHz on this axis.) Below roughly 1 MHz the intensity spectrum is ruled by technical noise — pump fluctuations, acoustics, mode competition — typically 10–40 dB above shot at kilohertz frequencies, and solid-state lasers usually add a relaxation-oscillation peak somewhere in the 0.1–1 MHz region. The quantum floor is where the spectrum finally flattens, often only at MHz frequencies. This is the moral that chapter 8 elevates to a rule — put your signal at a frequency where the floor is quantum, not technical. And the floor itself is not quite the end: squeezed light is the only way below it, and LIGO has run with squeezing since 2019 (chapter 9).

Flicker noise ($\alpha = 1$): the same power in every decade

Pink noise has one defining, slightly eerie property. Integrate $S(f) = h/f$ across one decade of frequency, starting anywhere:

$$ \int_{f_1}^{10 f_1} \frac{h}{f}\, df \;=\; h \ln 10 \;\approx\; 2.3\,h \qquad \text{independent of } f_1 . $$

The decade from 1 mHz to 10 mHz carries exactly as much power as the decade from 1 kHz to 10 kHz. There is no characteristic frequency, hence no characteristic time: flicker noise is scale-free. Zoom out by any factor, renormalize the vertical scale, and the trace is statistically indistinguishable from what you started with. Try it:

Flicker noise is embarrassingly universal: the low-frequency end of every transistor and resistor, the frequency of every quartz and laser oscillator, the sensitivity drift of SQUIDs and photodetectors, heartbeat intervals, and the volatility of the stock market all show $1/f$-like spectra over many decades. The honest statement is that there is no single universal microscopic origin. What there is instead is a generic recipe: a superposition of many independent relaxation processes — two-state fluctuators, charge traps, defects — each with its own rate, with the rates spread roughly uniformly on a log scale, produces a $1/f$ spectrum over the corresponding decades. In the classic example — a MOSFET — the fluctuators are charge traps in the gate oxide; the trapping time depends exponentially on the trap's depth, so a flat distribution of depths gives the required flat distribution of log-rates, and $1/f$ noise over many decades follows. Different devices, different fluctuators, same arithmetic — which is why the same slope appears everywhere.

This recipe is worth seeing, because its ingredient is the simplest random process after white noise: random telegraph noise, a signal that just hops between two values at Poisson-random times (a "jump process" — the same family as the discrete arrivals behind shot noise). A single fluctuator with total switching rate $\gamma$ forgets its state exponentially, $R(\tau) = a^2 e^{-\gamma|\tau|}$, so its PSD is a Lorentzian:

$$ S(f) \;=\; \frac{4 a^2 \gamma}{\gamma^2 + (2\pi f)^2} \;\;=\;\; \begin{cases} \text{flat (white)}, & f \ll \gamma/2\pi \\[2pt] \propto 1/f^2, & f \gg \gamma/2\pi . \end{cases} $$

No single fluctuator is ever $1/f$ — it is white below its corner and $1/f^2$ above. But now take many independent fluctuators and simply add their time traces, $x(t) = x_1(t) + x_2(t) + \cdots$. Independent noises add in power (chapter 3's rule), so the summed PSD is just the sum of the individual Lorentzians — and if the switching rates are spread evenly in $\log\gamma$, each Lorentzian hands over to the next as it rolls off, and the crossovers pile into a $1/f$ staircase. Watch it happen:

Random walk ($\alpha = 2$): white noise, integrated

Brown noise is not a new microscopic mechanism at all: it is what you get when something integrates white noise. If a velocity $v(t)$ is white, the position $x(t) = \int_0^t v\,dt'$ is a random walk. In chapter 3 we found its defining feature directly: the variance grows linearly with time, $\mathrm{Var}[x(t)] \propto t$, without bound. Such a process is non-stationary — it does not, strictly speaking, have a PSD at all, but any finite-length measurement of it yields a clean $1/f^2$ spectrum over the measured band, and that is how it shows up in practice. Each factor of $1/f$ in the PSD is one integration in time (a derivative, conversely, multiplies the PSD by $(2\pi f)^2$).

In the lab, look for random walks wherever an integral hides: the position of a mirror driven by white force noise, the accumulated angle of a gyroscope with white rate noise, and — the case that owns chapter 6 — the phase of an oscillator whose frequency noise is white. Frequency is the derivative of phase, so white frequency noise means random-walk phase. Keep that sentence; it is the skeleton key to phase noise.

Does averaging help? The punchline

Here is the question that makes $\alpha$ matter in practice. You measure a noisy quantity for a time $T$. Common sense says: measure longer, know it better. Does the spread of your record actually settle down as $T$ grows? The variance of what you observe is the integral of the PSD over the frequencies your record can see — roughly $1/T$ up to your sampling bandwidth:

$$ \sigma^2(T) \;\approx\; \int_{1/T}^{f_{\mathrm{hi}}} \frac{h_\alpha}{f^{\alpha}}\, df . $$

For white noise the integral converges at the low end: watching longer adds essentially nothing, and the sample std settles to a constant. For $\alpha \ge 1$ the integral diverges as the lower limit $1/T \to 0$: every doubling of the observation time lets in new, ever-lower frequencies carrying real power. For pink noise the growth is logarithmic in the variance — the std creeps up like $\sqrt{\ln T}$ — and for a random walk $\sigma^2 \propto T$, so the std grows as $\sqrt{T}$. The demo below measures exactly this: one long record of each color, and the sample std of its first $T$ samples as $T$ grows (averaged over 30 independent realizations so the curves are smooth).

Reading slopes like a pro

On log–log axes, $S(f) = h f^{-\alpha}$ is the straight line $\log S = \log h - \alpha \log f$: the slope is exactly $-\alpha$. Spectrum analyzers speak decibels, where a factor of 10 in power is 10 dB and a factor of 2 in frequency is an octave, so each unit of $\alpha$ is worth 10 dB per decade, or 3 dB per octave:

Real spectra are sums of these: a laser's frequency noise might read $1/f$ below 1 Hz (flicker), flat from 1 Hz to 100 kHz (white), with spikes on top (the 50 Hz comb, fans, your vacuum pump). Reading off the corner frequencies where one slope hands over to the next is 90% of noise diagnosis. Sweep the noise machine again with this table in mind — you can now name every value of the slider.

How the machine works: shaping a spectrum

Every colored trace on this page is made by one trick, straight from Noise.powerLaw in this site's library: take white noise, Fourier transform it, tilt the spectrum, transform back.

// Noise.powerLaw(n, alpha, rand) — the essential lines
const re = whiteGaussian(n), im = zeros(n);
FFT.fft(re, im);                    // -> flat spectrum, random phases
re[0] = 0; im[0] = 0;               // drop DC (no mean for alpha >= 1)
for (let k = 1; k <= n / 2; k++) {
  const g = Math.pow(k, -alpha / 2);   // amplitude gain  f^(-alpha/2)
  re[k] *= g;  im[k] *= g;             // (mirror bins too, to keep x real)
}
FFT.ifft(re, im);                   // back to time: PSD ∝ f^(-alpha)

Why is the exponent $\alpha/2$ and not $\alpha$? Because the loop multiplies the amplitude of each Fourier bin, and the PSD is the amplitude squared. An amplitude filter $|H(f)| = f^{-\alpha/2}$ shapes the power as $|H(f)|^2 = f^{-\alpha}$ — the same halving of exponents as in the ASD-vs-PSD callout above, now used as a tool instead of suffered as a trap.

Noise in two dimensions: pictures instead of traces

Nothing in this chapter cares that the axis was time. A camera's dark frame, a mirror's surface roughness, terrain, the temperature field across a vacuum chamber — these are noisy fields $x(\mathbf r)$, and the same machinery applies with time traded for position: a two-dimensional Fourier transform, a PSD $S(\mathbf k)$ over spatial frequency (cycles per image), and — for isotropic noise — a radially averaged spectrum $S(k)$ that is a curve you can read slopes off exactly like every other plot on this page.

Power laws are just as ubiquitous here, and your eye already knows them. Sweep the exponent below: $\alpha = 0$ is television static (every pixel independent — all structure at the finest scale); by $\alpha \approx 2$ the field is scale-free, structure at every size, and looks like clouds or terrain seen from above (this exact trick generates procedural landscapes in computer graphics); by $\alpha \approx 4$ only the largest blobs survive and the field is smooth. The eye turns out to be a decent spectrum analyzer.

The self-similarity section's punchline also survives the trip to 2-D: at $\alpha \approx 2$ (the two-dimensional analogue of "equal power per octave") a zoomed-in patch of the field is statistically indistinguishable from the whole — which is precisely why clouds and coastlines have no obvious size. And the diagnostic habit transfers whole: if a camera's dark frames show a radial PSD rising toward small $k$, the "noise" has structure — fixed-pattern nonuniformity, thermal gradients, interference fringes — and averaging frames will not remove all of it. Chapter 8 shows the frame-averaging version of that story.

Exercises