An interactive crash course for physicists
Making Sense of Noise
Power spectral densities, phase noise, Allan deviations, sensitivity in Hz/√Hz — the working language of every precision-measurement lab, and one of the most weirdly hard-to-learn corners of experimental physics. This course builds it from first principles: sines and cosines first, then modulation, then randomness, with live plots you can twiddle at every step.
Written for the smart experimental physicist who has stared at a spectrum analyzer, quoted a sensitivity, or read an Allan-deviation plot — and quietly felt that the underlying mathematics was a bit mysterious. No measure theory, no formal stochastic calculus. Every idea is introduced with a picture you can manipulate, every formula is checked against a simulation you can read the code of, and every chapter ends with exercises (with hidden solutions).
How to read this
Each chapter is self-contained but they build on each other; if you are new
to the subject, go in order. The demos are the point — drag every
slider and watch what moves. Hover over any plot to read off
values. All simulations run live in your browser; the code snippets shown
are essentially the code that generates the plots.
- CHAPTER 1 The pure tone Amplitude, frequency, phase. Phasors, the rotating-arrow picture, and what an ideal oscillator looks like in the time and frequency domains.
- CHAPTER 2 Wiggling the knobs: modulation Amplitude, phase and frequency modulation. Sidebands, the small-modulation limit, and why AM and PM look different on a spectrum analyzer.
- CHAPTER 3 From signals to noise: random processes Ensembles, averages, correlation functions. Stationarity and drift — and why "the variance" is sometimes not even defined.
- CHAPTER 4 The power spectral density The single most important tool in the book. Where PSDs come from, their units, one-sided vs two-sided, and how the area under the curve is the RMS noise you measure.
- CHAPTER 5 The colors of noise White, flicker (1/f), random walk (1/f²). A live noise machine: turn the spectral-slope knob and watch the time trace change character.
- CHAPTER 6 Noisy oscillators: amplitude, phase & frequency noise Sa(f), Sφ(f), Sν(f), ℒ(f) in dBc/Hz — how they relate, how to convert, and what each looks like on a spectrum analyzer.
- CHAPTER 7 Allan deviation and clock stability Why the ordinary variance fails for clocks, the two-sample fix, and reading the σy(τ) slopes: τ−1/2, flicker floor, random-walk rise, drift.
- CHAPTER 8 Sensitivity: what Hz/√Hz means Noise floors, integration time, and signal-to-noise. Why sensor sensitivities carry that strange per-root-hertz unit, with a magnetometer worked end to end.
- CHAPTER 9 Case study: LIGO and noise budgets Strain sensitivity curves, transfer functions, control loops, and how a real experiment stacks its noise sources into one plot.
- CHAPTER 10 Case study: phase-locked loops How a loop steals the best of two oscillators: block diagrams you can click, noise shaping by loop bandwidth, synthesizers and the 20 log N tax — and why an atomic clock is a lock loop with atoms as the reference.
What you will be able to do afterwards
- Read a power spectral density plot and know, at a glance, where the RMS noise actually lives.
- Convert between Sφ(f), Sν(f), ℒ(f) in dBc/Hz, and fractional-frequency PSDs without looking anything up.
- Read a lock-loop block diagram — a synthesizer, a laser lock, an atomic clock — component by component, and predict how the loop reshapes each noise source.
- Look at an Allan-deviation plot of a clock and identify white frequency noise, the flicker floor, random walk and drift from the slopes alone.
- Say precisely what a sensitivity of, e.g., 5 pT/√Hz or 1 Hz/√Hz means, and predict how measurement uncertainty improves (or stops improving) with integration time.
- Understand a LIGO-style sensitivity curve and a noise budget.
Lab note
Everything here runs offline from local files, except the equation
rendering (KaTeX) which loads from a CDN the first time.
References & further reading
The course follows the spirit (and often the examples) of a handful of sources, roughly one per layer of the subject:
- W. P. Robins, Phase Noise in Signal Sources: Theory and Applications, Peter Peregrinus / IET (1982) — the most first-principles treatment in print: sines, modulation and thermal noise first, then oscillators, multiplier chains, phase-locked loops and spurs. Chapters 2, 5–8 shaped chapters 2, 6 and 10 here.
- K. Jacobs, Stochastic Processes for Physicists: Understanding Noisy Systems, Cambridge University Press (2010) — the gentlest route into the trajectory-level view (Wiener increments, SDEs, Ornstein–Uhlenbeck, jump/telegraph processes) behind chapters 3 and 5, and how to simulate noisy systems correctly.
- E. Rubiola, Phase Noise and Frequency Stability in Oscillators, Cambridge University Press (2009) — the modern standard on ℒ(f), Sφ and oscillator noise mechanisms.
- W. J. Riley, Handbook of Frequency Stability Analysis, NIST Special Publication 1065 (2008) — everything Allan-family, free from tf.nist.gov.
- D. W. Allan, "Statistics of atomic frequency standards", Proc. IEEE 54, 221 (1966), and J. Rutman, "Characterization of phase and frequency instabilities in precision frequency sources", Proc. IEEE 66, 1048 (1978) — the original and the classic review.
- LIGO–Virgo Collaboration, "A guide to LIGO–Virgo detector noise and extraction of transient gravitational-wave signals", Class. Quantum Grav. 37, 055002 (2020), arXiv:1908.11170 — chapter 9's noise budgets, from the source.
- G. Heinzel, A. Rüdiger and R. Schilling, "Spectrum and spectral density estimation by the discrete Fourier transform (DFT), including a comprehensive list of window functions" (2002) — the free classic on PSD estimation practicalities (windows, ENBW, normalization).
- J. Vig, "Quartz crystal resonators and oscillators" tutorial — source of chapter 7's accuracy-versus-stability picture.