HN Debrief

An Interesting Fourier Transform – 1/f Noise (2007)

  • Signal Processing
  • Hardware
  • Mathematics

The post is an old DSP essay about 1/f noise, better known in audio as pink noise. It starts from a neat mathematical observation. As you shape spectra toward a 1/f form, the time-domain and frequency-domain pictures start to resemble each other in a way that makes 1/f look almost like its own Fourier transform. People liked the curiosity, but they did not buy it as an explanation for why pink noise appears all over physics and engineering.

If low-frequency drift is showing up in sensors, analog front ends, or audio analysis, treat it as an engineering constraint, not a curiosity. Design around it with modulation, calibration, and measurement methods that move your signal out of the bad band instead of assuming longer averaging will save you.

Discussion mood

Curious and appreciative, with a strong practical streak. People enjoyed the mathematical weirdness, but most of the energy went into grounding it in engineering reality: 1/f noise is real, common, poorly unified in theory, easy to misinterpret in spectra, and something you work around rather than explain away.

Key insights

  1. 01

    Offset cancellation beats low-frequency drift

    Treating 1/f noise as a slowly wandering baseline makes the mitigation strategy obvious. Techniques like correlated double sampling and chopper amplifiers repeatedly re-reference the measurement or shift it upward in frequency, where flicker noise is weaker and starts to look more like ordinary white noise.

    If your system is limited by low-frequency noise, change the measurement architecture before chasing more samples or better averaging. Add modulation, chopping, or explicit baseline reads in the analog front end and firmware.

      Attribution:
    • DoctorOetker #1
    • tiazumdove #1
  2. 02

    The hard part is identifying the source

    The unresolved piece is not whether 1/f exists. It is which physical process is actually producing it in a given system. Trapped charges may fit some semiconductor devices, convection and turbulence can fit others, and outside controlled lab setups the measurement itself becomes so fragile over long time spans that the plotted 1/f region is often part data and part modeling judgment.

    Do not stop at seeing a sloped low-frequency spectrum and labeling it pink noise. Audit drift sources, environment, and fitting assumptions before you pick a fix or write a root-cause memo.

      Attribution:
    • rcxdude #1
    • analog31 #1
  3. 03

    Pink noise generation is still mostly filtering

    Despite how famous pink noise is, there is no magical one-line sample generator that avoids shaping white noise. In practice you either filter white noise with a bank of lowpass sections or an FIR convolution, or you synthesize chunks in the Fourier domain and transform back. That is mundane DSP, but it is still the standard route.

    If you need synthetic 1/f noise for testing, plan for approximation quality, latency, and computational cost. Pick FIR or IIR style filtering for streaming systems and FFT-based synthesis for offline generation.

      Attribution:
    • cousin_it #1
    • stagger87 #1
  4. 04

    Stable distributions are a better mathematical lens

    Looking at 1/f through Lévy stable distributions gives a more grounded route than the article's Fourier-transform intuition. The key point is that Gaussian behavior is only the finite-variance corner of a larger family. Once heavy tails and effectively unbounded variance enter, self-similar behavior under summation and characteristic functions becomes much less surprising.

    If you are modeling systems with bursts, jumps, or heavy-tailed behavior, do not default to Gaussian assumptions. Check whether a stable-law model fits the data before building estimators or forecasts around variance that may not exist.

      Attribution:
    • abetusk #1

Against the grain

  1. 01

    Self-Fourier functions are not rare clues

    Making 1/f look special because it echoes itself under the Fourier transform oversells the math. There are many constructions with similar self-referential transform properties, so that symmetry is a neat artifact, not strong evidence for a physical explanation.

    Do not infer mechanism from transform symmetry alone. Treat this kind of mathematical elegance as a prompt for better models, not as a model by itself.

      Attribution:
    • mturmon #1
  2. 02

    Averaging is not actually doomed

    Saying 1/f noise 'kills averaging' is too blunt. The real limitation is naive averaging at the original baseband. Once you reframe the problem as low-frequency offset drift, you can move the signal to a cleaner band and recover the benefits of averaging there.

    When noise seems immune to longer measurement time, revisit your sampling scheme and signal placement. The failure mode may be baseband drift, not an absolute floor on achievable precision.

      Attribution:
    • threatripper #1
    • DoctorOetker #1

In plain english

1/f noise ↩
Noise whose power increases as frequency decreases, roughly in inverse proportion to frequency, so slow fluctuations dominate.
baseband ↩
The original low-frequency range where a signal lives before any modulation shifts it elsewhere.
correlated double sampling ↩
A measurement technique that samples a reference and a signal in close succession so slow offset drift and low-frequency noise cancel out.
DSP ↩
Digital signal processing, the field of analyzing and modifying signals such as audio, radio, and sensor data using digital computation.
FFT ↩
Fast Fourier transform, an efficient algorithm for computing the discrete Fourier transform.
Fourier transform ↩
A mathematical operation that rewrites a signal in terms of its frequency content instead of its values over time.
Gaussian ↩
The normal distribution, the familiar bell-shaped probability distribution that arises from many finite-variance random processes.
pink noise ↩
Another name for 1/f noise, often used in audio because it has equal power per octave and sounds more balanced than white noise.
white noise ↩
Noise with equal power per unit bandwidth across frequency, so each hertz contributes the same amount of energy.

Reference links

Background references

Probability and heavy-tail context