--- title: An Interesting Fourier Transform | daily.dev description: Power law functions have a remarkable property: their Fourier transforms are also power laws, following the relation t^α ↔ ω^-(α+1). The transform includes a... canonical: https://daily.dev/posts/an-interesting-fourier-transform-m9qkbpfcx twitter:card: summary_large_image twitter:site: @dailydotdev og:type: website og:site_name: daily.dev og:title: An Interesting Fourier Transform | daily.dev og:description: Power law functions have a remarkable property: their Fourier transforms are also power laws, following the relation t^α ↔ ω^-(α+1). The transform includes a... og:url: https://daily.dev/posts/an-interesting-fourier-transform-m9qkbpfcx og:image: https://api.daily.dev/og/posts/m9QKBPFCx.png og:image:alt: An Interesting Fourier Transform og:image:width: 1200 og:image:height: 630 og:locale: en --- > ## Documentation Index > Fetch the complete documentation index at: https://daily.dev/llms.txt > Use this file to discover all available pages before exploring further. # An Interesting Fourier Transform **[Hacker News](https://daily.dev/sources/hn)** · 17 min read · 0 upvotes · 0 comments ## Summary Power law functions have a remarkable property: their Fourier transforms are also power laws, following the relation t^α ↔ ω^-(α+1). The transform includes a unit step function, a Gamma function scaling factor, and a phase term. The special case α = -0.5 is particularly notable: both the time domain (t^-0.5) and frequency domain (ω^-0.5) have the same power law decay, making 1/f noise effectively its own Fourier transform in amplitude terms. This mathematical self-similarity is compared to the Gaussian curve, which is also its own Fourier transform and appears ubiquitously due to the Central Limit Theorem. Despite this elegant mathematical framing, the physical origin of 1/f noise — observed in electronics, traffic, music, and DNA — remains unexplained after 80+ years, partly because the phase of 1/f noise is unknown, meaning the corresponding time-domain waveform may not actually be t^-0.5. ## Full article daily.dev links to this article rather than hosting it. Read it at the original source: ## Questions this post answers ### What is the Fourier transform relationship for a power law function t^α? A power law in the time domain transforms to a power law in the frequency domain: t^α corresponds to ω^-(α+1). This relation is complicated by three factors: the time domain is one-sided due to the unit step function u(t), the frequency domain magnitude is accompanied by a nontrivial phase term, and there is a scaling factor given by the Gamma function Γ(α+1). _Anyone reasoning through Fourier transform edge cases can find deeper signal processing discussions curated on daily.dev._ ### Why is 1/f noise (flicker noise) considered mysterious and unexplained? 1/f noise, observed for over 80 years in electronics, traffic density, music loudness, and DNA coding, lacks a clear physical explanation despite its widespread occurrence. Its amplitude spectrum follows ω^-1/2, matching the special case α=-0.5 where time and frequency domain power laws are equal, but no scenario tying this to a physical mechanism has been confirmed, and the phase of the underlying signal remains unknown. _Developers puzzling over unresolved noise and signal mysteries can track related DSP discussions on daily.dev._ ### What happens to the Fourier transform of t^α as α approaches -1? As α approaches -1, the time domain function u(t)t^-α approaches a delta function δ(t), since the frequency domain magnitude flattens and the phase approaches zero, which is the transform pair of a delta function. Exactly at α=-1 the behavior is undefined because the Gamma function scaling factor Γ(α+1) has no defined value at that point. _Engineers working through Fourier transform singularities can find similar deep-dive math content on daily.dev._ ## Community take How the wider developer community reacted, aggregated from 1 discussion and 21 comments across hackernews (as of 2026-09-14). **TL;DR:** Commenters find the mathematical elegance of 1/f noise genuinely interesting and engage deeply with its physical origins, practical workarounds, and perceptual properties, while noting that the 'unsolved mystery' framing is somewhat overstated since partial explanations do exist. **Sentiment:** 55% positive · 35% mixed · 10% skeptical **The case for** - The self-similarity of 1/f noise as its own Fourier transform is seen as a genuinely elegant and surprising mathematical property. - Practical engineering workarounds like chopper amps and correlated double sampling show the phenomenon is manageable even without a full theoretical explanation. - Pink noise's perceptual flatness (equal energy per octave) is highlighted as a useful and underappreciated property in audio contexts. **The pushback** - The claim that 1/f noise is completely unexplained is contested — partial physical explanations (e.g., trapped charges, tunneling) exist in specific domains. - Generating pink noise sample-by-sample is surprisingly non-trivial algorithmically, which commenters find frustrating given how ubiquitous the phenomenon is. - The self-Fourier-transform property is noted to be less special than the post implies, as many such pairs exist and there is published literature on constructing them. **By community** - hackernews (mixed): Engaged and knowledgeable discussion that appreciates the mathematical framing but pushes back on the 'unexplained mystery' narrative, offering domain-specific partial explanations and practical engineering context. **Hottest debate:** Whether 1/f noise is truly 'unexplained' or merely lacks a single unified physical model despite having domain-specific partial explanations. **Open questions** - Is there a simple, direct algorithm for generating 1/f noise sample-by-sample, given how ubiquitous it is? - Is there a deeper connection between 1/f noise and entropy or thermodynamic principles? - Does the self-Fourier-transform property of 1/f noise actually provide a meaningful clue to its physical origin, or is that property too common to be informative? **Highlights** > It's not completely unexplained. Roughly speaking you can get that power spectrum in the limit when you are adding up many different events where the magnitude of the event is inversely proportional to its likelihood (and in practice, there is a limit to the magnitude of the events that will cause it to level off at some point, but for some processes this is not measurable even over decades). The main mystery in most cases is what exactly is the physical process that is causing it. For some electronics it looks like it is due to trapped charges sometimes tunneling around, but it doesn't explain every case of it in electronics let alone everything else. Convective thermal effects can also be a good candidate in a lot of systems, since turbulence also has 1/f noise properties. (Also, the low frequency range on a DFT can look weird for reasons other than noise: it'll also tend to rise up if there's any longer-term structure to the signal as well, so you need to be careful interpreting them blindly if you're trying to measure noise) > — [rcxdude on hackernews](https://news.ycombinator.com/item?id=49240825) > When I was learning DSP, I was surprised by the fact that generating pink (1/f) noise, sample by sample, is not mathematically easy at all. One practical approach is passing white noise (every sample independent random) through a "pinking filter", which is a sum of a bunch of lowpass filters at different frequencies, so that the sum of their cutoff "knees" approximates the frequency curve of pink noise. Another approach is generating in chunks using Fourier transform from a desired shape. Given the ubiquity of 1/f noise, you'd think there would be a simpler and more direct algorithm, but no. > — [cousin\_it on hackernews](https://news.ycombinator.com/item?id=49243475) > The piece ends with the observation that maybe the fact that 1/f noise is its own Fourier transform is a clue. Turns out this property is not unusual. There are many such pairs - there’s a reasonably well-known journal paper with a construction technique. > — [mturmon on hackernews · 1 comments](https://news.ycombinator.com/item?id=49240180) > 1/f noise does not truly kill averaging, observe how for increasing f, the noise spectrum looks "white" locally, but with a decreasing noise power for ever higher frequencies. the 1/f noise at significant power levels is restricted to the lower and lower frequencies, effectively a slowly varying reference ("0") voltage of the amplifier, measuring ADC ground every other sample effectively recalibrates the offset voltage of the amplifier, think of "correlated double sampling". Effectively measuring in sequence 0V, signal, 0V, signal, ... moves the signal of interest to a higher frequency band, where the 1/f noise is more tame. When a cliff blocks the way of a vehicle, we don't say "cliffs kill vehicle travel", insteas we just drive around it... > — [DoctorOetker on hackernews](https://news.ycombinator.com/item?id=49244967) > I think 1/f is the uniform measure for scaling rather than translations. It's to multiplication what the standard uniform measure is for addition. If you want to be boring you could call it the uniform distribution for log frequency. > — [shiandow on hackernews](https://news.ycombinator.com/item?id=49242923) **Source threads** - [hackernews](https://news.ycombinator.com/item?id=49205082) · 98 points · 21 comments ## Similar posts on daily.dev - [Notes on the Fourier Transform](https://daily.dev/posts/notes-on-the-fourier-transform-go3dssqrp) · Eli Bendersky · 0 upvotes · 0 comments - [Fourier Transforms](https://daily.dev/posts/fourier-transforms-93mxdwqm5) · Hacker News · 0 upvotes · 0 comments - [How the Fourier Transform Converts Sound Into Frequencies](https://daily.dev/posts/how-the-fourier-transform-converts-sound-into-frequencies-9uyp5rdfw) · Towards Data Science · 0 upvotes · 0 comments - [Notes on discrete-time Fourier series and transform](https://daily.dev/posts/notes-on-discrete-time-fourier-series-and-transform-zeurckvji) · Eli Bendersky · 0 upvotes · 0 comments --- Tags: [#data-science](https://daily.dev/tags/data-science), [#signal-processing](https://daily.dev/tags/signal-processing) [View this post on daily.dev](https://daily.dev/posts/an-interesting-fourier-transform-m9qkbpfcx) ```json {"@context":"https://schema.org","@graph":[{"@type":"Organization","@id":"https://daily.dev/#organization","name":"daily.dev","url":"https://daily.dev","logo":{"@type":"ImageObject","url":"https://daily.dev/apple-touch-icon.png","width":180,"height":180},"sameAs":["https://twitter.com/dailydotdev","https://github.com/dailydotdev","https://www.linkedin.com/company/daily-dev-ltd"]},{"@type":"WebSite","@id":"https://daily.dev/#website","url":"https://daily.dev","name":"daily.dev","publisher":{"@id":"https://daily.dev/#organization"},"potentialAction":{"@type":"SearchAction","target":{"@type":"EntryPoint","urlTemplate":"https://daily.dev/search?q={search_term_string}"},"query-input":"required name=search_term_string"}}]} {"@context":"https://schema.org","@type":"TechArticle","headline":"An Interesting Fourier Transform","url":"https://daily.dev/posts/an-interesting-fourier-transform-m9qkbpfcx","mainEntityOfPage":{"@type":"WebPage","@id":"https://daily.dev/posts/an-interesting-fourier-transform-m9qkbpfcx"},"datePublished":"2026-08-10T10:33:54.930Z","dateModified":"2026-09-14T07:30:18.485Z","description":"Power law functions have a remarkable property: their Fourier transforms are also power laws, following the relation t^α ↔ ω^-(α+1). 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