How the cochlea computes (2024)

man I need to finally learn what a Fourier transform is

3Blue1Brown has a really good explanation here: https://www.youtube.com/watch?v=spUNpyF58BY

It gave me a much better intuition than my math course.

Hahaha, I was working on learning these in second year uni… which was also exactly when I switched from an electrical engineering focussed degree to a software one!

Perhaps finally I should learn too…

It's an absolutely brilliant bit of maths that breaks a complex waveform into the individual components. Kind of like taking an orchestral song and then working out each individual instrument's contribution. Learning about this left me honestly aghast and in shock that it's not only possible but that someone (Joseph Fourier) figured it out and then shared it with the world.

This video does a great job explaining what it is and how it works to the layman. 3blue1brown - https://www.youtube.com/watch?v=spUNpyF58BY

the very simplest way to describe it: it is what turns a waveform (amplitude x time) to a spectrogram like on a stereo (amplitude x frequency)

It's a Copy>Paste Special>Transpose on a waveform, converting Rows/Columns that are time/amplitude (with wavelength embedded) into Rows/Columns that are frequency/amplitude (for a snapshot in time).

People love to go on about how brilliant it is and they're probably right but that's how I understand it.

And phase. People always forget about the phase as if it was purely imaginary.

Pretty much, but phase is also included. Which matters for some things.

But mostly not for ears it turns out!

Phase matters for some wideband signals, but most folks struggle to tell apart audio from hilbert-90-degree-shifted-audio

Phase is required if it is to be a reversible transform. Otherwise would just be a Functional.

Ha ha, as I understand it, phase is imaginary in a Fourier transform. Complex numbers are used and the imaginary portion does indeed represent phase.

I have been told that reversing the process — creating a time-based waveform — will not resemble (visually) the original due to this phase loss in the round-tripping. But then our brain never paid phase any mind so it will sound the same to our ears. (Yay, MP3!)

Read this (which is free): The Scientist's and Engineer's Guide to Digital Signal Processing* https://www.dspguide.com

It's very comprehensive, but it's also very well written and walks you through the mechanics of Fourier transforms in a way that makes them intuitive.

I'm glad someone picked up on my dumb joke :), I was getting worried.

That being said, round-tripping works just fine, axiomatically so, until you go out of your way to discard the imaginary component.

Actually, by the Kramers-Kronig relation you can infer the imaginary part just from the real parts, if given that your time signal is causal. So the phase isn’t actually lost in any way at all, if you assume causality.

Also, pedantic nit: phase would be the imaginary exponent of the spectrum rather than the imaginary part directly, i.e, you take the logarithm of the complex amplitude to get log-magnitude (real) plus phase (imag)

humble plug https://dsego.github.io/demystifying-fourier/

Even more complex and reflectively imaginative than the Fourier Transform is the mighty Cepstrum!

https://en.wikipedia.org/wiki/Cepstrum

It’s literally a "backwards spectrum", and the authors in 1963 were having such jolly fun they reversed the words too: quefrency => frequency, saphe => phase, alanysis => analysis, liftering => filtering

The cepstrum is the "spectrum of a log spectrum," where taking the complex logarithm turns multiplicative spectral features into additive ones, laying the foundation of cepstral alanysis, and later, the physiologically tuned Mel-frequency cepstrum used in audio compression and speech recognition.

https://en.wikipedia.org/wiki/Mel_scale

>The mel scale (after the word melody)[1] is a perceptual scale of pitches judged by listeners to be equal in distance from one another. [...] Use of the mel scale is believed to weigh the data in a way appropriate to human perception.

As Tukey might say: once you start doing cepstral alanysis, there’s no turning back, except inversely.

Skeptics said he was just going through a backwards phase, but it turned out to work! ;)

https://news.ycombinator.com/item?id=24386845

DonHopkins on Sept 5, 2020 | parent | context | favorite | on: Mathematicians should stop naming things after eac...

I love how they named the inverse spectrum the cepstrum, which uses quefrency, saphe, alanysis, and liftering, instead of frequency, phase, analysis and filtering. It should not be confused with the earlier concept of the kepstrum, of course! ;)

https://en.wikipedia.org/wiki/Cepstrum

>References to the Bogert paper, in a bibliography, are often edited incorrectly. The terms "quefrency", "alanysis", "cepstrum" and "saphe" were invented by the authors by rearranging some letters in frequency, analysis, spectrum and phase. The new invented terms are defined by analogies to the older terms.

>Thus: The name cepstrum was derived by reversing the first four letters of "spectrum". Operations on cepstra are labelled quefrency analysis (aka quefrency alanysis[1]), liftering, or cepstral analysis. It may be pronounced in the two ways given, the second having the advantage of avoiding confusion with "kepstrum", which also exists (see below). [...]

>The kepstrum, which stands for "Kolmogorov-equation power-series time response", is similar to the cepstrum and has the same relation to it as expected value has to statistical average, i.e. cepstrum is the empirically measured quantity, while kepstrum is the theoretical quantity. It was in use before the cepstrum.[12][13]

https://news.ycombinator.com/item?id=43341806

DonHopkins 7 months ago | parent | context | favorite | on: What makes code hard to read: Visual patterns of c...

Speaking of filters and clear ergonomic abstractions, if you like programming languages with keyword pairs like if/fi, for/rof, while/elihw, goto/otog, you will LOVE the cabkwards covabulary of cepstral quefrency alanysis, invented in 1963 by B. P. Bogert, M. J. Healy, and J. W. Tukey:

cepstrum: inverse spectrum

lifter: inverse filter

saphe: inverse phase

quefrency alanysis: inverse frequency analysis

gisnal orpcessing: inverse signal processing

https://en.wikipedia.org/wiki/Cepstrum

https://news.ycombinator.com/item?id=44062022

DonHopkins 5 months ago | parent | context | favorite | on: The scientific “unit” we call the decibel

At least the Mel-frequency cepstrum is honest about being a perceptual scale anchored to human hearing, rather than posing as a universally-applicable physical unit.

https://en.wikipedia.org/wiki/Mel-frequency_cepstrum

>Mel-frequency cepstral coefficients (MFCCs) are coefficients that collectively make up an MFC. They are derived from a type of cepstral representation of the audio clip (a nonlinear "spectrum-of-a-spectrum"). The difference between the cepstrum and the mel-frequency cepstrum is that in the MFC, the frequency bands are equally spaced on the mel scale, which approximates the human auditory system's response more closely than the linearly-spaced frequency bands used in the normal spectrum. This frequency warping can allow for better representation of sound, for example, in audio compression that might potentially reduce the transmission bandwidth and the storage requirements of audio signals.

https://en.wikipedia.org/wiki/Psychoacoustics

>Psychoacoustics is the branch of psychophysics involving the scientific study of the perception of sound by the human auditory system. It is the branch of science studying the psychological responses associated with sound including noise, speech, and music. Psychoacoustics is an interdisciplinary field including psychology, acoustics, electronic engineering, physics, biology, physiology, and computer science.