Fourier Analysis for Beginners (Thibos)

larry-thibos‘s coursenotes for V791: Quantitative Methods for Vision Research at the indiana-university School of Optometry. Sixth edition, 2014, copyright 1989–2014, 201 pages. Arrived as a PDF over Telegram on 2026-07-28 with no URL, was parked in the hub _inbox under signal-processing, and moved here with the split. No canonical web address is recorded, which makes the held PDF the only copy this hub has.

The epigraph is Galileo on the universe being written in the language of mathematics, which tells you the register: this is a methods course written by someone who thinks the maths is the point, for students who came for the eyes.

What it covers

Fourteen chapters running from mathematical preliminaries (vectors, phasors, complex numbers, Euler’s formula) through discrete and continuous Fourier analysis, inner products and orthogonality, the frequency domain, sampling theory, the Fourier transform and its properties, to signal analysis and Fourier optics — image formation treated as a linear system. Matlab is the working tool throughout, including its particular DFT convention. See fourier-analysis.

The three chapters that give away the audience

A pure-mathematics treatment would stop at the transform. This one continues into the inferential apparatus an experimentalist actually needs:

• statistics of Fourier coefficients
• hypothesis testing for Fourier coefficients
• directional data analysis, with a Rayleigh Z-statistic appendix

The bibliography is organized the same way — Fourier series and transforms, statistics of Fourier coefficients, directional data analysis, random signals and noise, probability theory and stochastic processes, signal detection theory, applications. That is a reading list for someone who has to publish a result, not prove a theorem.

Tier and limits

• T2. Sixth-edition coursenotes refined over 25 years by the faculty member who teaches the course, published under a university department’s name. Not peer-reviewed and not a press publication, so not T1; considerably more settled than a one-off explainer.
• Aimed at beginners, and says so. The title is the level. Preliminaries start at vector arithmetic and complex numbers, which places the audience precisely.
• The examples are optical. Roughly 85% of the content is general Fourier analysis and signal processing; the vision-science material concentrates in chapter 14 and the worked examples. That ratio is why it is here rather than in ../psychology-wiki.
• Matlab-bound in the implementation chapters. §4.H covers Matlab’s specific DFT implementation, which is a tool convention rather than mathematics.
• A page-number error survives into the sixth edition: the table of contents lists §5.A at page 58 inside a chapter starting at 61. Trivial, and a fair marker of how much copy-editing coursenotes get.
• No URL. If the PDF here is lost, the source is lost.

Cross-spoke context

• ../psychology-wiki — the closest call, and it was declined at parking. That spoke’s domain explicitly covers psychophysics and the empirical study of perception, and this is the quantitative-methods course for exactly that field. The fit is through the audience, not the content. Standing watch: if vision-science findings arrive they route there, and this book becomes their methods appendix.
• ../music-tech-wiki — the FFT under every filter, delay and spectrum analyser starts here; that spoke owns the instruments and effects, not the transform.
• ../loudspeaker-design-wiki — frequency-response and impedance measurement run on this maths.
• ../embedded-iot-wiki — on-device DSP is downstream of chapters 4 and 11.
• ../machine-learning-wiki — the linear-systems view of image formation in chapter 14 is the ancestor of convolution as a modelling primitive; that connection is not made in the book and is noted here as an adjacency, not a claim it supports.

Related

fourier-analysis · mathematics · larry-thibos · indiana-university · mcmullen-probability-theory · corral-vector-calculus · synthesis