Fourier series and the Fourier transform

Sinusoids as a basis, frequency as a dual coordinate.

The single mathematical idea most used across the bookshelf. A reasonable function can be expressed as a sum (for periodic functions) or an integral (for non-periodic) of sinusoids. The coefficients of that decomposition are the frequency-domain representation of the function. The decomposition is invertible, and the algebraic moves it enables — converting calculus to algebra, convolution to multiplication, energy-in-time to energy-in-frequency — are the technical core of every spectrogram, every linear filter, every transfer function, every cochlear-place-coding analysis on this site.

This chapter is the working reference, structured as four lessons that go from periodic-signal series through continuous-frequency transform, into the algebraic identities (convolution theorem and Parseval) that make Fourier methods so productive, and finally into the discrete-time DFT that signal processing actually computes.

• 7.1 Fourier series — periodic signals; orthonormality of complex exponentials; Gibbs phenomenon at jump discontinuities; the historical controversy that built modern analysis.
• 7.2 The Fourier transform — the non-periodic limit, operational properties (linearity, shift, scaling, differentiation), pairs table, uncertainty principle.
• 7.3 Convolution and Parseval — the two identities that turn the transform into the foundation of linear-systems theory. Convolution becomes multiplication; energy is preserved in either domain.
• 7.4 The DFT, sampling, and aliasing — the discrete version: Shannon’s sampling theorem, aliasing, the DFT formula, window functions, the FFT.

The convention used throughout: $\tilde f(\omega) = \int f(t)\, e^{-i \omega t}\, dt$ with the $1/(2\pi)$ on the inverse transform. Other conventions are equivalent; always check which one a textbook uses before applying any formula.