Fourier Series and Fourier Transforms: A Simple Guide

A Fourier series writes any periodic function as a sum of sines and cosines, like decomposing a musical chord into its pure notes. The Fourier transform does the same job for non-periodic signals, representing them as a continuous blend of frequencies instead of a discrete list.

This matters because the frequency view often reveals what the time view hides. Filtering noise, compressing images, solving the heat equation, and reading MRI scans all become simpler once a signal is expressed in frequencies.

Key Concepts

• Period and fundamental frequency: a function repeating every T has fundamental angular frequency omega0 = 2 pi / T; all other components are integer multiples, the harmonics.
• Fourier coefficients: a(n) and b(n) measure how much of cos(n omega0 t) and sin(n omega0 t) the signal contains, found by integrating the signal against each basis function.
• Orthogonality: sines and cosines of different frequencies are mutually orthogonal over one period, which is why each coefficient can be extracted independently.
• Fourier transform: F(omega) = the integral from minus infinity to infinity of f(t) e to the power (minus i omega t) dt, extending the idea to signals that never repeat.

Formulas

• Fourier series: f(t) = a0/2 + sum from n=1 to infinity of [a(n) cos(n omega0 t) + b(n) sin(n omega0 t)].
• Coefficients: a(n) = (2/T) times the integral over one period of f(t) cos(n omega0 t) dt; b(n) the same with sine.
• Fourier transform pair: F(omega) = integral of f(t) e to the power (minus i omega t) dt; f(t) = (1 / 2 pi) integral of F(omega) e to the power (i omega t) d omega.
• Square wave (odd, amplitude A, period 2 pi): f(t) = (4A / pi) [sin t + (sin 3t)/3 + (sin 5t)/5 and so on].

Worked Example

Use the square-wave series to verify it converges to the value 1 at t = pi/2, and recover a famous formula for pi.

1. Take amplitude A = 1. The series is f(t) = (4 / pi)[sin t + (sin 3t)/3 + (sin 5t)/5 and so on].
2. At t = pi/2: sin(pi/2) = 1, sin(3 pi/2) = minus 1, sin(5 pi/2) = 1, so the bracket becomes 1 minus 1/3 + 1/5 minus 1/7 and so on.
3. The square wave equals 1 at t = pi/2, so 1 = (4 / pi)[1 minus 1/3 + 1/5 minus 1/7 and so on].
4. Rearranging gives pi/4 = 1 minus 1/3 + 1/5 minus 1/7 and so on, the Leibniz formula for pi.
5. Numerically, three terms give 0.8667, and (4/pi) times that is about 1.104, already approaching 1 as more terms are added.

Common Misconceptions

• "Adding more terms removes the Gibbs overshoot." It does not. The overshoot shrinks in width but keeps its roughly 9 percent height at every jump, forever.
• "The series equals the function everywhere." At a discontinuity, the series converges to the midpoint of the jump, not to either one-sided value.
• "Fourier methods only work for periodic signals." The Fourier transform handles non-periodic signals, and even a single pulse has a perfectly good frequency spectrum.

Real-World Applications

• Audio processing: equalizers and noise removal work by adjusting frequency components.
• Medical imaging: MRI machines collect Fourier data directly and invert the transform to form images.

Fourier series and transforms move signals from the time domain to the frequency domain, where filtering, compression, and differential equations become easier: among the most-used tools in all of science.