Fourier series

Taylor series build a function from powers of \(x\), matching its behavior near a single point. Many signals in engineering are better described in a different way: as a sum of sines and cosines of different frequencies. The vibration of a machine, the current drawn by a rectifier, the sound of a musical instrument, the temperature cycle of a day and a year, and the voltage at the output of an inverter are all periodic, repeating after a fixed time, and each can be decomposed into a fundamental frequency and its harmonics. Joseph Fourier introduced this idea in 1807 to solve problems of heat conduction, and it has become one of the most widely used tools in science and engineering. This lesson shows how to compute the Fourier series of a periodic function, how such series converge, how symmetry simplifies them, and how they are used to analyze power quality, vibrations and filters.

Periodic functions and harmonics

A function is periodic with period \(T\) if \(f(t + T) = f(t)\) for all \(t\). The fundamental angular frequency is \(\omega_0 = 2\pi/T\), and the sinusoids \(\cos n\omega_0 t\) and \(\sin n\omega_0 t\) for \(n = 1, 2, 3, \ldots\) all repeat with period \(T\). The \(n\)-th harmonic has frequency \(n\) times the fundamental: on a 60 Hz power system, the third harmonic is 180 Hz and the fifth is 300 Hz. Fourier's claim, made precise later, is that a periodic function with reasonable behavior can be written as:

where \(a_0\) is the average value (the DC component) and \(a_n\), \(b_n\) give the amount of each harmonic.

Finding the coefficients

The coefficients can be found because sines and cosines of different harmonics are orthogonal over a period: the integral of the product of any two different ones over a full period is zero. For integers \(m \ne n\):

while \(\int_0^T \cos^2 n\omega_0 t\,dt = \int_0^T \sin^2 n\omega_0 t\,dt = T/2\). These follow from the product-to-sum identities of Lesson 1. Multiplying the series by \(\cos n\omega_0 t\) and integrating over a period therefore kills every term except one, and similarly for \(\sin n\omega_0 t\). The result is the Euler-Fourier formulas:

The integrals can be taken over any interval of length \(T\), for example \(-T/2\) to \(T/2\), whichever is more convenient. Each coefficient measures how strongly the function correlates with one harmonic: the same idea, comparing a signal with a reference sinusoid and averaging, is how lock-in amplifiers and spectrum analyzers work.

The square wave

Convergence and the Gibbs phenomenon

Dirichlet proved in 1829 that the Fourier series of a periodic function that is piecewise smooth (made of finitely many smooth pieces with finitely many jumps) converges to the function at every point where it is continuous, and to the average of the left and right values at each jump. The square wave series gives 0 at \(t = 0\), halfway between −1 and +1.

Near a jump, however, the partial sums overshoot. As more terms are added, the overshoot becomes narrower but does not get smaller: it approaches about 9 percent of the size of the jump. This Gibbs phenomenon appears whenever a signal with a sharp edge is represented with a limited number of frequencies, such as ringing near edges in compressed images and in digitally filtered signals. Smoother functions have Fourier coefficients that decrease faster: coefficients fall like \(1/n\) for functions with jumps, like \(1/n^2\) for continuous functions with corners, such as a triangle wave, and faster still for smooth functions. Fewer terms are then needed for a good approximation.

Using symmetry

Symmetry eliminates half the work. If \(f\) is even, \(f(-t) = f(t)\), then \(f(t)\sin n\omega_0 t\) is odd and integrates to zero over a symmetric interval, so all \(b_n = 0\) and the series contains only cosines (a cosine series). If \(f\) is odd, all \(a_n = 0\) (including \(a_0\)), and the series contains only sines. A further symmetry, half-wave symmetry, \(f(t + T/2) = -f(t)\), which most AC waveforms have because their positive and negative halves are mirror images, removes all even harmonics. This is why the current of a balanced three-phase rectifier or a saturated transformer contains odd harmonics (3rd, 5th, 7th) but not even ones.

Energy, RMS value and Parseval's theorem

Because the harmonics are orthogonal, the mean square value of a periodic function is the sum of the mean squares of its components, Parseval's theorem:

In electrical terms, the RMS value (Calculus I) of a distorted waveform is \(\sqrt{I_0^2 + I_{1,\text{RMS}}^2 + I_{2,\text{RMS}}^2 + \cdots}\), where each harmonic's RMS value is its amplitude divided by \(\sqrt{2}\). The power delivered to a resistor is the sum of the powers delivered by each harmonic separately.

Fourier series and linear systems

The great practical value of Fourier series lies in how linear systems respond to them. A linear system, such as an electrical circuit, a mechanical structure or a filter, responds to a sinusoidal input with a sinusoid of the same frequency, changed in amplitude and phase by amounts that depend on the frequency. By the principle of superposition, the response to a sum of sinusoids is the sum of the responses. So to find the response to any periodic input, decompose the input into harmonics, find the response to each using phasors (Lesson 9), and add.

The same reasoning explains resonance in structures and machines. A periodic force with harmonics near a natural frequency of the structure excites large vibrations even if the fundamental is far away, which is why engineers examine the full harmonic content of forces from rotating machinery, footfalls on floors and the blade-passing frequencies of fans and turbines.

Complex form and the road to the Fourier transform

Using Euler's formula, the sines and cosines can be combined into complex exponentials, giving the compact complex form:

Each \(c_n\) is a complex number whose modulus gives the amplitude and whose argument gives the phase of the \(n\)-th harmonic; a plot of \(|c_n|\) against frequency is the signal's spectrum. For signals that are not periodic, letting the period grow without bound turns the sum into an integral, the Fourier transform, which describes a signal by a continuous spectrum. Sampled data are analyzed with the discrete Fourier transform, computed by the fast Fourier transform (FFT) algorithm popularized in 1965, which made spectrum analysis routine in instruments, audio and image compression, wireless communication and vibration monitoring. The signals and systems courses in the ELE and CMP programs develop these ideas.