Fourier series – Scitura

What is it about?

A $2\pi$ -periodic function can be written as a Fourier series $f(x) \approx \frac{a_0}{2} + \sum_{k=1}^{N}(a_k \cos kx + b_k \sin kx)$ . Writing each term as $\hat A_k \sin(kx + \varphi_k)$ gives the amplitude spectrum $\hat A_k = \sqrt{a_k^2 + b_k^2}$ and the phase spectrum $\varphi_k$ . The square wave is $\frac{4}{\pi}(\sin x + \frac13 \sin 3x + \frac15 \sin 5x + \ldots)$ .

Phasors: each term is a phasor of length $\hat A_k$ turning $k$ times as fast as the fundamental; chained together, the height of the tip draws $S_N$ .
Symmetry: odd functions have sine terms only, even functions cosine terms only; if $f(x + \pi) = -f(x)$ , all even $k$ vanish.
Jumps vs. kinks: with jumps the amplitudes fall like $\frac1k$ , with kinks like $\frac{1}{k^2}$ .
Gibbs phenomenon: at a jump the partial sums overshoot by about $9\,\%$ of the jump (square wave: $1.179$ ), however large $N$ is. Fejér factors $\sigma_k = 1 - \frac{k}{N+1}$ remove the overshoot.
Parseval: $P = \frac{1}{2\pi}\int f^2\,dx = (\frac{a_0}{2})^2 + \frac12 \sum (a_k^2 + b_k^2)$ . The fundamental of the square wave carries $\frac{8}{\pi^2} \approx 81.06\,\%$ of the power.

Try it

Square wave: press “Add a harmonic” several times, then “Play the build-up”.
Phasors draw a square wave and stacked harmonics.
Gibbs phenomenon versus Fejér smoothing.
Clarinet versus flute – compare spectra and listen.
LED dimmer: change the duty cycle $d$ .

Tasks

Task 1: Symmetry

Why does the square wave series contain only sines and the triangle series only cosines? What happens if the square wave is shifted by $90°$ ?

Show solution

The square wave is odd ( $a_k = 0$ ), the triangle is even ( $b_k = 0$ ). Shifted by $\frac{\pi}{2}$ the square wave becomes even: cosine terms only, same amplitudes. Show

Task 2: Parseval

What share of the power of the square wave is contained in $S_1$ and in $S_3$ ? From which $N$ on is it at least $95\,\%$ ? Load task

Show solution

$S_1$ : $\frac{8}{\pi^2} \approx 81.06\,\%$ ; $S_3$ : $\frac{8}{\pi^2}(1 + \frac19) \approx 90.06\,\%$ ; $S_7$ : $94.96\,\%$ ; $S_9$ : $95.96\,\%$ – so from $N = 9$ on. Show

Task 3: LED dimmer

A pulse train (height $1$ , duty cycle $d = 25\,\%$ ) drives an LED. Find the DC part and the first four amplitudes. Which one is missing?

Show solution

DC part $d = 0.25$ ; $\hat A_k = \frac{2|\sin(\pi k d)|}{\pi k}$ : $0.450$ , $0.318$ , $0.150$ and $\hat A_4 = 0$ (every fourth harmonic vanishes). Show

Task 4: Full-wave rectifier

Mains at $50\,\text{Hz}$ gives $|\sin x|$ after the rectifier. Find the mean and the second harmonic (first overtone, $k = 2$ ). At which frequency does the power supply hum?

Show solution

Mean $\frac{2}{\pi} \approx 0.637$ ; only even $k$ : $a_2 = -\frac{4}{3\pi} \approx -0.424$ at $100\,\text{Hz}$ . Show

Ideas for the classroom

Start with “Play the build-up”, then add the values by hand in the stacked view; with “Values in the picture” off (info=0) the pictures work as exercises.
Link to acoustics (timbre of clarinet, violin, flute) and electronics (rectifier, pulse-width modulation).
Common stumbling blocks: the constant term is $\frac{a_0}{2}$ ; expecting the Gibbs overshoot to vanish; a shift changes only the phases, not the amplitudes.