05 / CHAPTER 11
Fourier thinking
Decompose a complex signal into simpler frequencies.
ANCHOR EQUATION
Fourier representation
In symbols: f(t) = Σ An cos(nω₀t + φn)
Complex signals can be understood as sums of simpler frequencies.
Fourier analysis turns shape into frequencies
A complicated periodic signal can be represented as a sum of sinusoids. This is useful because linear systems respond to each frequency separately, making resonance and filtering visible.
Project onto orthogonal basis functions
For period T, write f(t) = a₀/2 + Σ[a_n cos(nω₀t) + b_n sin(nω₀t)], with ω₀ = 2π/T. Multiply by cos(mω₀t) and integrate over one period; orthogonality makes all terms vanish except n = m, isolating a_m. The coefficients are measurements of correlation with each basis frequency.
A square wave of amplitude ±1 has only odd harmonics in its sine series: f(t) = (4/π)[sinωt + (1/3)sin3ωt + (1/5)sin5ωt + …]. The 3rd harmonic amplitude is 0.424 of the fundamental.
Finite bandwidth limits reconstruction
Truncating the series approximates sharp edges with ringing near discontinuities. Sampling also requires a rate above twice the highest represented frequency to avoid aliasing.
Work it: Approximate a square wave using the first and third harmonics at t = 0, T/8, T/4. Compare with the ideal values and explain the edge error.