Lecture 35 Wave spectrum·Fourierseries·Fourieranalysis·Fouriertransformation
Lecture 35 Wave spectrum • Fourier series • Fourier analysis • Fourier transformation 1
Fourier seriesAny periodic function can be decomposed into thesum of a set of simple oscillation functions.Ia osrot)+ bsihro) f(t) = ao +n=1Where w = 2π/T, a's and b's are numericalconstants which tell us how much of eachcomponent oscillation is present.Eg.cos(t + Φ) = cosΦ coswt + sin Φ sin wt2
Fourier series Any periodic function can be decomposed into the sum of a set of simple oscillation functions. 𝑓 𝑡 = 𝑎0 + 𝑛=1 ∞ [𝑎𝑛 cos 𝑛𝜔𝑡 + 𝑏𝑛 sin 𝑛𝜔𝑡 ] Where 𝜔 = 2𝜋/𝑇, 𝑎′𝑠 𝑎𝑛𝑑 𝑏 ′ 𝑠 are numerical constants which tell us how much of each component oscillation is present. Eg. cos 𝜔𝑡 + 𝜙 = cos𝜙 cos 𝜔𝑡 + sin 𝜙 sin 𝜔𝑡 2
SquareSquare wave4 sin 504 sin 04 sin 304 sin 70f(t)+3元5元7元元4sin0T4sin3e34sin505㎡4sin707
Square wave 𝑓 𝑡 = 4 sin 𝜃 𝜋 + 4 sin 3𝜃 3𝜋 + 4 sin 5𝜃 5𝜋 + 4 sin 7𝜃 7𝜋 + ⋯ 3
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Sawtooth wave2sine一n2sin202m2sin30-3㎡2sin404π2 sin 02 sin 202 sin 302 sin 40f(t)2元3元4元元5
Sawtooth wave 5 𝑓 𝑡 = − 2 sin 𝜃 𝜋 + 2 sin 2𝜃 2𝜋 − 2 sin 3𝜃 3𝜋 + 2 sin 4𝜃 4𝜋 + ⋯