Professor Brendan Morris, SEB 3216, brendan.morris@unlv.edu EE361: Signals and System II Fourier Series Highlights http://www.ee.unlv.edu/~b1morris/ee361/
2 Big Idea: Transform Analysis • Make use of properties of LTI system to simplify analysis • Represent signals as a linear combination of basic signals with two properties ▫ Simple response: easy to characterize LTI system response to basic signal ▫ Representation power: the set of basic signals can be use to construct a broad/useful class of signals
3 Normal Modes of Vibrating String • Consider plucking a string • Dividing the string length into integer divisions results in harmonics ▫ The frequency of each harmonic is an integer multiple of a “fundamental frequency” ▫ Also known as the normal modes • It was realized that the vertical deflection at any point on the string at a given time was a linear combination of these normal modes ▫ Any string deflection could be built out of a linear combination of “modes”
4 Fourier Series • Fourier argued that periodic • Harmonically related period signals (like the single period signals form family from a plucked string) were ▫ Integer multiple of actually useful fundamental frequency ▫ 𝜚 𝑙 𝑢 = 𝑓 𝑘𝑙𝜕 0 𝑢 for 𝑙 = ▫ Represent complex periodic signals 0, ±1, ±2, … • Examples of basic periodic signals • Fourier Series is a way to ▫ Sinusoid: 𝑦 𝑢 = 𝑑𝑝𝑡𝜕 0 𝑢 represent a periodic signal as a linear combination of ▫ Complex exponential: 𝑦 𝑢 = 𝑓 𝑘𝜕 0 t harmonics ▫ Fundamental frequency: 𝜕 0 ∞ 𝑏 𝑙 𝑓 𝑘𝑙𝜕 0 𝑢 ▫ 𝑦 𝑢 = 𝑙=−∞ 2𝜌 ▫ 𝑏 𝑙 coefficient gives the ▫ Fundamental period: 𝑈 = 𝜕 0 contribution of a harmonic (periodic signal of 𝑙 times frequency)
5 Sawtooth Example Harmonics: height given by coefficient signal 𝑏 0 𝑏 1 𝑏 2 𝑏 3 … Animation showing approximation as more harmonics added
6 Square Wave Example • Better approximation of • Aligned approximations square wave with more coefficients # 𝒃 𝒍 coeff 1 2 • Animation of FS 3 4 Note: 𝑇(𝑔) ~ 𝑏 𝑙
7 Arbitrary Examples • Interactive examples
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