MIT18_03S10_c21

MIT18_03S10_c21 - 18.03 Class 21 , March 29 Fourier series...

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Unformatted text preview: 18.03 Class 21 , March 29 Fourier series II [1] Review [2] Square wave [3] Piecewise continuity [4] Tricks [1] Recall from before break: A function f(t) is periodic of period 2L if f(t+2L) = f(t) . Theorem: Any decent periodic function f(t) of period 2pi has can be written in exactly one way as a *Fourier series*: f(t) = a_0/2 + a_1 cos(t) + a_2 cos(2t) + ... + b_1 sin(t) + b_2 sin(2t) + ... If the need arises, the "Fourier coefficients" can be computed as integrals: a_n = (1/pi) integral_{-pi}^{pi} f(t) cos(nt) dt , n geq 0 b_n = (1/pi) integral_{-pi}^{pi} f(t) sin(nt) dt , n > 0 [2] Squarewave: A basic example is given by the "standard squarewave," which I denote by sq(t) : it has period 2pi and sq(t) = 1 for 0 < t < pi = -1 for -pi < t < 0 = 0 for t = 0 , t = pi This is a standard building block for all sorts of "on/off" periodic signals. It's odd, so a_n = integral_{-pi}^pi odd . even dt = 0 for all n . If f(t) is an odd function of period 2pi, we can simplify the integral for bn a little bit. The integrand f(t) sin(nt) is even, so the integral is twice the integral from 0 to pi: bn = (2/pi) integral_0^pi f(t) sin(nt) dt Similarly, if f(t) is even then an = (2/pi) integral_0^pi f(t) cos(nt) dt In our case this is particularly convenient, since...
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This note was uploaded on 01/18/2012 for the course MATH 18.03 taught by Professor Vogan during the Spring '09 term at MIT.

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MIT18_03S10_c21 - 18.03 Class 21 , March 29 Fourier series...

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