18.03 Lecture Notes - Fourier Series, Periodic Function, Even And Odd Functions
Document Summary
[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. = 0 for t = 0 , t = pi. This is a standard building block for all sorts of "on/off" periodic signals.
