Fourier Series
Analysis_and_Synthesis_Fourier_Series.jpg
  • A Fourier series is an expansion of a periodic function $f(x)$ in terms of an infinite sum of sines and cosines.
  • period = $2\pi$

$f(x) = a_0 + \Sigma_{n=1}^{\infty}a_ncos(nx)+\Sigma_{n=1}^{\infty}b_nsin(nx)$
where
$a_0 = \frac{1}{2 \pi}\int_{-\pi}^{\pi}f(x)dx$
$a_n = \frac{1}{\pi}\int_{-\pi}^{\pi}f(x)cos(nx)dx$
$b_n = \frac{1}{\pi}\int_{-\pi}^{\pi}f(x)sin(nx)dx$

  • period = 2L

$f(x) = a_0+\Sigma_{n=1}^{\infty}a_ncos(n\frac{\pi}{L}x)+\Sigma_{n=1}^{\infty}b_nsin(n\frac{\pi}{L}x)$
or
$f(x) = a_0 + \Sigma_{n=1}^{\infty} D_n cos(nx+{\theta}_n)$
where
$a_0 = \frac{1}{2L}\int_{-L}^{L}f(x)dx$
$a_n = \frac{1}{L}\int_{-L}^{L}f(x)cos(n\frac{\pi}{L}x)dx$
$b_n = \frac{1}{L}\int_{-L}^{L}f(x)sin(n\frac{\pi}{L}x)dx$
$D_n \angle \theta_n = 2 c_n = a_n - i b_n$

  • period p

$f(x) = a_0 + \Sigma_{n=1}^{\infty}(a_n cos(n\omega_0x) + b_n sin(n\omega_0x))$, $\omega_0 = \frac{2\pi}{p}$

$a_0 = \frac{1}{p}\int_{-p/2}^{p/2} f(x) dx$
$a_n = \frac{2}{p}\int_{-p/2}^{p/2} f(x)cos(n\omega_0x) dx$
$b_n = \frac{2}{p}\int_{-p/2}^{p/2} f(x)sin(n\omega_0x) dx$

$f(x) = a_0 + \Sigma_{n=1}^{\infty}c_n cos(n\omega_0x + \delta_n)$, $a_n cos(n\omega_0x) + b_n sin(n\omega_0x) = c_n cos(n\omega_0x + \delta_n)$

$c_n = \sqrt{a_n^2+b_n^2}$, $\delta_n = tan^{-1}(-\frac{b_n}{a_n})$

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