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Thread: Exponential Fourier-series expansion

  1. #1
    Apr 2009

    Exponential Fourier-series expansion

    I need help in determining the exponential fourier series expansion for the half-wave rectified signal x(t)=cos(t). I am trying to find Xn, and have determined for the case where n=0, Xn=1/pi...but I'm having trouble finding the general case for just Xn. Any help would be appreciated.
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  2. #2
    Senior Member
    Apr 2009
    Atlanta, GA

    Fourier Series

    $\displaystyle f(x)=\frac{1}{2}a_0+\sum_{n=1}^\infty a_n\cos(nx)+\sum_{n=1}^\infty b_n\sin(nx)$

    $\displaystyle a_0=\frac{1}{\pi}\int_{-\pi}^\pi f(x)dx$

    $\displaystyle a_n=\frac{1}{\pi}\int_{-\pi}^\pi f(x)\cos(nx)dx$

    $\displaystyle b_n=\frac{1}{\pi}\int_{-\pi}^\pi f(x)\sin(nx)dx$

    Look very, very carefully at the function you are trying to expand here. $\displaystyle x(t)=cos(t)$, therefore $\displaystyle a_0=0$, $\displaystyle a_1=1$, $\displaystyle a_n=0$ for all $\displaystyle n>1$, and $\displaystyle b_n=0$ for all $\displaystyle n$.
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