Exponential Fourier Series.

Question

Determine the exponential Fourier series(which invovle exp(jkwt) terms) of the following.

x(t)=cos(t)+cos(2t)+0.5

I calculated C0 and got the following.

C0=0.5

however, I calculated Cm to be 0 for all m, I believe this is wrong as it contradicts with C0.

Any help is appreciated, thanks.

You can rewrite $2\cos(t) = \exp(jt) + \exp(-jt)$ and read off the Fourier coefficients from that. — DanielM, Sep 03 '13 at 09:16

score 0 · Answer 1 · answered Sep 03 '13 at 14:04

The given function $x(t)$ can be written as

$$x(t)=\frac{1}{2}\left(e^{2\pi it}+e^{-2\pi it}+e^{4\pi it}+e^{-4\pi it}\right)+\frac{1}{2}; $$

from this expression you can obtain

$$C_0=C_{\pm 1}=C_{\pm 2}=\frac{1}{2}$$

and $C_j=0$ for all $j\in \mathbb Z-\{0,\pm 1,\pm 2\}$ just looking at the definition of the Fourier expansion (of period $2\pi$) of $x(t)$.

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