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.
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 $x(t)$.