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If I have an equation such as $x(t) = \displaystyle \sum_{n=1}^N \left( a_n \cos(\omega_nt) + b_n \sin(\omega_n t) \right)$, how do I convert it to a sum of complex exponentials? In other words what do I do with the coefficients in front of the sine and cosine to turn them into the coefficient of each complex exponential.

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Write $\displaystyle \cos(w_n t) = \frac{e^{i w_n t} + e^{-i w_n t}}{2}$ and $\displaystyle \sin(w_n t) = \frac{e^{i w_n t} - e^{-i w_n t}}{2i}$ and rearrange to convert it to a sum of complex exponentials.

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so if I had 2cos(wt) + 4jsin(wt), what would be the resulting complex exponential? –  mike Aug 14 '11 at 6:48
    
It will be $$3e^{j\omega t} - e^{-j\omega t}$$ –  user17762 Aug 14 '11 at 7:37

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