- The Inverse Euler Formulas express the and functions in terms of complex exponentials:

- A signal formed from the summation of sinusoids is given by:

where*X*_{0}=*A*_{0}and . - Then by the inverse Euler relationships:

where is the complex-conjugate of*X*_{k}. - Each sinusoid is represented by two rotating phasors, one with a positive frequency
*f*_{k}and the other with -*f*_{k}. - The two-sided spectrum of a signal composed of sinusoids is then given by:

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