Fourier series synthesis

In the previous section we have seen that every periodic signal $x\left(t\right)$, with Fundamental period $T_0=1/F_0$, can be written as a sum of weighted harmonic related phasor components. In theory the summation can contain an infinite amount of components which leads to the following Fourier series expansion:

$$x\left(t\right)=\sum _{k=-\infty }^{\infty }{\alpha }_k{e}^{j2\pi k{F}_{0}t}.$$

Thus when the spectral weights, which are represented by the complex numbers ${\alpha }_k$, are known we can construct (synthesize) the periodic signal $x\left(t\right)$ according the above general expression.