Fourier Series (FS) and Relation to DFT

In continuous time, a periodic signal $x(t)$, with period $P$ seconds,^E.2 may be expanded into its Fourier series as

$$X(\omega_k) \isdef \frac{1}{P}\int_0^P x(t) e^{-j\omega_k t} dt, \quad k=0,\pm1,\pm2,\dots \protect$$ (E.5)

where $\omega_k \isdef 2\pi k/P$ is the $k$th harmonic frequency (rad/sec). The generally complex value $X(\omega_k)$ is called the $k$th Fourier series coefficient.