I want to derive the Discrete Fourier series: $$f(x) = \sum_{k=0}^{N-1} X_k e^{i2\pi xk}$$ from the Continuous Fourier Series formula: $$f(x) = \sum_{k=-\infty}^{\infty} X_k e^{i2\pi xk}$$
additionally I've heard that if the signal is real then the fourier coefficients are symmetrical and hence the discrete fourier series only sums from 0 to N - 1? can somebody explain how the first equation is derived from the second and what role the coefficient symmetry plays into all of this? thank in advance