Fourier Transform and Power Spectra

Fourier series analysis is performed to obtain the discrete spectrum representation of a given periodic signal (power signal) xp(t) which has finite periodic time T_o , finite average power and infinite energy, to describe its frequency components content (n/T_o), where n = 0, 1, 2, 3, 4, ... , by either using the real coefficients method to obtain the real coefficients a_n and b_n, Equations (2.2) and (2.3), to construct x_p(t), Eq.(2.1), or by using the complex coefficient method to obtain the complex coefficient C_n , Eq.(2.13) to construct the real value of x_p(t), Eq.(2.12), (chapter II). While Fourier transform analysis is performed to obtain the continuous spectrum representation of a given unperiodic signal (energy signal) x(t) which has infinite periodic time T_o , finite energy, and zero average power (chapter III). Fourier transform is also used in a limiting sense, to evaluate the frequency content of the periodic signal x_p(t). Moreover, there are some special periodic functions cannot be solved using Fourier series analysis such as the periodic Dirac delta function, in this case, Fourier transform is the only way to evaluate its frequency content.

Keywords: Spectral Analysis, Fourier transform, Power spectra, Periodic signals.