Fourier Transform and Energy Spectra

In the time domain, the signal of constant amplitude does not contain frequency components while the signal of variable amplitude do contain frequency components. Fourier transform (Fourier integrals) is used for evaluating the frequency contents of the unperiodic signals and in a limiting sense for the periodic signals. Fourier transform provides a means of analyzing and designing frequency selective filters for the separation of signals on the basis of their frequency contents. This chapter is involved for performing the Fourier transform to obtain the continuous frequency spectrum of a given unperiodic signal x(t) such as energy signals and deterministic signals which has infinite periodic time To , finite energy, and zero average power. Fourier transform maps the signal x(t) from the time domain to its frequency domain X(f) in terms of the exponential functions. The waveform x(t) and its Fourier transform X(f) are separate aspects, dependent of each other and are related by Fourier transform analysis. Also in a limiting sense, Fourier transform represents the periodic signals such as power signals and random signals as an infinite sum of discrete sinusoidal wave components (chapter V).

Keywords: Fourier transform, Energy spectra, Rayleigh`s Energy theorem, Energy spectral density.