ABSTRACT
The Fourier transform is a mathematical tool that is used to expand signals into a spectrum of sinusoidal components to facilitate the signal representation and the analysis of system performance. In certain applications, the Fourier transform is used for spectral analysis, and while in others it is used for spectrum shaping that adjusts the relative contributions of different frequency components in the filtered result. In certain applications, the Fourier transform is used for its ability to decompose the input signal into uncorrelated components, so that signal processing can be more effectively implemented on the
individual spectral components. Different forms of the Fourier transform, such as the continuous-time (CT) Fourier series, the CT Fourier transform, the discrete-time (DT) Fourier transform (DTFT), the discrete Fourier transform (DFT), and the fast Fourier transform (FFT), are applicable in different circumstances. The goal of this section is to clearly define the various Fourier transforms, to discuss their properties, and to illustrate how each form is related to the others in the context of a family tree of Fourier signal processing methods.
