ABSTRACT
The Fourier transform is a mathematical tool that is used to expand signals into a spectrum of sinusoidal components to facilitate signal representation and the analysis of system performance. Different forms of the Fourier transform, such as the continuous-time Fourier series, the continuous-time Fourier transform, the discrete-time Fourier transform the discrete Fourier Transform and the fast Fourier transform are applicable in different circumstances. The class of discrete-time (DT) Fourier methods is particularly useful as a basis for digital signal processing (DSP) because it extends the theory of classical Fourier analysis to DT signals and leads to many effective algorithms that can be directly implemented on general computers or special purpose DSP devices. Although the complex form of the Fourier series expansion is useful for complex periodic signals, the Fourier series can be more easily expressed in terms of real-valued sine and cosine functions for real-valued periodic signals.
