Discrete Fourier Analysis

TLDR: Since periodic discrete-time signals have a periodic and discrete-frequency transform the Fourier series is a special case of the DFT, which in turn is implemented very efficiently by the Fast Fourier Transform (FFT) algorithm.

Abstract: If the Z-transform of a signal or the transfer function of a system is defined on the unit circle then the Discrete-Time Fourier Transform (DTFT) of the signal or the frequency response of the system are obtained. Two computational disadvantages of the DTFT, being a function of a continuously varying frequency and requiring integration for the inversion, are removed by sampling in frequency and resulting in the Discrete Fourier Transform (DFT). Since periodic discrete-time signals have a periodic and discrete-frequency transform the Fourier series is a special case of the DFT. Circular representation, circular shift and circular convolution characterize the DFT. Thus, periodic or aperiodic signals can be represented and processed by the DFT, which in turn is implemented very efficiently by the Fast Fourier Transform (FFT) algorithm. Basic theory and application of the FFT are introduced. Fourier representation and processing of two-dimensional signals and systems are similar to those in one dimension. The use of transforms for data compression is illustrated by the discrete cosine transform, which represents the signal efficiently using real-valued coefficients. MATLAB is used for computation of the transforms and processing of one- and two-dimensional signals.