01-08-2012, 02:55 PM
DISCRETE FOURIER TRANSFORM & ITS PROPERTIES WITH EXAMPLE
,The discrete Fourier transform (DFT) is a specific kind of discrete transform, used in Fourier analysis. It transforms one function into another, which is called the frequency domain representation, or simply the DFT, ofthe original function (which is often a function in the time domain). The DFT requires an input function that is discrete. Such inputs are often created by sampling a continuous function, such as a person's voice. The discrete input function must also have a limited (finite) duration, such as one period of a periodic sequence or a windowed segment of a longer sequence. Unlike the discrete-time Fourier transform (DTFT), the DFT only evaluates enough frequency components to reconstruct the finite segment that was analyzed. The inverse DFT cannot reproduce the entire time domain, unless the input happens to be periodic. Therefore it is often said that the DFT is a transform for Fourier analysis of finite-domain discrete-time functions
The input to the DFT is a finite sequence of real or complex numbers (with more abstract generalizations discussed below), making the DFT ideal for processing information stored in computers. In particular, the DFT is widely employed in signal processing and related fields to analyze the frequencies contained in a sampled signal, to solve partial differential equations, and to perform other operations such asconvolutions or multiplying large integers. A key enabling factor for these applications is the fact that the DFT can be computed efficiently in practice using a fast Fourier transform (FFT) algorithm.