傅里叶变换综述

A Survey of Fourier TransA Survey of Fourier Trans AbstractAbstract: As a linear integral trans tool, Fourier trans is becoming more and more widely used. In this paper, the origin of Fourier trans and classification and discrete Fourier trans and fast Fourier trans are explained and found that both in the field of mathematics or engineering applications are of great significance, is a very common Tools to help people solve problems. Key wordsKey words: Fourier transFFTDFTapplications ONE.ONE. Fourier trans origin Fourier trans is a linear integral trans.[1-3]Because its basic idea first by the French scholar Joseph Fourier systematically put forward, so named by its name to commemorate. Fourier trans is a kind of linear integral trans which is widely used in the fields of physics, acoustics, optics, structural dynamics, number theory, combinatorial mathematics, probability theory, statistics, signal processing, cryptography, oceanography and communication. [4-7]Applications. Fourier transs can express certain functions satisfying certain conditions as a linear combination of trigonometric functions (sine and / or cosine functions) or their integrals. In different research fields, Fourier trans has many different variants, such as continuous Fourier trans and discrete Fourier trans. The initial Fourier analysis was proposed as a tool for analytical analysis of thermal processes. The sine basis function is the eigenfunction of the differential operator, so that the solution of the linear differential equation can be transed into algebraic equation with constant coefficients. In a linear time-invariant physical system, the frequency is invariant in nature, so that the response of the system to complex stimuli can be obtained by combining its response to sinusoidal signals of different frequencies. TWOTWO. Fourier trans classification According to the different types of the original signal, we can divide the Fourier trans into four categories:[8-9] 1 non-periodic continuous signal Fourier trans (Fourier Trans) 2 periodic continuous signal Fourier series (Fourier Series) 3 Discrete Time Discrete Time Fourier Trans (Discrete Time Fourier Trans) 4 periodic discrete signal Discrete Fourier Trans (Discrete Fourier Trans) These four Fourier transs are for infinite and negative infinity of the signal, that is, the length of the signal is infinite, we know that this is not possible for computer processing, then there is no limited for the length of the Fourier trans it? No. Since a sine-cosine wave is defined from negative infinity to positive infinity, we can not combine a signal of infinite length into a signal of finite length. Faced with this difficulty, the is to limit the length of the signal is expressed as an infinite length of the signal, the signal can be infinitely extended from left to right, the extension of the part to zero, s