解析傅里叶变换

本科毕业论文（设计）题目解析傅里叶变换2013年4月30日解析傅里叶变换西南大学电子信息工程学院，重庆400715摘要：傅里叶变换的实质就是将信号分解成不同频率复指数信号的叠加，由于复指数信号在LTI系统中的响应十分简单，且傅里叶变换具有多种极其有用的性质使得傅里叶变换在信号分析中得以广泛运用。信号的傅里叶变换有多种形式，且各种变换间具有很强联系：如非周期信号的周期性扩展，其傅里叶级数是原信号傅里叶变换的等间隔采样；若离散序列由连续信号周期取样得到，则离散序列的DTFT是连续信号FT以取样频率进行周期性扩展后的尺度变换。为了用计算机对信号进行傅里叶分析，而引入DFT，它是对信号时域和频域的采样。各种FFT算法的提出大大减少了DFT的运算次数，使DFT得以广泛应用，并极大促进了数字信号处理技术的发展。关键字：LTI系统；卷积；傅里叶变换；DFTAnalysisofFourierTransformLIUChaoyuanSchoolofElectronicandInformationEngineering,SouthwestChinaUniversity,Chongqing400715,ChinaAbstract:TheessenceoftheFouriertransformtosignalisdecomposedintodifferentfrequencyofcomplexexponentialsignals,duetotheresponseofcomplexexponentialsignalsinLTIsystemisverysimple,andtheFouriertransformhasmanyusefulpropertiesmakeFouriertransformcanbewidelyusedinsignalanalysis.SignalFouriertransformhasmanyforms,andhasastronglinkbetweeneachother:suchasperiodicextensionofnon-periodicsignal,theFourierseries(FS)istheoriginalsignalFouriertransform(FT)intervalsampling;ifthediscretesequenceobtainedbycontinuoussignalsampling,discretesequenceofDTFTisthescaletransformofthecontinuoussignalFTperiodicextensionbythesamplingfrequency.InordertousethecomputertoFourieranalysisofthesignal,thenintroductiontheDFT,whichisthesamplingofthesignalintimedomainandfrequencydomain.AVarietyofFFTalgorithmsproposedgreatlyreducingthenumberofDFTcalculations,soDFTcanbewidelyused,andgreatlypromotethedevelopmentofdigitalsignalprocessingtechnology.Keywords:LTIsystems;Convolution;FourierTransform;DFT目录第一章引言.....................................................................................................................................11.1历史背景...........................................................................................................................11.2本文所研究内容...............................................................................................................2第二章线性时不变系统（LTI系统）..........................................................................................32.1线性系统...........................................................................................................................32.2时不变系统.......................................................................................................................32.3LTI系统对信号的响应....................................................................................................42.3.1卷积.......................................................................................................................42.3.2LTI系统对复指数信号的响应............................................................................6第三章傅里叶变换.........................................................................................................................73.1连续与离散.......................................................................................................................73.1.1离散时间序列.......................................................................................................83.1.2连续时间复指数信号与离散时间复指数信号...................................................83.2连续时间周期信号与离散时间周期信号的傅里叶级数(FS).....................................103.3连续时间傅里叶变换（FT）与离散时间傅里叶变换（DTFT）.................................123.3.1连续时间傅里叶变换.........................................................................................123.3.2离散时间傅里叶变换.........................................................................................143.4连续时间傅里叶变换与离散时间傅里叶变换之间的关系.........................................163.5连续信号的离散化处理过程.........................................................................................183.6离散时间信号的采样与抽取.........................................................................................20第四章数字信号处理...................................................................................................................244.1周期序列的离散傅里叶级数.........................................................................................244.2离散傅里叶变换（DFT）...............................................................................................244.2.1频率域采样.........................................................................................................264.2.2循环卷积.............................................................................................................284.2.3基于DFT的信号频谱分析.................................................................................314.3快速傅里叶变换FFT......................................................................................................32第五章总结...................................................................................................................................37参考文献：.....................................................................................................................................38致谢：...............................................................................................................错误!未定义书签。1第一章引言1.1历史背景傅里叶变换是大家所熟悉的一种变换，又是一种令人感到陌生的变换！随着信号从模拟信号到数字信号，信号处理从模拟信号处理到数字信号处理，18世纪末和19世纪初诞生的傅里叶变换发生了巨大的变化。傅里叶变换的丰富和发展，极大地促进了信息科学的丰富和发展。现代的信息科学和技术也离不开傅里