[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-0-en-105":3,"doc-seo-268246-105":59,"doc-detail-268246-en":123},{"code":4,"msg":5,"data":6},0,"success",[7,13,18,23,28,33,38,43,48,51,55],{"id":8,"doc_module":4,"doc_module_name":9,"category_name":10,"show_sort_weight":11,"slug":12},1,"Document","Story & Novel",90,"story-novel",{"id":14,"doc_module":4,"doc_module_name":9,"category_name":15,"show_sort_weight":16,"slug":17},2,"Literature",80,"literature",{"id":19,"doc_module":4,"doc_module_name":9,"category_name":20,"show_sort_weight":21,"slug":22},4,"Exam",70,"exam",{"id":24,"doc_module":4,"doc_module_name":9,"category_name":25,"show_sort_weight":26,"slug":27},5,"Comic",60,"comic",{"id":29,"doc_module":4,"doc_module_name":9,"category_name":30,"show_sort_weight":31,"slug":32},6,"Technology",50,"technology",{"id":34,"doc_module":4,"doc_module_name":9,"category_name":35,"show_sort_weight":36,"slug":37},7,"Healthcare",40,"healthcare",{"id":39,"doc_module":4,"doc_module_name":9,"category_name":40,"show_sort_weight":41,"slug":42},8,"Research & Report",30,"research-report",{"id":44,"doc_module":4,"doc_module_name":9,"category_name":45,"show_sort_weight":46,"slug":47},9,"Religion & Spirituality",20,"religion-spirituality",{"id":46,"doc_module":4,"doc_module_name":9,"category_name":49,"show_sort_weight":46,"slug":50},"World Cup","world-cup",{"id":52,"doc_module":4,"doc_module_name":9,"category_name":53,"show_sort_weight":52,"slug":54},10,"Lifestyle","lifestyle",{"id":56,"doc_module":4,"doc_module_name":9,"category_name":57,"show_sort_weight":24,"slug":58},19,"General","general",{"code":4,"msg":60,"data":61},"ok",{"site_id":62,"language":63,"slug":64,"title":65,"keywords":66,"description":67,"schema_data":68,"social_meta":116,"head_meta":118,"extra_data":120,"updated_unix":122},105,"en","ee-261-the-fourier-transform-and-its-applications","EE 261 - The Fourier Transform and its Applications","","EE 261 lecture notes focus on the Fourier series, Fourier transform, and their applications in electrical engineering and signal processing. The material develops periodic phenomena, orthogonality, convergence behavior, and practical Fourier series analysis. It then extends to the Fourier transform, convolution and filtering, and connections to differential equations and probability via the central limit theorem. Later sections treat distributions, transforms of tempered distributions, and key theorems like shift and scaling, before covering sampling, interpolation, and the discrete Fourier transform including inversion and properties.",{"@graph":69,"@context":115},[70,84,106],{"@type":71,"itemListElement":72},"BreadcrumbList",[73,77,79,82],{"item":74,"name":75,"@type":76,"position":8},"https://docshare.wps.com","Home","ListItem",{"item":78,"name":9,"@type":76,"position":14},"https://docshare.wps.com/document/",{"item":80,"name":30,"@type":76,"position":81},"https://docshare.wps.com/document/technology/",3,{"item":83,"name":65,"@type":76,"position":19},"https://docshare.wps.com/document/ee-261-the-fourier-transform-and-its-applications/268246/",{"url":83,"name":65,"@type":85,"image":86,"author":91,"headline":65,"publisher":94,"fileFormat":97,"inLanguage":63,"description":67,"dateModified":98,"datePublished":99,"encodingFormat":97,"isAccessibleForFree":100,"interactionStatistic":101},"DigitalDocument",{"url":87,"@type":88,"width":89,"height":90},"https://docshare.wps.com/thumbnails/ee-261-the-fourier-transform-and-its-applications/268246.png","ImageObject",300,407,{"name":92,"@type":93},"Riley","Person",{"url":74,"name":95,"@type":96},"DocShare","Organization","application/pdf","2026-10-20","2026-09-14",true,{"@type":102,"interactionType":103,"userInteractionCount":105},"InteractionCounter",{"@type":104},"ViewAction",15,{"@type":107,"mainEntity":108},"FAQPage",[109],{"name":110,"@type":111,"acceptedAnswer":112},"What additional mathematical framework is introduced after the Fourier transform sections?","Question",{"text":113,"@type":114},"The notes introduce distributions and their Fourier transforms, including the Fourier transform of tempered distributions and operations on distributions. They also cover theorems such as shifts, scaling, and derivative-related results.","Answer","https://schema.org",{"og:url":83,"og:type":117,"og:title":65,"og:site_name":95,"og:description":67},"article",{"robots":119,"canonical":83},"index,follow",{"doc_id":121,"site_id":62},268246,1789429622,{"code":4,"msg":5,"data":124},{"doc_id":121,"user_id":125,"nickname":92,"user_avatar":126,"doc_module":4,"category_id":29,"category_name":30,"doc_title":65,"doc_description":67,"doc_content":127,"file_id":128,"file_url":129,"file_type":130,"file_size":131,"view_count":105,"is_deleted":4,"is_public":8,"is_downloadable":8,"audit_status":8,"page_count":132,"language":133,"language_code":63,"site_id":62,"html_lang":63,"table_of_contents":134,"faqs":135,"seo_title":136,"seo_description":67,"update_tm":122,"read_time":137},1374391975076,"https://ap-avatar.wpscdn.com/avatar/14000253ca4ec9f6853?x-image-process=image/resize,m_fixed,w_180,h_180&k=1783305029341752051","Lecture Notes for  \nEE 261  \nThe Fourier Transform and its Applications  \nProf. Brad Osgood  \nElectrical Engineering Department Stanford University  \nContents  \n1 Fourier Series 1  \n1.1 Introduction and Choices to Make ................................ 1  \n1.2 Periodic Phenomena ........................................ 2  \n1.3 Periodicity: Definitions, Examples, and Things to Come .................... 4  \n1.4 It All Adds Up ........................................... 9  \n1.5 Lost at c ............................................... 10  \n1.6 Period, Frequencies, and Spectrum ................................ 13  \n1.7 Two Examples and a Warning ................................... 16  \n1.8 The Math, the Majesty, the End ................................. 21  \n1.9 Orthogonality ............................................ 26  \n1.10 Appendix: The Cauchy-Schwarz Inequality and its Consequences ............... 33  \n1.11 Appendix: More on the Complex Inner Product ......................... 36  \n1.12 Appendix: Best L2 Approximation by Finite Fourier Series .................. 38  \n1.13 Fourier Series in Action ...................................... 39  \n1.14 Notes on Convergence of Fourier Series .............................. 50  \n1.15 Appendix: Pointwise Convergence vs. Uniform Convergence .................. 58  \n1.16 Appendix: Studying Partial Sums via the Dirichlet Kernel: The Buzz Is Back ........ 59  \n1.17 Appendix: The Complex Exponentials Are a Basis for L2 ([0 , 1]) ................ 61  \n1.18 Appendix: More on the Gibbs Phenomenon ........................... 62  \n2 Fourier Transform 65  \n2.1 A First Look at the Fourier Transform .............................. 65  \n2.2 Getting to Know Your Fourier Transform ............................ 75  \n3 Convolution 95  \n3.1 A ∗ is Born ............................................. 95  \n3.2 What is Convolution, Really? ................................... 99  \n3.3 Properties of Convolution: It’s a Lot like Multiplication .................... 101  \nii CONTENTS  \n3.4 Convolution in Action I: A Little Bit on Filtering ........................ 102  \n3.5 Convolution in Action II: Differential Equations ......................... 106  \n3.6 Convolution in Action III: The Central Limit Theorem ..................... 116  \n3.7 The Central Limit Theorem: The Bell Curve Tolls for Thee .................. 128  \n3.8 Fourier transform formulas under different normalizations ................... 130  \n3.9 Appendix: The Mean and Standard Deviation for the Sum of Random Variables ...... 131  \n3.10 More Details on the Central Limit Theorem ........................... 132  \n3.11 Appendix: Heisenberg’s Inequality ................................ 133  \n4 Distributions and Their Fourier Transforms 137  \n4.1 The Day of Reckoning ....................................... 137  \n4.2 The Right Functions for Fourier Transforms: Rapidly Decreasing Functions ......... 142  \n4.3 A Very Little on Integrals ..................................... 148  \n4.4 Distributions ............................................ 152  \n4.5 A Physical Analogy for Distributions ............................... 164  \n4.6 Limits of Distributions ....................................... 165  \n4.7 The Fourier Transform of a Tempered Distribution ....................... 168  \n4.8 Fluxions Finis: The End of Differential Calculus ........................ 174  \n4.9 Approximations of Distributions ................................. 177  \n4.10 The Generalized Fourier Transform Includes the Classical Fourier Transform ........ 178  \n4.11 Operations on Distributions and Fourier Transforms ...................... 179  \n4.12 Duality, Changing Signs, Evenness and Oddness ........................ 179  \n4.13 A Function Times a Distribution Makes Sense .......................... 182  \n4.14 The Derivative Theorem ...................................... 185  \n4.15 Shifts and the Shift Theorem ................................... 186  \n4.16 Scaling and the Stretch Theorem ..........","cbCaiuI7pzv6Slwp","https://ap.wps.com/l/cbCaiuI7pzv6Slwp","pdf",31520130,428,"English","# Contents\n## Fourier Series\n## Fourier Transform\n## Convolution\n## Distributions and Their Fourier Transforms\n## III, Sampling, and Interpolation\n## Discrete Fourier Transform","[{\"question\":\"What additional mathematical framework is introduced after the Fourier transform sections?\",\"answer\":\"The notes introduce distributions and their Fourier transforms, including the Fourier transform of tempered distributions and operations on distributions. They also cover theorems such as shifts, scaling, and derivative-related results.\"}]","EE 261 - The Fourier Transform and its Applications | PDF",1079]