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Unformatted text preview: 18791 Lecture #17INTRODUCTION TO THE FAST FOURIER TRANSFORM ALGORITHMDepartment of Electrical and Computer EngineeringCarnegie Mellon UniversityPittsburgh, Pennsylvania 15213Phone: +1 (412) 2682535FAX: +1 (412) 2683890rms@cs.cmu.eduhttp://www.ece.cmu.edu/~rmsOctober 24, 2005Richard M. SternCarnegieMellonSlide 2ECE DepartmentIntroductionToday we will begin our discussion of the family of algorithms known as Fast Fourier Transforms, which have revolutionized digital signal processingWhat is the FFT?A collection of tricks that exploit the symmetry of the DFT calculation to make its execution much fasterSpeedup increases with DFT sizeToday  will outline the basic workings of the simplest formulation, the radix2 decimationintime algorithmThursday  will discuss some of the variations and extensionsAlternate structuresNonradix 2 formulationsCarnegieMellonSlide 3ECE DepartmentIntroduction, continuedSome dates:~1880  algorithm first described by Gauss1965  algorithm rediscovered (not for the first time) by Cooley and Tukey In 1967 (spring of my freshman year), calculation of a 8192point DFT on the topofthe line IBM 7094 took .~30 minutes using conventional techniques~5 seconds using FFTsCarnegieMellonSlide 4ECE DepartmentMeasures of computational efficiencyCould considerNumber of additionsNumber of multiplicationsAmount of memory requiredScalability and regularityFor the present discussion well focus most on number of multiplications as a measure of computational complexityMore costly than additions for fixedpoint processorsSame cost as additions for floatingpoint processors, but number of operations is comparableCarnegieMellonSlide 5ECE DepartmentComputational Cost of DiscreteTime FilteringConvolution of an Npoint input with an Mpoint unit sample response ....
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 Spring '12
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 Algorithms

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