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**Unformatted text preview: **EE264 Digital Signal Processing Lecture 15 FFT and Spectrum Analysis November 12, 2008 Ronald W. Schafer Department of Electrical Engineering Stanford University STANFORD UNIVERSITY, EE264 Administrative HW 8 due on Tuesday, Nov. 18 by 5pm in EE264 drawer on 2 nd floor Packard. HW 7 due by 5pm Friday, Nov. 21. Mid-term exam: I still have a few papers. Pick them up after class. Average 89, about half above 90 READ: Chapter 9 and 10 Review Session: Thursdays 4:15 - 5pm Gates B03 (recording available online) Office Hours: RWS: Mon./Weds 10-11, and 12:15-12:45 Raunaq: Mon. 5-7pm, Tues. 9-11am Rahim: Friday 4-6pm Graders: Pegah Afshar, Ramin Miri STANFORD UNIVERSITY, EE264 Overview of Lecture Review: Decimation-in-Time FFT Decimation-in-frequency FFT Some FFT details Introduction to spectrum analysis using the DFT and FFT Anti-aliasing Sampling Windowing DFT STANFORD UNIVERSITY, EE264 Two ways of thinking about the DFT Exact representation of a periodic sequence Representation of a finite-length sequence X [ k ] = x [ n ] e j (2 / N ) kn k = N 1 = X [ k + N ] x [ n ] = 1 N X [ k ] e j (2 / N ) kn k = N 1 = x [ n + N ] X [ k ] = x [ n ] e j (2 / N ) kn k = N 1 k = 0,1, , N 1 x [ n ] = 1 N X [ k ] e j (2 / N ) kn k = N 1 n = 0,1, , N 1 STANFORD UNIVERSITY, EE264 Computation of the DFT In order for the DFT to be useful for linear filtering, nonlinear filtering, spectrum analysis, etc., we need efficient computation algorithms for Using the above directly requires N complex multiplications and N-1 complex additions for each of the N DFT values. X [ k ] = x [ n ] W N kn n = N 1 k = 0,1, , N 1 N = 1024 ( N ) 10 6 Number of multiplications = ( N ) = N 2 ) / 2 ( N j N e W = STANFORD UNIVERSITY, EE264 Decimation-in-Time FFT Algorithms STANFORD UNIVERSITY, EE264 Decimatation-in-Time Consider dividing the input sequence into two parts each of length N /2 samples. Then the computation of G [ k ] and H [ k ] would require only ( N /2) 2 operations each. Since x [ n ] can be expressed in terms of...

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