lecture14-f08

lecture14-f08 - Administrative EE264 Digital Signal...

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EE264 Digital Signal Processing Lecture 14 Introduction to the FFT November 10, 2008 Ronald W. Schafer Department of Electrical Engineering Stanford University STANFORD UNIVERSITY, EE264 Administrative • HW 6 due on Tuesday, Nov. 11 by 5pm in EE264 drawer on 2 nd floor Packard. HW 7 due by 5pm Friday, Nov. 21. New HW 8 due Tuesday, Nov. 18 by 5pm . • Mid-term exam: – I still have a few papers. Pick them up after class. – Average 89, about half above 90 • READ: Chapter 8 (except 8.8). • 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 • Grader: Pegah Afshar, Ramin Miri STANFORD UNIVERSITY, EE264 Overview of Lecture • Review: DFT is sampled DTFT – Properties of DFT – Circular shift • Circular (i.e., periodic) convolution • Computing ordinary convolution by circular convolution • Computation of the DFT – The Fast Fourier Transform – Decimation-in-time STANFORD UNIVERSITY, EE264 The Discrete-Time Fourier Transform (DTFT)
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STANFORD UNIVERSITY, EE264 Review of the DTFT • Definition: • Inverse transform: • Periodicity: • Convolution theorems: X ( e j ω ) = x [ n ] e j n n =−∞ = X ( z ) z = e j x [ n ] = 1 2 π X ( e j ) e j n d X ( e j ( + 2 ) ) = X ( e j ) y [ n ] = x [ n ] h [ n ] Y ( e j ) = X ( e j ) H ( e j ) y [ n ] = w [ n ] x [ n ] Y ( e j ) = 1 2 W ( e j ) X ( e j ) STANFORD UNIVERSITY, EE264 The Discrete Fourier Transform (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 = 0 N 1 = ˜ X [ k + N ] ˜ x [ n ] = 1 N ˜ X [ k ] e j (2 / N ) kn k = 0 N 1 = ˜ x [ n + N ] X [ k ] = x [ n ] e j (2 / N ) kn k = 0 N 1 k = 0,1, , N 1 x [ n ] = 1 N X [ k ] e j (2 / N ) kn k = 0 N 1 n = 0,1, , N 1 STANFORD UNIVERSITY, EE264 Basic Properties of the DFT • If the DFT is evaluated outside of , it repeats periodically as • Circular shift • Circular convolution 0 n N 1 ˜ x [ n ] = x [(( n )) N ].
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lecture14-f08 - Administrative EE264 Digital Signal...

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