lecture13-f08

lecture13-f08 - Administrative EE264 Digital Signal...

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EE264 Digital Signal Processing Lecture 13 The Discrete-Fourier Transform November 5, 2008 Ronald W. Schafer Department of Electrical Engineering Stanford University STANFORD UNIVERSITY, EE264 Administrative • HW 5 due on Tuesday, Nov. 4 by 5pm in EE264 drawer on 2 nd floor Packard. • Mid-term exam: – I’m rusty! I’ll try to challenge you more on the final. – Average 89, about half above 90 • READ: Chapter 7. • 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 • Wrap up: FIR design by mini-max approximation – the Parks-McClellan algorithm • Summary comparison of filter designs • The discrete Fourier transform (DFT) – Exact representation of periodic sequence – Exact representation of finite-length sequence • DFT is sampled DTFT • Properties of DFT • Circular (i.e., periodic) convolution STANFORD UNIVERSITY, EE264 The Parks-McClellan Algorithm
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STANFORD UNIVERSITY, EE264 Linear Phase Type I FIR Filter • Zero-phase impulse response: • Frequency response: • Causal version: h e [ n ] = h e [ n ] L n L h [ n ] = h e [ n L ] H ( e j ω ) = A e ( e j ) e j L A e ( e j ) = h e [ n ] e j n n =− L L = h e [0] + h e [ n ] e j n + h e [ n ] e j n ( ) n = 1 L = h e + 2 h e [ n ]cos n n = 1 L = a k (cos ) k k = 0 L STANFORD UNIVERSITY, EE264 The Alternation Theorem • Weighted approximation error: • Minimize the maximum error over a set of frequencies; e.g., for a lowpass filter: • The optimum approximation alternates between + δ and - δ at least L +2 times in F . The maximum number of alternations is L +3. E ( ) = W ( ) H d ( e j ) A e ( e j ) [ ] E = max F E ( ) [] δ = min h e [ n ] E {} F = :0 p and s π { } STANFORD UNIVERSITY, EE264 Weighted Approximation Error E ( ) = W ( ) H d ( e j ) A e ( e j ) [ ] W ( ) = 2 1 0 p 1 s ) ( of ns" alternatio " 3 or 2 E L L + + STANFORD UNIVERSITY, EE264 Remez Demo
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STANFORD UNIVERSITY, EE264 Parks-McClellan Lowpass STANFORD UNIVERSITY, EE264 Parks-McClellan Lowpass STANFORD UNIVERSITY, EE264 A Design Example • The C-T specifications are ( 1/T= 2000 Hz): • The corresponding D-T specifications are: 0.99 | H eff ( Ω ) | 1.01, | Ω | 2 π (400) | H eff ( Ω )| 0.001, 2 (600) | Ω | 2 (1000) 0.99 | H ( e j ω 1.01, | | 0.4 | H ( e j 0.001, 0.6 | | i.e., Ω p = 2 (400) and Ω s = 2 (600). i.e., p p T = 0.4 and s s T = 0.6 . STANFORD UNIVERSITY, EE264 Design Formula • Kaiser obtained the following design formula by curve fitting many examples: • MATLAB example: » [M,Fo,Mo,W] = firpmord( [.4,.6], [1,0], [0.01,0.001], 2 ); » [h,delta]=firpm(M,Fo,Mo,W); M = 10log 10 δ 1 2 () 13 2.324 Δ
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STANFORD UNIVERSITY, EE264 Parks McClellan Lowpass Design M = 27 STANFORD UNIVERSITY, EE264 Parks McClellan D/A Compensated Filter
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lecture13-f08 - Administrative EE264 Digital Signal...

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