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# lec12 - 6.003 Signals and Systems CT Frequency Response and...

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6.003: Signals and Systems CT Frequency Response and Bode Plots October 22, 2009

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Mid-term Examination #2 Wednesday, October 28, 7:30-9:30pm, Walker Memorial. No recitations on the day of the exam. Coverage: cumulative with more emphasis on recent material lectures 1–12 homeworks 1–7 Homework 7 includes practice problems for mid-term 2. It will not collected or graded. Solutions will be posted. Closed book: 2 pages of notes ( 8 1 2 × 11 inches; front and back). Designed as 1-hour exam; two hours to complete. Review sessions Monday 5-6pm and 8-9pm in 32-044. Conflict? Contact [email protected] by Friday, October 23, 5pm.
Last Time Complex exponentials are eigenfunctions of LTI systems. H ( s ) e s 0 t H ( s 0 ) e s 0 t H ( s 0 ) can be determined graphically using vectorial analysis. H ( s 0 ) = K ( s 0 z 0 )( s 0 z 1 )( s 0 z 2 ) · · · ( s 0 p 0 )( s 0 p 1 )( s 0 p 2 ) · · · z 0 z 0 s 0 z 0 s 0 s -plane s 0 Response of an LTI system to an eternal cosine is an eternal cosine: same frequency, but scaled and shifted. H ( s ) cos( ω 0 t ) | H ( 0 ) | cos ( ω 0 t + H ( 0 ) )

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Frequency Response: H ( s ) | s s -plane σ ω 5 5 5 5 H ( s ) = s z 1 5 0 5 5 | H ( ) | 5 5 π/ 2 π/ 2 H ( )
Frequency Response: H ( s ) | s s -plane σ ω 5 5 5 5 H ( s ) = 9 s p 1 5 0 5 5 | H ( ) | 5 5 π/ 2 π/ 2 H ( )

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Frequency Response: H ( s ) | s s -plane σ ω 5 5 5 5 H ( s ) = 3 s z 1 s p 1 5 0 5 5 | H ( ) | 5 5 π/ 2 π/ 2 H ( )
Poles and Zeros Thinking about systems as collections of poles and zeros is an im- portant design concept. simple: just a few numbers characterize entire system powerful: complete information about frequency response Today: poles, zeros, frequency responses, and Bode plots.

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Asymptotic Behavior: Isolated Zero The magnitude response is simple at low and high frequencies. σ ω 5 5 5 5 H ( ) = z 1 5 0 5 5 | H ( ) | 5 5 π/ 2 π/ 2 H ( )
Asymptotic Behavior: Isolated Zero The magnitude response is simple at low and high frequencies. σ ω 5 5 5 5 H ( ) = z 1 5 0 5 5 | H ( ) | z 1 5 5 π/ 2 π/ 2 H ( )

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Asymptotic Behavior: Isolated Zero The magnitude response is simple at low and high frequencies. σ ω 5 5 5 5 H ( ) = z 1 5 0 5 5 | H ( ) | z 1 ω 5 5 π/ 2 π/ 2 H ( )
Asymptotic Behavior: Isolated Zero Two asymptotes provide a good approxmation on log-log axes. H ( s ) = s z 1 5 0 5 5 | H ( ) | 2 1 0 1 2 2 1 0 log ω z 1 1 log | H ( ) | z 1 lim ω 0 | H ( ) | = z 1 lim ω →∞ | H ( ) | = ω

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Asymptotic Behavior: Isolated Pole The magnitude response is simple at low and high frequencies. σ ω 5 5 5 5 H ( s ) = 9 s p 1 5 0 5 5 | H ( ) | 9 p 1 9 ω 5 5 π/ 2 π/ 2 H ( )
Asymptotic Behavior: Isolated Pole Two asymptotes provide a good approxmation on log-log axes. H ( s ) = 9 s p 1 5 0 5 5 | H ( ) | 2 1 0 1 2 0 1 2 log ω p 1 1 log | H ( ) | 9 /p 1 lim ω 0 | H ( ) | = 9 p 1 lim ω →∞ | H ( ) | = 9 ω

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Check Yourself Compare log-log plots of the frequency-response magnitudes of the following system functions: H 1 ( s ) = 1 s + 1 and H 2 ( s ) = 1 s + 10 The former can be transformed into the latter by 1. shifting horizontally
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lec12 - 6.003 Signals and Systems CT Frequency Response and...

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