MIT6_003S10_lec10_handout

# MIT6_003S10_lec10_handout - 6.003 Signals and Systems...

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Unformatted text preview: 6.003: Signals and Systems Lecture 10 March 9, 2010 6.003: Signals and Systems CT Frequency Response and Bode Plots March 9, 2010 Frequency Response: H ( s ) | s ← jω | H ( jω ) | H ( s ) = 3 s − z 1 5 s − p 1 ω 5 s-plane − 5 5 ∠ H ( jω ) 5 σ π/ 2 − 5 − 5 5 − 5 − π/ 2 Asymptotic Behavior: Isolated Zero The magnitude response is simple at low and high frequencies. | H ( jω ) | H ( jω ) = jω − z 1 ω 5 ω 5 z 1 − 5 5 ∠ H ( jω ) − 5 5 σ π/ 2 − 5 5 − 5 − π/ 2 Last Time Complex exponentials are eigenfunctions of LTI systems. H ( s ) e s t H ( s ) e s t H ( s ) can be determined graphically using vectorial analysis. H ( s ) = K ( s − z )( s − z 1 )( s − z 2 ) ··· ( s − p )( s − p 1 )( s − p 2 ) ··· z z s − z s s-plane s Response of an LTI system to an eternal cosine is an eternal cosine: same frequency, but scaled and shifted. H ( s ) cos( ω t ) | H ( jω ) | cos ( ω t + ∠ H ( jω ) ) 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. Asymptotic Behavior: Isolated Zero Two asymptotes provide a good approxmation on log-log axes. H ( s ) = s − z 1 log | H ( jω ) | | H ( jω ) | 2 z 1 5 1 1 log ω − 5 5 − 2 − 1 1 2 z 1 lim | H ( jω ) | = z 1 ω → lim | H ( jω ) | = ω ω →∞ 1 6.003: Signals and Systems Lecture 10 March 9, 2010 Asymptotic Behavior: Isolated Pole Asymptotic Behavior: Isolated Pole Two asymptotes provide a good approxmation on log-log axes. The magnitude response is simple at low and high frequencies. 9 9 9 | H ( jω ) | H ( s ) = s − p 1 H ( s ) = 5 ω s − p 1 9 log | H ( jω ) | ω 9 /p 1 5 p 1 | H ( jω ) | 5 − 5 5 − 1 − 1 ∠ H ( jω ) − 2 5 σ π/ 2 − 5 log ω − 5 5 − 2 − 1 1 2 p 1 − 5 5 9 lim | H ( jω ) | = ω → p 1 − 5 − π/ 2 9 lim | H ( jω ) | = ω →∞ ω Check Yourself Asymptotic Behavior of More Complicated Systems Compare log-log plots of the frequency-response magnitudes of the following system functions:...
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MIT6_003S10_lec10_handout - 6.003 Signals and Systems...

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