MIT6_003S10_lec10

MIT6_003S10_lec10 - 6.003 Signals and Systems CT Frequency...

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6.003: Signals and Systems CT Frequency Response and Bode Plots March 9, 2010

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

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Frequency Response: H ( s ) | s | H ( ) | 9 H ( s )= 5 s p 1 ω 5 s -plane 5 0 5 H ( ) 5 σ π/ 2 5 5 5 5 2
Frequency Response: H ( s ) | s H ( s )=3 s z 1 | H 5 ( ) | s p 1 ω 5 s -plane 5 0 5 H ( ) 5 σ π/ 2 5 5 5 5 2

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

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

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

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Asymptotic Behavior: Isolated Pole Two asymptotes provide a good approxmation on log-log axes. 9 H ( s )= s p 1 log | H 9 ( /p ) | 1 | H ( ) | 0 5 1 2 1 log ω 5 0 5 2 1 0 1 2 p 1 9 lim | H ( ) | = ω 0 p 1 9 lim | H ( ) | = ω →∞ ω
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 2. shifting and scaling horizontally 3. shifting both horizontally and vertically 4. shifting and scaling both horizontally and vertically 5. none of the above

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Check Yourself Compare log-log plots of the frequency-response magnitudes of the following system functions: 1 1 H 1 ( s )= and H 2 ( s s +1 s +10 log | H ( ) | 0 | H 1 ( ) | 1 | H 2 ( ) | 1 2 log ω 2 1 0 1 2
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 3 1. shifting horizontally 2. shifting and scaling

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MIT6_003S10_lec10 - 6.003 Signals and Systems CT Frequency...

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