MIT6_003S10_lec17

MIT6_003S10_lec17 - 6.003: Signals and Systems CT Fourier...

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6.003: Signals and Systems CT Fourier Transform April 8, 2010
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CT Fourier Transform Representing signals by their frequency content. X ( )= x ( t ) e jωt dt (“analysis” equation) −∞ 1 x ( t )= 2 π −∞ X ( ) e (“synthesis” equation) generalizes Fourier series to represent aperiodic signals. equals Laplace transform X ( s ) | s =j ω if ROC includes axis. inherits properties of Laplace transform. complex-valued function of real domain ω . simple ”inverse” relation more general than table-lookup method for inverse Laplace. “duality.” filtering . applications in physics .
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Filtering Notion of a filter. LTI systems cannot create new frequencies. can only scale magnitudes and shift phases of existing components. Example: Low-Pass Filtering with an RC circuit + v i + v o R C
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Lowpass Filter Calculate the frequency response of an RC circuit. KVL: v i ( t )= Ri ( t )+ v o ( t ) R + C: i ( t ) = Cv ˙ o ( t ) + v i C Solving: v i ( t RCv ˙ o ( t v o ( t ) v o V i ( s ) = (1 + sRC ) V o ( s ) H ( s V o ( s ) = 1 V i ( s ) 1+ sRC 1 0 . 1 | H ( ) 0 . 01 ω 0 . 01 0 . 1 1 10 100 1 /RC 0 π 2 ω 0 . 01 0 . 1 1 10 100 1 /RC
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Lowpass Filtering Let the input be a square wave. t 1 2 1 2 0 T 1 2 π x ( t )= e 0 kt ; ω 0 = k odd jπk T 1 | X ( ) 0 . 1 0 . 01 ω 0 . 01 0 . 1 1 10 100 1 /RC 0 π 2 ω 0 . 01 0 . 1 1 10 100 1 /RC
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Lowpass Filtering Low frequency square wave: ω 0 << 1 /RC . t 1 2 1 2 0 T 1 2 π x ( t )= e 0 kt ; ω 0 = k odd jπk T 1 | H ( ) 0 . 1 0 . 01 ω 0 . 01 0 . 1 1 10 100 1 /RC 0 π 2 ω 0 . 01 0 . 1 1 10 100 1 /RC
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Lowpass Filtering Higher frequency square wave: ω 0 < 1 /RC . t 1 2 1 2 0 T 1 2 π x ( t )= e 0 kt ; ω 0 = k odd jπk T 1 | H ( ) 0 . 1 0 . 01 ω 0 . 01 0 . 1 1 10 100 1 /RC 0 π 2 ω 0 . 01 0 . 1 1 10 100 1 /RC
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Lowpass Filtering Still higher frequency square wave: ω 0 =1 /RC . t 1 2 1 2 0 T 1 0 kt ; 2 π x ( t )= e ω 0 = k odd jπk T 0 . 01 0 . 1 1 ω | H ( ) π 2 0 . 01 0 . 1 1 10 100 1 /RC 0 ω 0 . 01 0 . 1 1 10 100 1 /RC
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Lowpass Filtering High frequency square wave: ω 0 > 1 /RC . t 1 2 1 2 0 T 1 0 kt ; 2 π x ( t )= e ω 0 = k odd jπk T 0 . 01 0 . 1 1 ω | H ( ) π 2 0 . 01 0 . 1 1 10 100 1 /RC 0 ω 0 . 01 0 . 1 1 10 100 1 /RC
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Source-Filter Model of Speech Production Vibrations of the vocal cords are “filtered” by the mouth and nasal cavities to generate speech. buzz from throat and speech vocal cords nasal cavities
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Filtering LTI systems “filter” signals based on their frequency content. Fourier transforms represent signals as sums of complex exponen- tials. 1 x ( t )= 2 π −∞ X ( ) e jωt Complex exponentials are eigenfunctions of systems. e H ( ) e systems “filter” signals by adjusting the amplitudes and phases of each frequency component.
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This note was uploaded on 12/14/2011 for the course EE 6.003 taught by Professor Freeman during the Fall '11 term at MIT.

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MIT6_003S10_lec17 - 6.003: Signals and Systems CT Fourier...

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