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

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