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Unformatted text preview: 6.003: Signals and Systems CT Fourier Transform November 12, 2009 Midterm Examination #3 Wednesday, November 18, 7:309:30pm, Walker Memorial (this exam is after drop date). No recitations on the day of the exam. Coverage: cumulative with more emphasis on recent material lectures 1–18 homeworks 1–10 Homework 10 will not collected or graded. Solutions will be posted. Closed book: 3 page of notes ( 8 1 2 × 11 inches; front and back). Designed as 1hour exam; two hours to complete. Review sessions during open office hours. Conflict? Contact [email protected] by Friday, November 13, 2009. CT Fourier Transform Representing signals by their frequency content. X ( jω )= Ú ∞ −∞ x ( t ) e − jωt dt (“analysis” equation) x ( t )= 1 2 π Ú ∞ −∞ X ( jω ) e jωt dω (“synthesis” equation) • generalizes Fourier series to represent aperiodic signals. • equals Laplace transform X ( s )  s = ω if ROC includes jω axis. → inherits properties of Laplace transform. • complexvalued function of real domain ω . • simple ”inverse” relation → more general than tablelookup method for inverse Laplace. → “duality.” • filtering. • applications in physics Visualizing the Fourier Transform Fourier transform is a function of a single variable: frequency ω . Time representation: − 1 1 x 1 ( t ) 1 t Frequency representation: 2 π X 1 ( jω ) = 2 sin ω ω ω Check Yourself Signal x 2 ( t ) and its Fourier transform X 2 ( jω ) are shown below. − 2 2 x 2 ( t ) 1 t b ω X 2 ( jω ) ω Which is true? 1. b = 2 and ω = π/ 2 2. b = 2 and ω = 2 π 3. b = 4 and ω = π/ 2 4. b = 4 and ω = 2 π 5. none of the above Check Yourself Find the Fourier transform. X 2 ( jω ) = Ú 2 − 2 e − jωt dt = e − jωt − jω 2 − 2 = 2 sin 2 ω ω = 4 sin 2 ω 2 ω 4 π/ 2 ω Check Yourself Signal x 2 ( t ) and its Fourier transform X 2 ( jω ) are shown below. − 2 2 x 2 ( t ) 1 t b ω X 2 ( jω ) ω Which is true? 3 1. b = 2 and ω = π/ 2 2. b = 2 and ω = 2 π 3. b = 4 and ω = π/ 2 4. b = 4 and ω = 2 π 5. none of the above Fourier Transforms Stretching time compresses frequency. − 1 1 x 1 ( t ) 1 t 2 π X 1 ( jω ) = 2 sin ω ω ω − 2 2 x 2 ( t ) 1 t 4 π/ 2 X 2 ( jω ) = 4 sin 2 ω 2 ω ω Fourier Transforms Stretching time compresses frequency....
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This note was uploaded on 08/24/2011 for the course EECS 6.003 taught by Professor Dennism.freeman during the Spring '11 term at MIT.
 Spring '11
 DennisM.Freeman

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