lec18-handout-6up

lec18-handout-6up - 6.003: Signals and Systems Lecture 18...

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Lecture 18 November 12, 2009 1 6.003: Signals and Systems CT Fourier Transform November 12, 2009 Mid-term Examination #3 Wednesday, November 18, 7:30-9:30pm, 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 1-hour exam; two hours to complete. Review sessions during open office hours. Conflict? Contact freeman@mit.edu by Friday, November 13, 2009. CT Fourier Transform Representing signals by their frequency content. X ( )= Ú −∞ x ( t ) e jωt dt (“analysis” equation) x ( t )= 1 2 π Ú −∞ X ( ) e jωt (“synthesis” equation) generalizes Fourier series to represent aperiodic signals. equals Laplace transform X ( s ) | s = 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 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 ( ) = 2 sin ω ω ω Check Yourself Signal x 2 ( t ) and its Fourier transform X 2 ( ) are shown below. 2 2 x 2 ( t ) 1 t b ω 0 X 2 ( ) ω Which is true? 1. b = 2 and ω 0 = π/ 2 2. b = 2 and ω 0 = 2 π 3. b = 4 and ω 0 = π/ 2 4. b = 4 and ω 0 = 2 π 5. none of the above Check Yourself Stretching time compresses frequency. Find a general scaling rule.
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lec18-handout-6up - 6.003: Signals and Systems Lecture 18...

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