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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 118 homeworks 110 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 freeman@mit.edu by Friday, November 13, 2009. CT Fourier Transform Representing signals by their frequency content. X ( j )= x ( t ) e jt dt (analysis equation) x ( t )= 1 2 X ( j ) e jt d (synthesis equation) generalizes Fourier series to represent aperiodic signals. equals Laplace transform X ( s )  s = if ROC includes j axis. inherits properties of Laplace transform. complexvalued function of real domain . 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 . Time representation: 1 1 x 1 ( t ) 1 t Frequency representation: 2 X 1 ( j ) = 2 sin Check Yourself Signal x 2 ( t ) and its Fourier transform X 2 ( j ) are shown below. 2 2 x 2 ( t ) 1 t b X 2 ( j ) Which is true? 1. b = 2 and = / 2 2. b = 2 and = 2 3. b = 4 and = / 2 4. b = 4 and = 2 5. none of the above Check Yourself Find the Fourier transform. X 2 ( j ) = 2 2 e jt dt = e jt j 2 2 = 2 sin 2 = 4 sin 2 2 4 / 2 Check Yourself Signal x 2 ( t ) and its Fourier transform X 2 ( j ) are shown below. 2 2 x 2 ( t ) 1 t b X 2 ( j ) Which is true? 3 1. b = 2 and = / 2 2. b = 2 and = 2 3. b = 4 and = / 2 4. b = 4 and = 2 5. none of the above Fourier Transforms Stretching time compresses frequency. 1 1 x 1 ( t ) 1 t 2 X 1 ( j ) = 2 sin 2 2 x 2 ( t ) 1 t 4 / 2 X 2 ( j ) = 4 sin 2 2 Fourier Transforms Stretching time compresses frequency....
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 Spring '11
 DennisM.Freeman

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