MIT6_003S10_lec15_handout

MIT6_003S10_lec15_handout - 6.003: Signals and Systems...

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± 6.003: Signals and Systems Lecture 15 April 1, 2010 6.003: Signals and Systems Fourier Series April 1, 2010 Mid-term Examination #2 Wednesday, April 7, 7:30-9:30pm . No recitations on the day of the exam. Coverage: Lectures 1–15 Recitations 1–15 Homeworks 1–8 Homework 8 will not collected or graded. Solutions will be posted. Closed book: 2 pages 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. Last Time: Describing Signals by Frequency Content Harmonic content is natural way to describe some kinds of signals. Ex: musical instruments (http://theremin.music.uiowa.edu/MIS) piano piano t k violin violin t k bassoon bassoon t k π T π T Last Time: Fourier Series Determining harmonic components of a periodic signal. a k = 1 x ( t ) e j 2 kt dt (“analysis” equation) T T ² 2 x ( t )= x ( t + T )= a k e j kt (“synthesis” equation) k = −∞ We can think of Fourier series as an orthogonal decomposition . π T π T π T π T Orthogonal Decompositions Vector representation of 3-space: let r ¯ represent a vector with components { x , y , and z } in the { x ˆ , y ˆ , and z ˆ } directions, respectively. x r · x ˆ y r · y ˆ (“analysis” equations) z r · z ˆ r ¯ = xx ˆ + yy ˆ + zz ˆ (“synthesis” equation) Fourier series: let x ( t ) represent a signal with harmonic components { a 0 , a 1 , ... , a k } for harmonics { e j 0 t , e j 2 t , , e j 2 kt } respectively. a k = 1 x ( t ) e j 2 kt dt (“analysis” equation) T T ² 2 x ( t )= x ( t + T a k e j kt (“synthesis” equation) k = −∞ π T π T π T π T π T π T Orthogonal Decompositions Integrating over a period sifts out the k th component of the series.
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MIT6_003S10_lec15_handout - 6.003: Signals and Systems...

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