MIT6_003S10_lec15

MIT6_003S10_lec15 - 6.003 Signals and Systems Fourier...

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6.003: Signals and Systems Fourier Series April 1, 2010

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Mid-term Examination #2 Wednesday, April 7, 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. 1 8 Closed book: 2 pages of notes ( 2 × 11 inches; front and back). Designed as 1-hour exam; two hours to complete. Review sessions during open oﬃce hours. 7:30-9:30pm.
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 t k bassoon

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Last Time: Fourier Series Determining harmonic components of a periodic signal. a k = 1 x ( t ) e j 2 T π kt dt (“analysis” equation) T T ± 2 π x ( t )= x ( t + T )= a k e j T (“synthesis” equation) k = −∞
Last Time: Fourier Series Determining harmonic components of a periodic signal. a k = 1 x ( t ) e j 2 T π kt dt (“analysis” equation) T T ± 2 π x ( t )= x ( t + T )= a k e j T (“synthesis” equation) k = −∞ We can think of Fourier series as an orthogonal decomposition .

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a k = 1 T T x ( t ) e j 2 π T kt dt (“analysis” equation) x ( t )= x ( t + T )= ± k = −∞ a k e j 2 π T (“synthesis” equation) 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)
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 π t , , e j 2 T π kt } respectively. a k = 1 x ( t ) e j 2 T π dt (“analysis” equation) T T ± 2 π x ( t )= x ( t + T )= a k e j T (“synthesis” equation) k = −∞

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Orthogonal Decompositions ˆ + yy ˆ + zz 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.
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This note was uploaded on 12/14/2011 for the course EE 6.003 taught by Professor Freeman during the Fall '11 term at MIT.

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MIT6_003S10_lec15 - 6.003 Signals and Systems Fourier...

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