MIT6_003S10_lec14

MIT6_003S10_lec14 - 6.003: Signals and Systems Fourier...

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6.003: Signals and Systems Fourier Representations March 30, 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.
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Fourier Representations Fourier series represent signals in terms of sinusoids . leads to a new representation for systems as filters .
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Fourier Series Representing signals by their harmonic components. DC fundamental second ha rmonic third ω ω 0 2 ω 0 3 ω 0 4 ω 0 5 ω 0 6 ω 0 0 1 2 3 4 5 6 harmonic # fourth fifth sixth
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bassoon t Musical Instruments Harmonic content is natural way to describe some kinds of signals. Ex: musical instruments (http://theremin.music.uiowa.edu/MIS) piano cello bassoon t t t oboe horn altosax t t t violin t 1 252 seconds
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Musical Instruments Harmonic content is natural way to describe some kinds of signals. Ex: musical instruments (http://theremin.music.uiowa.edu/MIS) piano cello bassoon k k k oboe horn altosax k k k violin k
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Musical Instruments 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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Harmonics Harmonic structure determines consonance and dissonance. octave (D+D’) fifth (D+A) D+E ± –1 0 1 0123456789101112 time(periods of "D") D' A E 01234567 0123456789 D D harmonics D
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Harmonic Representations What signals can be represented by sums of harmonic components? ω ω 0 2 ω 0 3 ω 0 4 ω 0 5 ω 0 6 ω 0 T = 2 π ω 0 T = 2 π ω 0 t t Only periodic signals: all harmonics of ω 0 are periodic in T =2 π/ω 0 .
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Harmonic Representations Is it possible to represent ALL periodic signals with harmonics? What about discontinuous signals? 2 π ω 0 t 2 π ω 0 t Fourier claimed YES even though all harmonics are continuous! Lagrange ridiculed the idea that a discontinuous signal could be written as a sum of continuous signals. We will assume the answer is YES and see if the answer makes sense.
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± ± Separating harmonic components Underlying properties. 1. Multiplying two harmonics produces a new harmonic with the same fundamental frequency: jkω 0 t jlω 0 t = e j ( k + l ) ω 0 t e × e . 2. The integral of a harmonic over any time interval with length equal to a period T is zero unless the harmonic is at DC: =0 t 0 + T 0 t dt 0 t dt = 0 ,k ± t 0 e T e T, k = [ k ]
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Separating harmonic components Assume that x t is periodic in T and is composed of a weighted sum ( ) of harmonics of ω 0 =2 π/T . ± x ( t )= x ( t + T a k e 0 kt k = −∞ Then ± x ( t ) e jlω 0 t dt = a k e 0 e 0 lt dt T T k = −∞ = ± a k e 0 ( k l ) t dt k = −∞ T ± = a k [ k l ]= Ta l k = −∞ Therefore 1 1 x ( t ) e j π T 2 x ( t ) e 0 dt = = dt a k T T T T
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Fourier Series Determining harmonic components of a periodic signal.
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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_lec14 - 6.003: Signals and Systems Fourier...

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