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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.
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 harmonic third harmonic 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 harmonic fifth harmonic sixth harmonic
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
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 0 1 2 3 4 5 6 7 8 9 101112 0 1 2 3 4 5 6 7 8 9 101112 0 1 2 3 4 5 6 7 8 9 101112 time(periods of "D") D' A E 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 D D harmonics D
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.
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 jkω 0 t dt jkω 0 t dt = 0 , k t 0 e T e T, k = 0 = [ 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 .
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MIT6_003S10_lec14 - 6.003 Signals and Systems Fourier...

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