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Unformatted text preview: 6.003: Signals and Systems Lecture 16 November 5, 2009 1 6.003: Signals and Systems Fourier Series November 5, 2009 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 t piano k violin t violin k bassoon t bassoon k Last Time: Fourier Series Determining harmonic components of a periodic signal. a k = 1 T T x ( t ) e j 2 T kt dt (analysis equation) x ( t )= x ( t + T ) = k = a k e j 2 T kt (synthesis equation) Separating harmonic components Underlying properties. 1. Multiplying two harmonics produces a new harmonic with the same fundamental frequency: e jk t e jl t = e j ( k + l ) t . Closure: the set of harmonics is closed under multiplication. 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: t + T t e jk t dt T e jk t dt = , k = 0 T, k = 0 = T [ k ] Separating harmonic components Underlying properties. 1. Multiplying two harmonics produces a new harmonic with the same fundamental frequency: e jk t e jl t = e j ( k + l ) t . Closure: the set of harmonics is closed under multiplication. 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: t + T t e jk t dt T e jk t dt = , k = 0 T, k = 0 = T [ k ] Orthogonality: harmonics are orthogonal ( ) to each other....
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This note was uploaded on 08/24/2011 for the course EECS 6.003 taught by Professor Dennism.freeman during the Spring '11 term at MIT.
 Spring '11
 DennisM.Freeman
 Frequency

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