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Unformatted text preview: 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 Ú T x ( t ) e − j 2 π T kt dt (“analysis” equation) x ( t )= x ( t + T ) = ∞ Ø k = −∞ a k e j 2 π T kt (“synthesis” equation) Separating harmonic components Underlying properties. 1. Multiplying two harmonics produces a new harmonic with the same fundamental frequency: e jkω t × e jlω t = e j ( k + l ) ω 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: Ú t + T t e jkω t dt ≡ Ú T e jkω t dt = , k Ó = 0 T, k = 0 = Tδ [ k ] Separating harmonic components Underlying properties. 1. Multiplying two harmonics produces a new harmonic with the same fundamental frequency: e jkω t × e jlω t = e j ( k + l ) ω 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: Ú t + T t e jkω t dt ≡ Ú T e jkω t dt = , k Ó = 0 T, k = 0 = Tδ [ k ] Orthogonality: harmonics are orthogonal ( ⊥ ) to each other. Fourier Series as Orthogonal Decompositions Analogy with vectors in 3space. Let ˆ x , ˆ y , and ˆ z represent direction vectors in 3space. Vector ¯ r can be expressed as sum of components x ˆ x + y ˆ y + z ˆ z where x = ¯ r · ˆ x y = ¯ r · ˆ y z = ¯ r · ˆ z Similarly for Fourier series (where basis functions are φ k ( t ) = e j 2 π T kt ), a signal can be expressed as a sum of orthogonal components:...
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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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