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lec16-handout-6up - 6.003 Signals and Systems Lecture 16...

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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 π T kt dt (“analysis” equation) x ( t )= x ( t + T ) = 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 0 t × e jlω 0 t = e j ( k + l ) ω 0 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 0 + T t 0 e jkω 0 t dt T e jkω 0 t dt = 0 , k = 0 T, k = 0 = [ k ] Separating harmonic components Underlying properties. 1. Multiplying two harmonics produces a new harmonic with the same fundamental frequency: e jkω 0 t × e jlω 0 t = e j ( k + l ) ω 0 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 0 + T t 0 e jkω 0 t dt T e jkω 0 t dt = 0 , k = 0 T, k = 0 = [ k ] Orthogonality: harmonics are orthogonal ( ) to each other. Fourier Series as Orthogonal Decompositions Analogy with vectors in 3-space. Let ˆ x , ˆ y , and ˆ z represent direction vectors in 3-space.

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lec16-handout-6up - 6.003 Signals and Systems Lecture 16...

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