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Unformatted text preview: 6.003: Signals and Systems Fourier Series November 3, 2009 Fourier Representations Representations based on sinusoids. system signal in signal out To date, we have focused primarily on timedomain techniques, especially with transient signals (e.g., impulse response). The primary focus for the next few weeks will be frequencydomain techniques (e.g., frequency response) which concern eternal signals. Fourier Series Today: Fourier series represent signals in terms of sinusoids. This new representation for signals leads to a new representation for systems as filters . Harmonics Representing signals by the amplitudes and phases of harmonic com ponents. 2 3 4 5 6 1 2 3 4 5 6 harmonic # DC fundamental secondharmonic thirdharmonic fourthharmonic fifthharmonic sixthharmonic Musical Instruments Harmonic content is natural way to describe some kinds of signals. Ex: musical instruments (http://theremin.music.uiowa.edu/MIS) piano t cello t bassoon t oboe t horn t altosax t violin t bassoon t 1 252 seconds Musical Instruments Harmonic content is natural way to describe some kinds of signals. Ex: musical instruments (http://theremin.music.uiowa.edu/MIS) piano k cello k bassoon k oboe k horn k altosax 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 t piano k violin t violin k bassoon t bassoon k Harmonics Harmonic structure determines consonance and dissonance. octave (D+D) fifth (D+A) D+E time(periods of "D") harmonics 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 1 1 D D' D A D E 0 1 2 3 4 5 6 7 8 9 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 1 2 3 4 5 6 7 Harmonic Representations What signals can be represented by sums of harmonic components? 2 3 4 5 6 T = 2 t T = 2 t Only periodic signals: all harmonics of are periodic in T = 2 / . Harmonic Representations Is it possible to represent ALL periodic signals with harmonics? What about discontinuous signals? 2 t 2 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: e jk t e jl t = e j ( k + l ) t . 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 Assume that x ( t ) is periodic in T and is composed of a weighted sum of harmonics of = 2 /T ....
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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

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