lec15 - 6.003: Signals and Systems Fourier Series November...

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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 time-domain techniques, especially with transient signals (e.g., impulse response). The primary focus for the next few weeks will be frequency-domain 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.

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lec15 - 6.003: Signals and Systems Fourier Series November...

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