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lec15-handout-6up - 6.003 Signals and Systems Lecture 15...

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6.003: Signals and Systems Lecture 15 November 3, 2009 1 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. ω ω 0 2 ω 0 3 ω 0 4 ω 0 5 ω 0 6 ω 0 0 1 2 3 4 5 6 harmonic # DC fundamental second harmonic third harmonic fourth harmonic fifth harmonic sixth harmonic 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
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6.003: Signals and Systems Lecture 15 November 3, 2009 2 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 0 1 D D' D A D E 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 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 Harmonic Representations What signals can be represented by sums of harmonic components?
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