Intro to Fourier Series BY JORDAN KEARNS (W&L ‘14) & JON ERICKSON (STILL HERE )

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Presentation transcript:

Intro to Fourier Series BY JORDAN KEARNS (W&L ‘14) & JON ERICKSON (STILL HERE )

220 Hz (A3) Why do they sound different? Instrument 1 Instrument 2Sine Wave

Waveform Piano Guitar Sine Wave ToTo

Overtones and Music Perception Overtones occur at integer multiples of the fundamental frequency when an object vibrates. The addition of these tones at regular intervals is musical to the human ear. Example: Fundamental (1 st Harmonic): 220Hz 1 st Overtone (2 nd Harmonic): 440Hz 2 nd Overtone (3 rd Harmonic): 660Hz Video produced by Brandon Pletsch Univ. of Georgia Medical School URL:

PianoGuitar Frequency Spectrum

Frequency Decomposition: Pure Sine Wave T = 2ms f = 1/T f = 500Hz

Frequency Decomposition: Pure Sine Wave T = 1ms f = 1/T f = 1000Hz

Composite Wave I

Composite Wave II

Why you should change strings A quick experiment with a spectrogram Old New

Orthogonality m = 2, n = 3, T = 0.2 m = 1, n = 2, T = 0.2

More orthogonality m = 3, n = 1, T = 0.2 m = 2, n = 2, T = 0.2 Integrate over one period: m = n is the only case where any of these is non-zero. Allows us to extract a n ’s and b n ’s

Waveform Piano Guitar Sine Wave

Piano: Component Sine Waves Time Microphone Signal Amplitude

Piano: Component Sine Waves Composite Wave (From Previous Slide) Original Piano Wave Look how close with only three sine waves!!! Try it yourself:

Frequency Spectra for Different Instruments Same pitch played, but TIMBRE is entirely unique

Biomedical Example GASTROINTESTINAL RHYTHMIC ACTIVITYGI ELECTRICAL SIGNALS (VOLTAGE VS. TIME) V(t) t

Fourier series representation EXAMPLE RECORDED SIGNAL FOURIER SERIES REPRESENTATION Peaks occur at: 0.059, 0.293, 0.527, 0.820, 1.113, … Hz Frequency (Hz) 2|c n |

C major chord Piano C chord (2 nd inversion) G4 (388) C5 E5 (657) G5 (775)

Modes of Vibration: Standing Waves

Harmonic Motion in Guitar

Spectrogram: Piano

Fourier Series and Superposition