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

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Presentation on theme: "Intro to Fourier Series BY JORDAN KEARNS (W&L ‘14) & JON ERICKSON (STILL HERE )"— Presentation transcript:

1

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

3 220 Hz (A3) Why do they sound different? Instrument 1 Instrument 2Sine Wave

4 Waveform Piano Guitar Sine Wave ToTo

5 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: https://www.youtube.com/watch?v=PeTriGTENoc

6 PianoGuitar Frequency Spectrum

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

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

9 Composite Wave I

10 Composite Wave II

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

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

13 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

14 Waveform Piano Guitar Sine Wave

15 Piano: Component Sine Waves Time Microphone Signal Amplitude

16 Piano: Component Sine Waves Composite Wave (From Previous Slide) Original Piano Wave Look how close with only three sine waves!!! Try it yourself: http://lpsa.swarthmore.edu/Fourier/Series/WhyFS.html

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

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

19 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 |

20 C major chord Piano C chord (2 nd inversion) G4 (388) C5 E5 (657) G5 (775) 1171 1314 1564

21 Modes of Vibration: Standing Waves

22 Harmonic Motion in Guitar

23 Spectrogram: Piano

24 Fourier Series and Superposition

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