Physics 145 Introduction to Experimental Physics I Instructor: Karine Chesnel Office: N319 ESC Tel: 801- 422-5687 Office hours: on appointment.

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Physics 145 Introduction to Experimental Physics I Instructor: Karine Chesnel Office: N319 ESC Tel: Office hours: on appointment Class website:

Lab 12 Fourier Transform

Resonators Spring – mass resonator Tuning fork

Time – frequency Pure sine wave Time space Frequency space

Fourier Transform Joseph Fourier french mathematician Decomposition of functions in linear combination of sine waves Discrete Fourier series Example: N = 3 N = 10

Fourier Transform Discrete Fourier series Using sine functions Using complexe notation Fourier’s trick where

Fourier Transform Continuous Fourier transforms Integration over time Integration over frequency range

Square wave Fourier Transform Time space Frequency space

Modulated wave Fourier Transform Time space Frequency space  t  

Power spectrum

Nyquist-Shannon criterion A periodic signal needs to be sampled at least at twice the frequency to be properly measured /reconstructed

Lab 12: Fourier Transform A. Computer generated waveforms L12.1: open Labview Fourier-waveform.vi generate different waveform examine the time functions and the frequency spectra Sine wave Square wave Modulated wave

Lab 12: Fourier Transform C. Fourier spectra of sound-wave L12.2: open Labview Fourier-sound.vi plug microphone + headset speakers to computer sample yourself whistling… sampling at 20kHz for 1s L12.3: Record notes produced by tuning forks look at fundamental frequency f 0 and harmonics compare fundamental frequency to nominal value L12.4: Test the Nyquist criterion - use sine wave from tuning fork (f 0 = 1kHz) - sample at different frequencies from 1kHz to 10kHz… - observe what happens to the time and frequency spectra L12.5: Generate Fourier spectra from different abrupt sounds: -clapping, yelling, popping balloons… -Print spectra

Lab 12: Fourier Transform C. Application: vowel sound recognition L12.6: generate Fourier spectra from vowels: a, e, o, u (hold the note steady for entire acquisition) L12.7: print series of spectra from different persons play to guess which spectrum correspond to which vowel L12.8: Record vocal input (sentences, etc…) - increase the sampling interval to several seconds at 20kHz - turn the frequency filter ON (band pass) - compare unfiltered (left) and filtered (right) signals L12.9: Play with parameters of band-pass filter ( low band-pass: Hz…. High band-pass 1kHz and more) listen to the resulting filtered signal, print spectra D. Application: frequency filter to vocal input