Physics 145 Introduction to Experimental Physics I
Instructor: Karine Chesnel Office: N319 ESC
Tel: 801- 422-5687 [email protected]
Office hours: on appointment
Class website:
http://www.physics.byu.edu/faculty/chesnel/physics145.aspx
Lab 12Fourier Transform
Resonators
Spring – mass resonator Tuning fork
Time – frequency
Pure sine wave
Time space Frequency space
Fourier Transform
Joseph Fourier1768 - 1830french mathematician
Decomposition of functions in linear combination of sine waves
sin( )nn
f t c n t Discrete Fourier series
Example:
0
1sin( )
nN
n
f t n tn
N = 3 N = 10 N
Fourier Transform
sin( )nn
f t c n t
Discrete Fourier series
Using sine functions
in tn
n
f t c e
Using complexe notation
Fourier’s trick
/2
/2
1 Tin t
nT
c f t e dtT
where2T
Fourier Transform
Continuous Fourier transforms
12
i tf t F e d
12
i tF f t e dt
Integration over time
Integration over frequency range
Square wave
Fourier Transform
Time space Frequency space
Modulated wave
Fourier Transform
Time space
2/0( ) cos( )tf t Ae t
Frequency space
2 2
0 0( ) ( )4 4F A e e
Dt D 1/
Power spectrum
2( )P F
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 f0 and harmonics
compare fundamental frequency to nominal value
• L12.4: Test the Nyquist criterion- use sine wave from tuning fork (f0 = 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: 100-200Hz…. High band-pass 1kHz and more) listen to the resulting filtered signal, print spectra
D. Application: frequency filter to vocal input
