Physics 145
Introduction to Experimental Physics I
Instructor: Karine Chesnel
Office: N319 ESC
Tel: 801- 422-5687
kchesnel@byu.edu
Office hours: on appointment
Class website:
http://www.physics.byu.edu/faculty/chesnel/physics145.aspx
Lab 12
Fourier Transform
Resonators
Spring – mass resonator
Tuning fork
Time – frequency
Pure sine wave
Time space
Frequency space
Fourier Transform
Joseph Fourier
1768 - 1830
french mathematician
Decomposition of functions in linear combination of sine waves
f  t    cn sin(nt )
Discrete Fourier series
n
Example:
N
f t   
n 0
 1
n
n
sin( nt )
N=3
N = 10
N 
Fourier Transform
Discrete Fourier series
f  t    cn sin(nt )
Using sine functions
n
f t  

int
c
e
 n
Using complexe notation
n 
Fourier’s trick
T /2
1
 int
cn 
f
t
e
  dt

T T /2
where
T
2

Fourier Transform
Continuous Fourier transforms
1
F   
2
1
f t  
2


f  t eit dt
Integration over time




F  eit d
Integration over
frequency range
Fourier Transform
Square wave
Time space
Frequency space
Fourier Transform
Dt  
D  1/
Modulated wave


   ( 0 ) 
 ( 0 ) 
4
4




F    A  e
e


2
f (t )  Ae
 t / 
2
cos(0t )
Time space
Frequency space
2




Power spectrum
P( )  F  
2
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
D. Application: frequency filter to vocal input
• 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