Physics 145 Introduction to Experimental Physics I Instructor: Karine Chesnel

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 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