EXPLORATIONS OF SIGNAL ANALYSIS IN HIGH SCHOOL AND BEYOND
Russell Herman Gabriel LugoUniversity of North Carolina at Wilmington
Outline
Projects – High School through College Tools Theory Typical Project Examples
Projects
Sounds Music Birds Dolphins
EKG-ECG-EEG Ocean Waves
Tools
Sound Editors Cool Edit Gold Wave Audacity PPC Applications
Theory Support Mathcad, Maple, Matlab VB Modules Web Applets
Cool Edit
Gold Wave
Addition of Sines
Adding Sine WavesBirds\Untitled (8.2).wav
Birds\tone.wav
Birds\pt.wav
Theory
Fourier Series Sum of Sines
Fourier Analysis Spectrum Analysis Harmonic Analysis
+ =
Fourier Series
Fourier SeriesFourier Series
Eigenfunctions:
L 2 c x n( ) cosn xL
s x n( ) sinn xL
Function: f x( ) sin x( ) 2 cos 2x( ) x 0 .1 L
Fourier Coefficients:
a012L L
Lxf x( )
dN 8 n 1 N
a n( )1L L
Lxf x( ) c x n( )
d
b n( )1L L
Lxf x( ) s x n( )
d
Reconstruction
Reconstruction
2
0
a n( )
81 n
2
0
b n( )
81 n
2
0
a n( )2 b n( )2
81 n
F x( ) a 0n
a n( ) c x n( ) b n( ) s x n( )( )Fourier Expansion:
3
3
f x( )
F x( )
2 0 x
Comparison between f(x)and F(x)
Power Spectrum
Typical Project
Addition of Sinusoidal Functions Explorations Using Sound Editors Picking Out Dominant Frequencies Reconstruction and Fine Tuning
Example #1 - Birds
Birds: Reconstruction
Birds\Recbird.wav
Birds\firstbird.wav
More Birds
Birds\owl2.wav
Birds\newowl.wav
-0.025991*cos(2*pi*1327.0833*t)+0.033382*cos(2*pi*1329.8423*t)+-0.043522*cos(2*pi*1332.6014*t)+ 0.061318*cos(2*pi*1335.3604*t)+-0.081395*cos(2*pi*1338.1194*t)+0.09432*cos(2*pi*1340.8784*t)+ -0.096081*cos(2*pi*1343.6374*t)+0.090427*cos(2*pi*1346.3964*t)+-0.058718*cos(2*pi*1349.1554*t)+ 0.047292*cos(2*pi*1354.6734*t)+-0.090452*cos(2*pi*1357.4324*t)+0.091498*cos(2*pi*1360.1914*t)+ -0.044081*cos(2*pi*1362.9505*t)+-0.03472*cos(2*pi*1365.7095*t)+0.075536*cos(2*pi*1368.4685*t)+ -0.039251*cos(2*pi*1371.2275*t)+-0.058875*cos(2*pi*1373.9865*t)+0.12712*cos(2*pi*1376.7455*t)+-0.077859*cos(2*pi*1379.5045*t)+-0.062965*cos(2*pi*1382.2635*t)+0.13712*cos(2*pi*1385.0225*t)+ 0.12768*cos(2*pi*1390.5405*t)+0.081571*cos(2*pi*1393.2995*t)+0.070206*cos(2*pi*1396.0586*t)+ -0.022393*cos(2*pi*1398.8176*t)+-0.11979*cos(2*pi*1401.5766*t)+0.094213*cos(2*pi*1407.0946*t)+ 0.06029*cos(2*pi*1409.8536*t)+-0.076122*cos(2*pi*1412.6126*t)+-0.13165*cos(2*pi*1415.3716*t)+ -0.043479*cos(2*pi*1418.1306*t)+0.074439*cos(2*pi*1420.8896*t)+0.13719*cos(2*pi*1423.6486*t)+ 0.08474*cos(2*pi*1426.4077*t)+-0.060982*cos(2*pi*1431.9257*t)+-0.049597*cos(2*pi*1434.6847*t)+ -0.036721*cos(2*pi*1445.7207*t)+-0.024337*cos(2*pi*1448.4797*t)+0.032224*cos(2*pi*1456.7568*t)+ .033543*cos(2*pi*1459.5158*t)+0.024255*cos(2*pi*1465.0338*t)
Vibrating Beams
0 20 40 60 80 100 120 140-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
0 2 4 6 8 10 12 140
0.002
0.004
0.006
0.008
0.01
0.012
0.014
0.016
0.018
0.02
Frequency
Pow
er
Periodogram
Top Related