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