What do these and…
Spring 2006 – Week 8 PHY 131
… these have in common?
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Spring 2006 – Week 8 PHY 131 6
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What do these and ….
Spring 2006 – Week 8 PHY 131
...these have in common?
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Building Blocks for Periodic Waveforms
Sine-function
Cosine-function
Spring 2006 – Week 8 PHY 131 9
Spectrum Synthesis
(a.k.a. Fourier Synthesis)
Any periodic waveform can be expressed as a sum of sine and cosine functions of harmonics: x ( t )
= a o + a
1
2
+ b
1 cos(2 sin(2
π f t )
+ a
2
π f t )
+ b
2 cos(2
π
2 f t )
+ a sin(2
π
2 f t )
+ b
3
3 cos(2
π
3 f t )
+
...
sin(2
π
3 f t )
+
...
Pattern: only multiples of the fundamental frequency f are needed!
Spring 2006 – Week 8 PHY 131 10
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Spring 2006 – Week 8
Adding Waveforms
Wave
1
Wave
1
Pick a point on one waveform and determine its value
Wave
2
Wave
2
Determine the value of the equivalent point on the other waveform
Add the values of both waveforms to find one point of the sum waveform
Sum
+ = + = – = =
PHY 131 11
Examples of Spectrum Synthesis
• Manually, using a spreadsheet
• Using a Java Applet http://projects.cbe.ab.ca/sss/science/physics/map_north/applets/ fourier/fouriersynthesis.html
• Another Java Applet http://www.falstad.com/fourier/
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Spectrum Analysis
(a.k.a. Fourier Analysis)
The Process is reversible. Given a periodic waveform x(t) we can figure out how much of each f how many of its harmonics need to be used to reproduce this waveform.
x ( t )
= a o
+ a
1
2
+ b
1 cos(2 sin(2
π f t )
+ a
π f t )
+ b
2
2 cos(2
π
2 f t )
+ a sin(2
π
2 f t )
+ b
3
3 cos(2
π
3 f t )
+
...
sin(2
π
3 f t )
+
...
Spring 2006 – Week 8 PHY 131 13
Spectrum Analyzer x ( t )
= a o + a
1
2
+ b
1 cos(2 sin(2
π f t )
+ a
π f t )
+ b
2
2 cos(2
π
2 f t )
+ a sin(2
π
2 f t )
+ b
3
3 cos(2
π
3 f t )
+
...
sin(2
π
3 f t )
+
...
Waveform x ( t )
Spectrum
Spectrum
Analyzer
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Example of Spectrum Analyzer
• LabView Program http://audacity.sourceforge.net/
• Audactiy
Free software to record and edit sound http://audacity.sourceforge.net/
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