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What do these and…

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… these have in common?

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What do these and ….

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...these have in common?

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Building Blocks for Periodic Waveforms

Sine-function

Cosine-function

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Spectrum Synthesis (a.k.a. Fourier Synthesis)

Any periodic waveform can be expressed as asum of sine and cosine functions of harmonics:

x(t) = ao2+ a1 cos(2π ft) + a2 cos(2π2 ft) + a3 cos(2π 3 ft) + ...

+ b1 sin(2π ft) + b2 sin(2π2 ft) + b3 sin(2π 3 ft) + ... Pattern: only multiples of the fundamentalfrequency f are needed!

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20 1.05 2.1 3.15 4.2 5.25 6.3

Wave1

Wave2

Adding Waveforms

+ = + = – = =

Pick a point on onewaveform and determine itsvalue

Determine the value of theequivalent point on the otherwaveform

Add the values of bothwaveforms to find one pointof the sum waveform

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20 1.05 2.1 3.15 4.2 5.25 6.3

Wave1

Wave2

Sum

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Examples of Spectrum Synthesis

• Manually, using a spreadsheet

• Using a Java Applethttp://projects.cbe.ab.ca/sss/science/physics/map_north/applets/fourier/fouriersynthesis.html

• Another Java Applethttp://www.falstad.com/fourier/

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Spectrum Analysis (a.k.a. Fourier Analysis)

The Process is reversible. Given a periodicwaveform x(t) we can figure out how much ofeach f how many of its harmonics need to beused to reproduce this waveform.

x(t) = ao2+ a1 cos(2π ft) + a2 cos(2π2 ft) + a3 cos(2π 3 ft) + ...

+ b1 sin(2π ft) + b2 sin(2π2 ft) + b3 sin(2π 3 ft) + ...

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Spectrum Analyzer

SpectrumAnalyzer

x(t) = ao2+ a1 cos(2π ft) + a2 cos(2π2 ft) + a3 cos(2π 3 ft) + ...

+ b1 sin(2π ft) + b2 sin(2π2 ft) + b3 sin(2π 3 ft) + ...

Spectrumx(t)

Waveform

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Example of Spectrum Analyzer

• LabView Program

• AudactiyFree software to record and edit soundhttp://audacity.sourceforge.net/

http://audacity.sourceforge.net/