Fourier Transform

Saturday, 23 March 2019

2:30 PM

Previously, we derived the Fourier series for periodic functions.
What about non-periodic functions?

Consider when 𝑇→∞

We can `period-ise` our function, allowing us to use the Fourier series

In the frequency domain,

Series Representation
𝑓﷐𝑡﷯=﷐𝑛=−∞﷮∞﷮﷐𝑐﷮𝑛﷯﷐𝑒﷮﷮𝑗𝑛﷐ω﷮0﷯𝑡﷯﷯
=﷐𝑛=−∞﷮∞﷮﷐﷐1﷮𝑇﷯﷐−﷐𝑇﷮2﷯﷮﷐𝑇﷮2﷯﷮𝑓﷐𝑡﷯﷐𝑒﷮−𝑗𝑛﷐𝜔﷮0﷯𝑡﷯﷯﷯﷐𝑒﷮﷮𝑗𝑛﷐ω﷮0﷯𝑡﷯﷯
=﷐𝑛=−∞﷮∞﷮﷐﷐Δ𝜔﷮2𝜋﷯﷐−﷐𝑇﷮2﷯﷮﷐𝑇﷮2﷯﷮𝑓﷐𝑡﷯﷐𝑒﷮−𝑗𝑛﷐𝜔﷮0﷯𝑡﷯﷯﷯﷐𝑒﷮﷮𝑗𝑛﷐ω﷮0﷯𝑡﷯﷯
=﷐1﷮2𝜋﷯﷐𝑛=−∞﷮∞﷮﷐﷐−﷐𝑇﷮2﷯﷮﷐𝑇﷮2﷯﷮𝑓﷐𝑡﷯﷐𝑒﷮−𝑗𝑛﷐𝜔﷮0﷯𝑡﷯﷯﷯﷐𝑒﷮﷮𝑗𝑛﷐ω﷮0﷯𝑡﷯﷯Δ𝜔

As 𝑇→∞:
* ﷐𝑛=−∞﷮∞﷮ ﷯→﷐−∞﷮∞﷮ ﷯
* (harmonic spacing) Δ𝜔→𝑑𝜔 (differential separation)
* (discrete) 𝑛﷐ω﷮0﷯→𝜔 (continuous)

𝑓﷐𝑡﷯=﷐1﷮2𝜋﷯﷐−∞﷮∞﷮﷐﷐−∞﷮∞﷮𝑓﷐𝑡﷯﷐𝑒﷮−𝑗𝜔𝑡﷯﷯𝑑𝑡﷯﷐𝑒﷮𝑗𝜔𝑡﷯𝑑𝑡﷯

𝐹﷐ω﷯=ℱ﷐𝑓﷐𝑡﷯﷯=﷐−∞﷮∞﷮𝑓﷐𝑡﷯﷐𝑒﷮−𝑗𝜔𝑡﷯﷯𝑑𝑡
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Ink Drawings
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𝐹﷐ω﷯=ℱ﷐𝑓﷐𝑡﷯﷯=﷐−∞﷮∞﷮𝑓﷐𝑡﷯﷐𝑒﷮−𝑗𝜔𝑡﷯﷯𝑑𝑡
A transformation from a time domain function to a frequency domain representation

Inverse Fourier Transform
𝑓﷐𝑡﷯=﷐ℱ﷮−1﷯﷐𝐹﷐𝜔﷯﷯=﷐1﷮2𝜋﷯﷐−∞﷮∞﷮𝐹﷐ω﷯﷐𝑒﷮𝑗𝜔𝑡﷯﷯𝑑𝜔

Conditions:
The Fourier Transform only exists when ﷐−∞﷮∞﷮﷐𝑓﷐𝑡﷯﷯𝑑𝑡﷯<0 converges

Created with Microsoft OneNote 2016.