Properties of the Fourier Transform

Saturday, 23 March 2019

4:35 PM

Linearity
ℱ﷐﷐a﷮1﷯﷐f﷮1﷯﷐t﷯+﷐a﷮2﷯﷐f﷮2﷯﷐t﷯﷯=﷐a﷮1﷯﷐F﷮1﷯﷐ω﷯+﷐a﷮2﷯﷐F﷮2﷯﷐ω﷯
Proof
ℱ﷐﷐𝑎﷮1﷯﷐𝑓﷮1﷯﷐𝑡﷯+﷐𝑎﷮2﷯﷐𝑓﷮2﷯﷐𝑡﷯﷯=﷐∞﷮∞﷮﷐﷐𝑎﷮1﷯﷐𝑓﷮1﷯﷐𝑡﷯+﷐𝑎﷮2﷯﷐𝑓﷮2﷯﷐𝑡﷯﷯﷐𝑒﷮−𝑗𝜔𝑡﷯﷯𝑑𝑡
=﷐𝑎﷮1﷯﷐−∞﷮∞﷮﷐𝑓﷮1﷯﷐𝑡﷯﷐𝑒﷮−𝑗𝜔𝑡﷯﷯𝑑𝑡+﷐𝑎﷮2﷯﷐−∞﷮∞﷮﷐𝑓﷮2﷯﷐𝑡﷯﷐𝑒﷮−𝑗𝜔𝑡﷯﷯𝑑𝑡
=﷐𝑎﷮1﷯﷐𝐹﷮1﷯﷐ω﷯+﷐𝑎﷮2﷯﷐𝐹﷮2﷯﷐ω﷯

Time Scaling

Proof
ℱ﷐𝑓﷐𝑎𝑡﷯﷯=﷐−∞﷮∞﷮𝑓﷐𝑎𝑡﷯﷐𝑒﷮−𝑗𝜔𝑡﷯﷯𝑑𝑡
let 𝜏=𝑎𝑡
ℱ﷐𝑓﷐𝑎𝑡﷯﷯=﷐−∞﷮∞﷮𝑓﷐τ﷯﷐𝑒﷮−𝑗﷐﷐𝜔﷮𝑎﷯﷯𝜏﷯﷯ 𝑑﷐﷐τ﷮𝑎﷯﷯
=﷐1﷮𝑎﷯𝐹﷐﷐𝜔﷮𝑎﷯﷯

ℱ﷐𝑓﷐𝑎𝑡﷯﷯=﷐1﷮﷐𝑎﷯﷯𝐹﷐﷐ω﷮𝑎﷯﷯
Time Shifting
ℱ﷐𝑓﷐𝑡−﷐𝑡﷮0﷯﷯﷯=﷐𝑒﷮−𝑗ω﷐𝑡﷮0﷯﷯𝐹﷐ω﷯
Proof
ℱ﷐𝑓﷐𝑡−﷐𝑡﷮0﷯﷯﷯=﷐−∞﷮∞﷮𝑓﷐𝑡−﷐𝑡﷮0﷯﷯﷐𝑒﷮−𝑗𝜔𝑡﷯﷯𝑑𝑡
let τ=𝑡−﷐𝑡﷮0﷯
ℱ﷐𝑓﷐𝑡−﷐𝑡﷮0﷯﷯﷯=﷐−∞﷮∞﷮𝑓﷐𝜏﷯﷐𝑒﷮−𝑗𝜔﷐𝜏+﷐𝑡﷮0﷯﷯﷯﷯𝑑𝜏
=﷐𝑒﷮−𝑗𝜔﷐𝑡﷮0﷯﷯﷐−∞﷮∞﷮𝑓﷐τ﷯﷐𝑒﷮𝑗𝜔𝜏﷯﷯𝑑𝜏
=﷐𝑒﷮−𝑗𝜔﷐𝑡﷮0﷯﷯𝐹﷐𝜔﷯

Note - magnitude in a time shift is the same, only the phase changes

Frequency Shifting
ℱ﷐𝑓﷐𝑡﷯﷐𝑒﷮𝑗﷐ω﷮0﷯𝑡﷯﷯=𝐹﷐𝜔−﷐ω﷮0﷯﷯
Proof
ℱ﷐𝑓﷐𝑡﷯﷐𝑒﷮𝑗﷐ω﷮0﷯𝑡﷯﷯=﷐−∞﷮∞﷮𝑓﷐𝑡﷯﷯﷐𝑒﷮𝑗﷐ω﷮0﷯𝑡﷯﷐𝑒﷮−𝑗𝜔𝑡﷯𝑑𝑡
=﷐−∞﷮∞﷮𝑓﷐𝑡﷯﷯﷐𝑒﷮−𝑗(ω−﷐ω﷮0﷯)𝑡﷯ 𝑑𝑡
=𝐹﷐𝜔−﷐ω﷮0﷯﷯

Differentiation in Time
ℱ﷐﷐𝑓﷮′﷐𝑡﷯﷯﷯=𝑗𝑤𝐹﷐𝜔﷯
Proof
𝑓﷐𝑡﷯=﷐1﷮2𝜋﷯﷐−∞﷮∞﷮𝐹﷐ω﷯﷐𝑒﷮𝑗𝜔𝑡﷯﷯𝑑𝜔
Differentiate both sides…
﷐𝑑𝑓﷐𝑡﷯﷮𝑑𝑡﷯=﷐1﷮2𝜋﷯﷐𝑑﷮𝑑𝑡﷯﷐﷐−∞﷮∞﷮𝐹﷐ω﷯﷐𝑒﷮𝑗𝜔𝑡﷯﷯𝑑𝜔﷯

﷐𝑑𝑓﷐𝑡﷯﷮𝑑𝑡﷯=﷐1﷮2𝜋﷯﷐𝑑﷮𝑑𝑡﷯﷐﷐−∞﷮∞﷮𝐹﷐ω﷯﷐𝑒﷮𝑗𝜔𝑡﷯﷯𝑑𝜔﷯
=﷐1﷮2𝜋﷯﷐−∞﷮∞﷮𝐹﷐ω﷯﷐𝑑﷮𝑑𝑡﷯﷐𝑒﷮𝑗𝜔𝑡﷯﷯𝑑𝜔
=﷐1﷮2𝜋﷯﷐−∞﷮∞﷮𝐹﷐ω﷯𝑗𝜔﷐𝑒﷮𝑗𝜔𝑡﷯﷯𝑑𝜔
=﷐ℱ﷮−1﷯﷐𝑗𝜔𝐹﷐𝜔﷯﷯

So ℱ﷐﷐𝑑𝑓﷐𝑡﷯﷮𝑑𝑡﷯﷯=𝑗𝜔𝐹﷐𝜔﷯

Differentiation in Frequency
ℱ﷐−𝑗𝑡𝑓﷐𝑡﷯﷯=﷐𝐹﷮′﷯﷐𝜔﷯
Proof
𝐹﷐ω﷯=﷐−∞﷮∞﷮𝑓﷐𝑡﷯﷐𝑒﷮−𝑗𝜔𝑡﷯﷯𝑑𝑡
Differentiate both sides
﷐𝑑𝐹﷐ω﷯﷮𝑑𝜔﷯=﷐𝑑﷮𝑑𝜔﷯﷐﷐−∞﷮∞﷮𝑓﷐𝑡﷯﷐𝑒﷮−𝑗𝜔𝑡﷯𝑑𝑡﷯﷯
=﷐−∞﷮∞﷮𝑓﷐𝑡﷯﷐𝑑﷮𝑑𝜔﷯﷐𝑒﷮−𝑗𝜔𝑡﷯﷯𝑑𝑡
=﷐−∞﷮∞﷮𝑓﷐𝑡﷯﷐−𝑗𝑡﷯﷐𝑒﷮−𝑗𝜔𝑡﷯﷯𝑑𝑡
=ℱ﷐−𝑗𝑡𝑓﷐𝑡﷯﷯

Duality
ℱ﷐𝑓﷐𝑡﷯﷯=𝐹﷐ω﷯ ⇒ ℱ﷐𝐹﷐𝑡﷯﷯=2𝜋𝑓﷐−𝜔﷯

New symbol - convolution operator *

Convolution
Define 𝑦﷐𝑡﷯=ℎ﷐𝑡﷯∗𝑥(𝑡)
=﷐−∞﷮∞﷮ℎ﷐τ﷯ 𝑥(𝑡−𝜏)﷯𝑑𝜏

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