# OpenStax_CNX

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# Duality

Module by: Richard Baraniuk. E-mail the author

Fiω=fte(iωt)d t F ω t f t ω t
(1)
ft=12πFiωeiωtd ω f t 1 2 ω F ω ω t
(2)
Both formulas are symmetrical except for the ±± sign and 12π 1 2 . Useful! Say ft Fiω f t F ω through CTFT, and Giω=fiω G ω f ω , that is: the same form as some time-domain signal f f . Then what is gt g t ?
gt=12πGiωeiωtd ω =12πfiωeiωtd ω g t 1 2 ω G ω ω t 1 2 ω f ω ω t
(3)
Let τ=ω τ ω , and i in fiω f ω doesn't matter, then:
gt=12πfτeiτtd τ =fτe(iτ(t))d τ =12πF(it) g t 1 2 τ f τ τ t τ f τ τ t 1 2 F t
(4)
In other words: gt g t takes the form of FT F F (reversed and scaled, Figure 1).

## Example 2

CTFT of t :gt=1 t g t 1 (Figure 2).

### Direct approach:

Giω=1e(iωt)d t G ω t 1 ω t
(5)

### Duality approach:

We know Figure 3.

So by duality we get Figure 4.

## Exercise 1: FT of a complex sinusoid

ft=ei ω 0 t f t ω 0 t

## Exercise 5: FT of unit step

Uiω=πδω+1iω U ω δ ω 1 ω .

See Lathi pp. 250-1 for derivation.

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