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# Useful Fourier Transforms

Module by: Richard Baraniuk. E-mail the author

## Square Pulse / Box Car / Unit Gate / Square Window

Riω=rte(iωt)d t =TT1e(iωt)d t =e(iωt)(iω)|TT=(1iω)(e(iωT)eiωT) R ω t r t ω t t T T 1 ω t T T ω t ω 1 ω ω T ω T
(1)
since e(iωT)eiωT=(2isinωT) ω T ω T 2 ω T we get:
Riω=2sinωTω=2TsinωTωT R ω 2 ω T ω 2 T ω T ω T
(2)
which is a sinc!

### recall:

Moving average impulse responce, p. B560.
Figure 2.

## Triangle

Biω=bte(iωt)d t B ω t b t ω t
(3)

## Delta function

Δiω=δte(iωt)d t =1 Δ ω t δ t ω t 1
(4)
Figure 5.
• Delta function contains an equal mix of all frequencies.
• It is in some sense "white" (like white light is made up of colors of all frequencies).

## Shifted Impulse

Fiω=δt t 0 e(iωt)d t =e(iω t 0 ) F ω t δ t t 0 ω t ω t 0
(5)
which is a complex sinusoid in frequency (Figure 7).

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