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# Existence of CTFT: Dirichlet Conditions

Module by: Richard Baraniuk. E-mail the author

## A

If ft f t is L2R L 2 , in other words f22=|ft|2d t < 2 f 2 t f t 2 , then CTFT equations hold a.e. That is: with Fiω=fte(iωt)d t F ω t f t ω t and ft=12πFiωeiωtd ω f t 1 2 ω F ω ω t a.e.

## B

12πFiωeiωtd t =ft 1 2 t F ω ω t f t everywhere except at points of discontinuity if

1. fLR f L : f1=|ft|d t < 1 f t f t .
2. f f has a finite number of local maxima and minima.
3. f f has a finite number of discontinuities.
(sufficient condition only, example: sintt t t ).

## Example 1

Find FT of ft=e(at)ut f t a t u t .

### Direct method

Fiω=fte(iωt)d t =0e(at)e(iωt)d t =0e((a+iω)t)d t =1a+iωe((a+iω)t)|0 F ω t f t ω t t 0 a t ω t t 0 a ω t 1 a ω 0 a ω t
(1)

### Note:

e((a+iω)t)|0 0 a ω t evaluates to 01 0 1 since a>0 a 0 by Dirichlet conditions.
Fiω=1a+iω=aiωa2+ω2 F ω 1 a ω a ω a 2 ω 2
(2)
|Fiω|=1 2 a2+ω2 F ω 1 2 a 2 ω 2
(3)
Fiω=arctanωa F ω ω a
(4)

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