From Fourier Series to

Fourier Transforms

Recall that

n

Txninecxf /2)(

where ,...2,1,0,1 /2

2/

2/

ndxexf

Tc Txni

T

T

n

Now let T become large ...

andT

nπω

2

so ω becomes small ...

deFxf xi

2

1)(

dxexfF xi )(

Fourier Transform

of f(x)

Inverse Fourier Transform of F(ω).

Example 1Determine the Fourier Transform of

113 tututf

113 tututf

1

1

3

tiei

dtetfF tiωω )(

1

1

3 dteF ti

sin6

ii eei

3

ii eei

3

sin23

ii

sinc6)( F)(sinc

sin

113 tututf ω

ωω

sin6F

Note: F(ω) is REAL in this example.

These are the graphs of f(t) and F(ω):

Example 2Determine the Fourier Transform of

tetutf 2

i

2

1

dtetfF tiωω )( tetutf 2

0

2 dteeF tit

0

2 dte ti

0

2

2

1 tiei

102

1

i

Note: F(ω) is COMPLEX in this example.

Draw the graph of the modulus of F(ω) (the amplitude spectrum).

tetutf 2 ω

ωi

F

2

1

24

1

2

1

2

1

iiF

tetutf 2 F

Even Functions

If f is an even function, then tftf

0

cos2)( dtttfdtetfF ti ωω ω

This result arises because cosine is even ...

... and so is even ... ttf ωcos

Example 3Determine the Fourier Transform of

1111 tututtututtf

Even function!

1

0

cos12 dtttF

1cos22

1

0

1

0

sin12

sin12 dt

ttt

1

0

sin2

dtt

002

1

0

cos2

t AAA 22 sincos2cos

A2sin21AA 2sin22cos1

2

2)(

F 2/sin2 2 2/sin

)2/(

1 22

2/sinc2

Odd Functions

If f is an odd function, then tftf

0

sin2)( dtttfidtetfF ti ωω ω

This result arises because sine is odd ...

... and so is even ... ttf ωsin

Example 4Determine the Fourier Transform of

2424 tututututf

Odd function!

2

0

sin42 dttiF

2424 tututututf

0

sin2 dtttfiF ωω

2sin16i

2

0

cos8

t

i

12cos8

i

2sin28

i

FIm

Summary:

Example f(t) F(ω)

1 Even Real

2 Neither odd nor even

Complex

3 Even Real

4 Odd Imaginary

113 tutuω

ωsin6

tetu 2

ωi2

1

...11 tutut

2sinc2 ω

...24 tutuω

ω2sin16i

Special Case 0,2

aetf at

Use this known result:

πdxe x2

Substitute yax

πdyea ay2

Now use:a

ity

2

ω

atiat

a

itaay

42

22

22

a

dte atiat πωω

4

22

Hence:

atiat ea

dte 4

2

2ω

ω πor

set at iofTransformFourier2 ae

a4

2

Now look at Tutorial 1