Файл: Ersoy O.K. Diffraction, Fourier optics, and imaging (Wiley, 2006)(ISBN 0471238163)(427s) PEo .pdf

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PROPERTIES OF THE FOURIER TRANSFORM

13

Property 4: Modulation

 

 

 

 

 

 

If gðx; yÞ ¼ u1ðx; yÞu2ðx; yÞ, then

 

 

 

 

 

 

Gð fx; fyÞ ¼ F1ð fx; fyÞ F2ð fx; fyÞ

ð2:5-7Þ

Property 5: Separable function

 

 

 

 

 

 

If gðx; yÞ ¼ u1ðxÞu2ðyÞ, then

 

 

 

 

 

 

Gð fx; fyÞ ¼ U1ð fxÞU2ð fyÞ

ð2:5-8Þ

Property 6: Space shift

 

 

 

 

 

 

If gðx; yÞ ¼ uðx x0; y y0Þ, then

 

 

 

 

 

Gð fx; fyÞ ¼ e j2pð fxx0þfyy0Þ Uð fx; fyÞ

ð2:5-9Þ

Property 7: Frequency shift

 

 

 

 

 

 

If gðx; yÞ ¼ e j2pð fx0xþfy0yÞ uðx; yÞ, then

 

Gð fx; fyÞ ¼ Uð fx fx 0; f2 fy 0Þ

ð2:5-10Þ

Property 8: Differentiation in space domain

 

If gðx; yÞ ¼ @k=@xk @‘=@y‘ uðx; yÞ, then

 

Gð fx; fyÞ ¼ ð2p jfxÞkð2p jfyÞ‘Uð fx; fyÞ

ð2:5-11Þ

Property 9: Differentiation in frequency domain

 

If gðx; yÞ ¼ ð j2pxÞkð j2pyÞ‘uðx; yÞ, then

 

 

 

@k @‘

 

Gð fx; fyÞ ¼

 

 

 

Uð fx; fyÞ

ð2:5-12Þ

@fxk

@fy‘

Property 10: Parseval’s theorem

 

 

 

 

 

 

1

1

ð Uð fx; fyÞG ð fx; fyÞdfxdfy

 

ð ð uðx; yÞg ðx; yÞdxdy ¼

ð

ð2:5-13Þ

1

1

 

 

 

 

Property 11: Real uðx; yÞ

 

 

 

 

 

 

Uð fx; fyÞ ¼ U ð fx; fyÞ

ð2:5-14Þ

Property 12: Real and even uðx; yÞ

Uð fx; fyÞ is real and even

14

 

 

 

 

 

 

LINEAR SYSTEMS AND TRANSFORMS

Property 13: Real and odd uðx; yÞ

 

 

 

 

 

 

 

 

Uð fx; fyÞ is imaginary and odd

 

Property 14: Laplacian in the space domain

 

@2

 

@2

 

 

 

 

 

 

If gðx; yÞ ¼

 

þ

 

uðx; yÞ, then

 

 

 

@x2

@y2

 

 

 

 

 

 

Gð fx; fyÞ ¼ 4p2ð fx2 þ fy2ÞUð fx; fyÞ

ð2:5-15Þ

Property 15: Laplacian in the frequency domain

 

If gðx; yÞ ¼ 4p2ðx2 þ y2Þuðx; yÞ, then

 

 

 

 

 

 

Gð fx; fyÞ ¼

@2

þ

@2

!Uð fx; fyÞ

ð2:5-16Þ

 

 

 

@fx2

@fy2

Property 16: Square of signal

 

 

 

 

 

If gðx; yÞ ¼ juðx; yÞj2, then

 

 

 

 

 

 

 

 

Gð fx; fyÞ ¼ Uð fx; fyÞ U ð fx; fyÞ

ð2:5-17Þ

Property 17: Square of spectrum

 

 

 

 

 

If gðx; yÞ ¼ uðx; yÞ u ðx; yÞ, then

 

 

 

 

 

 

 

 

 

Gð fx; fyÞ ¼ jUð fx; fyÞj2

ð2:5-18Þ

Property 18: Rotation of axes

 

 

 

 

 

If gðx; yÞ ¼ uð x; yÞ, then

 

 

 

 

 

 

 

 

 

Gð fx; fyÞ ¼ Uð fx; fyÞ

ð2:5-19Þ

The important properties of the FT are summarized in Table 2-1.

EXAMPLE 2.2 Find the 1-D FT of

1ð

gðxÞ ¼

uðx; yÞdy

1

as a function of the FT of uðx; yÞ.

PROPERTIES OF THE FOURIER TRANSFORM

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

15

Table 2.1. Properties of the Fourier transform (a, b,

fx0

and fy0

are real nonzero

constants; k and l are nonnegative integers).

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Property

 

 

gðx; yÞ

 

 

 

 

 

 

 

 

 

 

 

 

 

Gðfx; fyÞ

 

Linearity

au1ðx; yÞ þ bu2ðx; yÞ

 

aU1ð fx; fyÞ þ bU2ð fx; fyÞ

Convolution

u1ðx; yÞ u2ðx; yÞ

 

 

U1ð fx; fyÞU2ð fx; fyÞ

Correlation

u1ðx; yÞ u2ðx; yÞ

 

 

U1ð fx; fyÞU2 ð fx; fyÞ

Modulation

 

u1ðx; yÞu2ðx; yÞ

 

 

U1ð fx; fyÞ U2ð fx; fyÞ

Separable function

 

 

u1ðxÞu2ðyÞ

 

 

 

 

 

 

U1ð fxÞU2ð fyÞ

 

Space shift

uðx

 

x0; y y0Þ

 

e

j2pð fxx0þfy y0 Þ Uð fx; fyÞ

Frequency shift

gðx; yÞ ¼ e j2pð fx0xþfy0yÞ uðx; yÞ

Gð fx; fyÞ ¼ Uð fx

fx0; f2

fy0Þ

 

 

 

@k

 

@‘

 

 

 

ð2pjfxÞkð2pjfyÞ‘Uð fx; fyÞ

Differentiation in

 

 

 

 

 

 

 

uðx; yÞ

 

 

 

@xk

@y‘

 

space domain

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

@k

 

 

@‘

 

 

 

 

 

ð j2pxÞkð j2pyÞ‘uðx; yÞ

 

 

 

 

 

 

 

 

 

 

 

Differentiation in

 

 

 

 

 

 

 

 

 

Uð fx; fyÞ

 

 

 

 

 

@f k

@f ‘

 

frequency domain

 

@2

 

 

@2

 

 

 

 

 

 

 

 

x

y

 

 

 

 

Laplacian in the space

 

 

 

 

 

 

 

 

 

2

 

2

 

 

2

 

 

 

@x2 þ @y2

uðx; yÞ

 

 

4p ð fx

þ fy ÞUð fx; fyÞ

domain

 

 

Laplacian in

4p2ðx2 þ y2Þuðx; yÞ

 

 

@2

þ

@2

!Uð fx; fyÞ

 

 

 

@fx2

@fy2

the frequency domain

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Square of signal

 

 

 

j

ð

 

Þj

2

 

 

U

ð

fx

; fy

Þ

U

ð

fx; fy

Þ

 

 

 

 

u x; y

 

 

 

 

 

 

 

 

 

Square of spectrum

 

uðx; yÞ u ðx; yÞ

 

 

 

 

 

 

 

jUð fx; fyÞj2

 

Rotation of axes

1

uð x; yÞ

1

 

 

 

 

 

Uð fx; fyÞ

 

 

 

 

 

 

 

 

 

 

ð Uð fx; fyÞG ð fx; fyÞdfxdfy

Parseval’s theorem

ð ð uðx; yÞg ðx; yÞdxdy ¼

ð

 

1

 

 

 

 

 

 

 

1

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Real uðx; yÞ

 

 

 

 

 

 

 

 

 

Uð fx; fyÞ ¼ U ð fx;

 

 

fyÞ

 

 

 

 

 

 

 

 

 

Real and even uðx; yÞ

 

 

 

 

 

 

 

 

 

Uð fx; fyÞis real and even

 

 

 

 

 

 

 

 

 

Real and odd uðx; yÞ

 

 

 

 

 

 

 

 

Uð fx; fyÞis imaginary and odd

 

 

 

 

 

 

Solution: gðxÞ and dðxÞ are considered as 2-D functions gðxÞ 1 and dðxÞ 1, respectively. Then, gðxÞ can be written as

gðxÞ 1 ¼ uðx; yÞ ½dðxÞ 1&

xÞdx1dy1

¼

ðð

uðx1; y1Þdðx1

 

1

 

 

 

1

 

 

 

1

 

 

¼

ð

uðx; yÞdy

ð2:5-20Þ

1

16

LINEAR SYSTEMS AND TRANSFORMS

Computing the FT of both sides of Eq. (2.5-20) and using the convolution theorem and property 5 of separable functions gives

Gð fxÞdð fyÞ ¼ Uð fx; fyÞdð fyÞ

or

Gð fxÞ ¼ Uð fx; 0Þ

EXAMPLE 2.3 Find the FT of

gðx; yÞ ¼ uða1x þ b1y þ c1; a2x þ b2y þ c2Þ

as a function of the Fourier transform of uðx; yÞ.

Solution:

 

ðð uða1x þ b1y þ c1; a2x þ b2y þ c2Þe j2pð f1xþf2yÞdxdy

 

Gð fx; fyÞ ¼

 

 

 

1

 

 

 

 

 

 

 

 

 

 

 

 

1

 

 

 

 

 

 

 

 

 

 

Letting x1 ¼ a1x þ b1y þ c1 gives

 

1

þ b2y þ c2 e j2p fx

a1 þfyy

ja11j

Gð fx; fyÞ ¼

ðð u x1; a2 x1

a11

 

 

1

 

 

b y

c

 

 

 

 

x1 a1y c1

 

dx dy

 

1

 

 

 

 

 

 

 

 

 

 

 

1 ¼

a

2

 

a1

 

þ

b

2

y

þ

c

2

gives

 

Also letting y

 

x1 b1y c1

 

 

 

 

 

Gðfx; fyÞ ¼ jDj

ðð

uðx1; y1Þe j2p½x1fx0 þy1fy0 þT1fxþT2fy&dx1dy1

ð2:5-21Þ

 

 

 

1

1

 

 

 

 

 

 

 

 

 

 

1

where

D ¼ a2b1 a1b2

fx0 ¼ D1 ð a1fx þ a2fyÞ

fy0 ¼ 1 ðb1fx a1fyÞ D

T1 ¼ b1c2 b2c1

D

T2 ¼ a2c1 a1c2

D

PROPERTIES OF THE FOURIER TRANSFORM

17

Equation (2.5-21) is the same as

Gðfx; fyÞ ¼ j1j e j2pðT1 fxþT2 fyÞUð fx0; fy0Þ

D

In other words, when the input signal is scaled, shifted, and skewed, its transform is also scaled, skewed, and linearly phase-shifted, but not shifted in position.

EXAMPLE 2.4 Find the FT of the ‘‘one-zero’’ function defined by

u x; y

Þ ¼

1

h1ðxÞ < y < h2

x

Þ

;

ð

0

otherwise

ð

 

where h1ðxÞ and h2ðxÞ are given single-valued functions of x. Determine Uðfx; fyÞ and Uðfx; 0Þ.

Solution: uðx; yÞ is as shown in Figure 2.2. Its FT is given by

ðð1

Uðfx; fyÞ ¼ uðx; yÞe j 2pð fxxþfyyÞdxdy

1

1ð h2ðxÞ

¼

e j 2p fx xdx

e j 2p fy ydy

 

 

1 h1ðxÞ

Figure 2.2. A typical zero-one function.

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