Fourier analysis in two dimensions University of Illinois at Chicago ECE 427, Dr. D. Erricolo University of Illinois at Chicago ECE 427, Dr. D. Erricolo

Fourier analysis in two dimensions

University of Illinois at Chicago

ECE 427, Dr. D. Erricolo



Definitions:

Given a (complex-valued) function g of two independent variables x and y, we define Fourier transform of g:

2 ( )( ) [ ] ( ) x yj f x f y

x yG f f g g x y e dx dy

F

The transform G is usually complex-valued, depends on two new independent variables, fx and fy, which are normally called frequencies. The inverse transformation is given by:

2 ( )1( ) [ ] ( ) x yj f x f y

x y x yg x y G G f f e df df

F

(1.1)

(1.2)





For a function g to have a Fourier transform, we request that (sufficient condition):g(x,y) must be absolutely integrable over the whole (x,y) plane.

This is easy to understand because if g(x,y) is absolutely integrable it implies the existence of

Using the inequality:

one obtains that:

and the last integral exists by assumption.

( )g x y dx dy

( ) ( )h x y dx dy h x y dx dy

2 ( )( ) ( ) ( )x yj f x f yg g x y e dx dy g x y dx dy

F

(1.3)

(1.4)

(1.5)





A very important 2D function that we will often use is the Dirac delta function (this should be called a generalized function or a distribution, but we will skip these details). We use the Dirac delta function to represent an idealized point source of light and we may define it as the limit of the following sequence:

Since

We obtain that

2 2 2 2( ) lim expNx y N N x y

2 2

2

2 2 2 2exp exp x yf f

NN N x y

F

2 2

2( ) lim exp 1x yf f

N Nx y

F

(1.6)

(1.7)

(1.8)

Fourier transform as a decomposition





One possible way to interpret the meaning of the Fourier transform is to start from the inverse transformation relationship

and recognize that it implies a superposition of elementary waveforms of the kind:

that contain as parameters the frequencies fx and fy. The values of fx and fy are related to:

1) The direction in the (x,y) plane of the lines with constant phase of the elementary waveforms. In fact, the phase of the elementary waveform is:

2 ( )( ) ( ) x yj f x f y

x y x yg x y G f f e df df

2 ( )x yj f x f ye

2 ( )x yf x f y

(1.10)

(1.9)

(1.11)





By setting = constant we obtain the lines of constant phase:

For example, the lines that have phase either zero or an integer multiple of 2 radians are:

So the elementary waveforms may be regarded as “directed” at an angle (with respect to the x axis) given by:

2 ( )x yf x f y

arctan x

y

ff (1.12)

x

y y

f ny x

f f





2) The spatial period (the distance between zero-phase lines) is given by

2 2

1

x yf fL

(1.13)





Fourier Transform theorems

(Repeat proofs in the Appendix of your textbook as an exercise)

real numbers

g, h : functions with Fourier transforms

Linearity theorem

Similarity theorem

so a strech of the coordinates in the (x,y) plane corresponds to a contraction of the coordinates in the (fx, fy) plane and viceversa. The overall spectrum amplitude is also affected.

]g hF[ ], F[

[ ] [ ] [ ]g h g h F F F

1[ ( )] yxffg x y G F (1.15)

(1.14)





Shif theorem

Rayleigh’s theorem (Parseval’s theorem)

Convolution theorem

The convolution theorem is very important with linear systems.

[ ( )] ( ) exp[ 2 ( )]x y x yg x y G f f j f f F

22( ) ( )x y x yg x y dx dy G f f df df

Energy associated with g(x,y) Energy density

( ) ( ) ( ) ( )x y x yg h x y d d G f f H f f

F

(1.18)

(1.17)

(1.16)





Autocorrelation theorem

and

This theorem may be looked upon as a special case of the convolution theorem where we convolve g(x,y) with g*(-x,-y)

Fourier integral theorem

2( ) *( ) ( )x yg g x y d d G f f

F

2( ) ( ) *( )x yg x y G G f f d d

F

1 1( ) ( ) ( )g x y g x y g x y F F F F (1.21)

(1.20)

(1.19)

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Fourier analysis in two dimensions University of Illinois at Chicago ECE 427, Dr. D. Erricolo University of Illinois at Chicago ECE 427, Dr. D. Erricolo