信息光学

Sufficient existence conditions



g must be absolutely integrable over the infinite (x,y) plane. g must be only a finite number of discontinuities and a finite number of maxima and minima in any finite rectangle. g must have no infinite discontinuities.

g (at) exp(i2ft )dt
F {g (at)} g (u) exp[i 2f (u / a)]d(u / a) 1 g (u) exp[i(2f / a)u]du a

G ( f / a) / a
If a < 0, the limits flip when we change variables, introducing a minus sign, hence the absolute value.
Linear system theory

The great advantage afforded by linearity is the ability to express the response to a complicated stimulus in terms of the responses to certain “elementary” stimuli.
decomposed A linear combination of elementary stimuli A known response of convenient form Linear combination Response of the initial signal
Signal
Fourier Series & The Fourier Transform
The Fourier Transform of a sum of two functions
Consider the sum of two sine waves (i.e., harmonic waves) of different flting wave is periodic, but not harmonic. Most waves are anharmonic.
Linear Systems

Definition

If f(y,z) optical system 1. af(y,z) optical system
g(Y,Z), Then ag(Y,Z)
2. af1(y,z)+ bf2(y,z) optical system bg2(Y,Z)
ag1(Y,Z)+
Physical possibility is a valid sufficient condition for the existence of a transform.
The conditions above could be violated

For example

Dirac delta function (t)


The Fourier Transform of (t) is 1:


(t )ei 2ft dt ei 2f [0] 1
And the Fourier Transform of 1 is (f):


1 e
i 2ft
df ( f )
The Fourier transform of exp(i2f0t)
g ( x, y) exp[ j 2 ( f
X
X
x fY y )]dxdy
(2 1) (2 2)
g ( x, y ) F -1{G}


G( f
, fY ) exp[ j 2 ( f X x fY y )]df X dfY

G is the Fourier transform (Fourier spectrum or frequency spectrum) of function g of two independent variables x and y fX and fY are frequencies The inverse Fourier transform is sometimes referred to as the Fourier integral representation of a function g(x,y).
What is the Fourier Transform? Anharmonic Waves
Fourier Cosine Series for even functions
Fourier Sine Series for odd functions The continuous limit: the Fourier transform (and its inverse)
Fourier decomposing functions
Here, we write a square wave as a sum of sine waves.
2.1 Fourier analysis in two dimensions

G ( f X , f Y ) F { g}



Such phenomena are called linear, and the property they share is called linearity.

The propagation of light through most media leads to regard optical imaging operations as linear mappings of “object” light distributions into “image” light distributions.
The Fourier transform of cos(2f0t)
F {cos(2f 0t )} cos(2f 0t )ei 2ft dt

cos(2f0t)
0
1 i 2f 0t [e e i 2f 0t ]e i 2ft dt 2 1 i 2 ( f f 0 ) t 1 i 2 ( f f 0 ) t e dt e dt 2 2 1 [ ( f f 0 ) ( f f 0 )] 2 F {cos(2f t)}
Introduction


Many physical phenomena are found experimentally to share the basic property that their response to several stimuli acting simultaneously is identically equal to the sum of the responses that each component stimulus would produce individually. Such phenomena are called linear, and the property they share is called linearity. The propagation of light through most media leads to regard optical imaging operations as linear mappings of “object” light distributions into “image” light distributions.
Linear Systems

A linear system is space invariant if it possesses the property of stationarity. Many physical phenomena are found experimentally to share the basic property that their response to several stimuli acting simultaneously is identically equal to the sum of the responses that each component stimulus would produce individually.
Chapter 2 Analysis of Two-Dimensional Signals and Systems
Chunming Tang
College of Information & Communication Engineering Harbin Engineering University
Lord Kelvin on Fourier’s theorem

Fourier’s theorem is not only one of the most beautiful results of modern analysis, but it may be said to furnish an indispensable instrument in the treatment of nearly every recondite question in modern physics. Lord Kelvin
F {e
i 2f 0t
} e



i 2f 0t i 2ft