The Fourier transform, indicated by the operator F, constructs a spectrum A(k x,k y ) = F {E(x,y)} from a spatial distribution E(x,y):

Transcription

1 1 4. FOURIER TRANSFORMS (vs. 3.0) DEFINITIONS GENERAL PROPERTIES linearity conjugate functions multiplication and convolution multiplication and correlation Parseval's relation time shifting frequency shifting scaling duality areas differentiation integration separable functions periodic functions Hilbert transform POLAR COORDINATES separable functions circular symmetry scaling DEFINITIONS The Fourier transform, indicated by the operator F, constructs a spectrum A(k x,k y ) = F {E(x,y)} from a spatial distribution E(x,y): A(K x,k y ) = - E(x,y)exp(-jk x x-jk y y)dxdy (4.1) - The inverse Fourier transform F -1 restores the spatial distribution E(x,y) = F -1 {A(k x,k y )}: E(x,y) = ( 1 2π )2. A(k x,k y ;)exp(+jk x x+jk y y)dk x dk y (4.2) - - The sequence of creation and restoration may be summed up in:

2 2 F -1 F {E(x,y)} = E(x,y) (4.3) It should not be thought that the forward transform F must exclusively operate on the spatial distribution and the inverse transform on the spectrum. For example in the relation: F F -1 {E(x,y)} = E(x,y) (4.4) the order of operations is reversed. EX4.1 However, the fundamental relations between the spatial distribution E and the spectrum A are defined by (4.1) and (4.2). (This is not a universal definition, so when reading a paper, conventions must be carefully checked.) In general then, the mathematical difference between forward and inverse transforms is indicated by the sign of the exponent, not by the particular variable used in the integration. Note that the factor (1/2π) 2 is connected with the variables k x and k y, not with the transform's being forward or inverse. There is a clear analogy to electronic communications theory. Apart from the difference in dimensions, the spatial spectrum domain (k x,k y ) corresponds to the temporal spectrum domain (ω) and the space domain (x,y) to the time domain (t). Sometimes we want to use one-dimensional Fourier transforms or inverse transforms. In that case the integrals in (4.1) and (4.2) become single integrals, integrated over the appropriate variable. In (4.2) factor (1/2π) 2 must be replaced by (1/2π) To avoid confusion, we shall indicate onedimensional Fourier transforms by F x, F x -1 or F ky, F ky -1, depending on the integration variable used, i.e x, y, k x or k y. It is possible to formulate the transforms in terms of f x = k x /2π, f y = k y /2π, just as in communications theory f = ω/2π. The variables f x and f y are called spatial frequencies to complete the analogy with the temporal frequency f in communications.

3 3 For simplicity, if no confusion results, we often write symbolic Fourier transforms as E(x,y) = F -1 {A(k x,k y )} = F -1 A(k x,k y ) = F -1 A, etc. A(fx,fy) =. E(x,y)exp(-j2πfxx-j2πfyy)dxdy) (4.5) - - E(x,y) =. A(fx,fy)exp(+j2πfxx+j2πfyy)dfxdfy (4.6) - - GENERAL PROPERTIES 1 linearity F {a 1 E 1 + a 2 E 2 } = a 1 A 1 + a 2 A 2 (4.7) 2 conjugate functions F {E*(x)} = A*(-k x,-k y ) (4.8) Thus if E is a real function then A(k x,k y ) = A*(-k x,-k y ) (4.9) 3 multiplication and convolution * represents convolution F {E 1 E 2 } = ( 1 2π )2 A 1* A 2 (4.10) F {E 1* E 2 } = A 1 A 2 (4.11) 4 multiplication and correlation represents correlation

4 4 F {E 1 E 2 * } = ( 1 2π )2 A 1 A 2 (4.12) F {E 1 E 2 } = A 1 A 2 * ( Parseval's relation A very useful relation is obtained, if in (4.12) we choose E 1 =E 2, let k x and k y 0 and then replace k x ' by k x and k y ' by k y : - E 1 (x,y)e 1 *(x,y) dxdy = - ( 1 2π )2 - A 1 (k x,k y )A 1 *(k x,k y )dk x dk y (4.14) - Eq. (4.14) is called Parseval's theorem. 6 time shifting F {E(x-a,y-b)} = exp(-jk x a-jk y b)a(k x,k y ) (4.15) 7 frequency shifting F {E(x,y)exp(+jk a x+jk b y)} = A(k x -k a,k y -k b ) (4.16) 8 scaling F {E(ax,by)} = 1 ab A(k x a,k y b ) (4.17) 9 duality F {A(x,y)} = 2π E(-k x,-k y ) (4.18)

5 5 10 areas - E(x,y)dxdy = A(0,0) (4.19) A(kx.k y )dk x dk y = E0,0) (4.20) - 11 differentiation F { E x } = jk xa, similar for y (4.21) 12 integration x F { E(x',y)dx'} = 1 A(k jk x,k y ) x 2 A(0,k y)δ(k x ) (4.22) similar for y. δ denotes the delta function x y F { E(x',y')dx'dy' } = -1 A(k k - - x k x,k y ) + 1 y 4 A(0,0)δ(k x)δ(k y ) (4.23) 13 separable functions If F x {E x (x) } = A kx (k x ) and F y {E y (y) } = A ky (k y ) then F {E x (x)e y (y)} = F x {E x (x)}f y {E y (y)} = A kx (k x )A ky (k y ) (4.24) 14 periodic funtions If E x (x) is periodic in x with period L, i.e., if E x (x) can be written as a Fourier series:

6 6 E x (x) = c m exp(j2πmx/l) (4.25) m=- where L/2 c m = 1 E L x (x)exp(-j2πmx/l)dx (4.26) -L/2 then F {E x (x)} = 2π c m δ(k x -m2π/l) (4.27) m=- 15 Hilbert transform If H b {E x (x)} denotes the Hilbert transform of E x (x) then F H b {E x (x)} = -jsgn(k x ) F {E x (x)} =-jsgn(k x )A kx (k x ) (4.28) where sgn denotes the signum function. Similarly for y POLAR COORDINATES If x = rcosθ, y = rsinθ, k x = κcosυ, k y = κsinυ then A(κ,υ) = 0 2π E(r,θ)exp[-jκrcos(θ υ)]rdrdθ (4.29) 0 16 separable functions in polar coordinates If E(r,θ) = E r (r)e θ (θ) and E θ (θ) is a periodic function in θ with period 2π, i.e if E θ (θ) may be written in a Fourier Series: E θ (θ) = c m exp(jmθ) m=- (4.30)

7 7 where c m = 1 2π 2π E θ (θ)exp(-jmθ)dθ (4.31) 0 then F {E(r,θ)} = c m (-j) m exp(jmυ)h m {E r (r)} m=- (4.32) where H m is a Hankel transform of order m: H m {E r (r)} = 0 re r (r)j m (rκ)dr (4.33) and J m is the mth order Bessel function 17 circular symmetry If E(r,θ) = E r (r) then F {E(r,θ)} = B {E r (r)} = A κ (κ) = 2π re r (r)j 0 (rκ)dr (4.34) o and E r (r) = 1 2π κa κ (κ)j 0 (rκ)dκ (4.35) o where B is the Fourier Bessel transform or Hankel transform of zero order. 18 scaling in circular symmetry If B{E r (r)} = A κ (κ) then

8 8 B {E r (ar)} = 1 a 2 A κ( κ a ) (4.36) FOURIER TRANSFORM PAIRS 19 delta function F x {δ(x-a)} = exp(-jk x a) (4.37) 20 complex exponential function F x {exp(jkx)} = 2π δ(k x -K) (4.38) 21 comb function comb(x) = δ(x-m) m=- F x {comb(x)} = comb( k x 2π ) (4.39) whence F x { δ(x-ma)} = 2π a m=- m=- 22 annular delta function δ(k x - m 2π a ) (4.40) B {δ(r-a)} = 2πaJ 0 (κa) (4.41) 23 signum function sgn (x) = 1 if x>0-1 if x<0

9 9 F x {sgn(x)} = 1 jk x (4.42) 24 unit step function u(x) = 1 if x>0 1 if x=0 2 0 if x<0 F x {u(x)} = π δ(k x ) + 1 jk x (4.43) 25 rectangle function rect(x) = 1 for < x < elsewhere F x {rect(x)} = sinc(k x /2π) (4.44) 26 circle function circ(r) = 1 for r<1. B {circ(r)} = 2π J 1(κ) κ (4.45) 27 sinc function sinc(x) = sin(πx) πx F x {sinc(x)} = rect( k x 2π ) (4.46) 28 triangle function Λ(x) = 1- x for x <1

10 10 0 otherwise F x {Λ(x)} = sinc 2 ( k x 2π ) (4.47)