.? (x i, y i )? (x(t), y(t))? (x(t)) (X(ω)) Wiener-Khintchine 35/97
. : x(t) = X(ω)e jωt dω () π X(ω) = x(t)e jωt dt () X(ω) S(ω) = lim (3) ω S(ω)dω X(ω) : F of x : [X] [ = ] [x t] Power spectral density : [S] = X = [ x t ] [ ] Power spectrum : [S dω] = X t t = [ x ] [ Energy spectral density : X ] = [ x t ] 36/97
. : f i (i (,,N)) E[f] N N f i (4) i= σ f E[ (f E[f]) ] (5) : (x i, y i ) C = E [ x y ], or r = E[x y ] E [ x ] E [ y ] (6) y y (x i = x i E[x], y i = y i E[y]) x x No correlation Positive correlation 37/97
.3 x, y E[x] lim x(t)dt = x(t) t ( ) (7) e.g. x(t): y(t): x(t) t y(t) t 38/97
(cross-correlation function) C xy (τ) = lim x(t)y(t+τ)dt = x(t)y(t+τ) t (8) (y(t) = x(t)) (auto-correlation function) C(τ) = lim ) x(t)x(t+τ)dt = x(t)x(t+τ) t (9) ( R(τ) = C(τ) C(0) = x(t)x(t+τ) t x(t), R(0) = t 39/97
(e.g. ) x(t) = Acosω t C(τ) = lim (Acosω t)(acosω (t+τ))dt = lim = A lim [ = A lim ( cos ω tcosω τ cosω tsinω tsinω τ ) dt +cosω t dtcosω τ } {{ } = ( (e.g. ) x(t) = Asinω t = Acos ω t π ) [ C(τ) = A lim sinω t dt }{{} =0 +cos ( ( ω t π )) dtcosω τ (Acosω t)(acosω (t+τ))dt ] sinω τ = A cosω τ R(τ) = cosω τ sin ( ( ω t π )) ] dtsinω τ = A cosω τ R(τ) = cosω τ t 40/97
( ) (C( τ) = C(τ)) C( τ) = lim = lim = lim τ x(t)x(t τ)dt τ x(t +τ)x(t )dt τ = 0 x(t +τ)x(t )dt = C(τ) (x(t)±x(t+τ)) dt 0 LHS = x (t)dt+ x (t+τ)dt }{{} } {{ } C(0) 0 C(0) 0 = (C(0)±C(τ)) 0 = RHS C(0) C(τ) ± (t = t τ) ( ) τ x(t)x(t+τ)dt } {{ } C(τ) 4/97
C(τ) C (τ) = dc dτ = lim x(t) x(t+τ) } τ {{} =x (t+τ) dt = x(t)x (t+τ) t (Replacing t+τ = ξ dt = dξ,because (t+τ) τ = ) +τ = lim x(ξ τ)x (ξ)dξ +τ = lim x(ξ τ)x (ξ)dξ = x(t τ)x (t) t x(t) x (t) C(τ) C (τ) = d C dτ = x(t)x (t+τ) t ( t τ = η ( (t τ) τ = )) ( = lim x (η)x (η +τ) ) dη = x (t)x (t+τ) t x(t) x (t) x (t) 4/97
.4 : (x(t) R) { 0 x(t) : t = 0 elsewhere : C(τ) = lim = lim = lim π = π x(t)x(t+τ)dt : x(t) = ( X(ω)e jωt dω = ) X (ω)e jωt dω π π X(ω) = x(t)e jωt dt [ ][ ] X (ω)e jωt dω X(ω )e jω (t+τ) dω dt ( ) X (ω)e jωτ X(ω ) e j(ω ω)t dt dω dω π }{{} X (ω)x(ω) lim } {{ } =S(ω) δ(ω ω) e jωτ dω = S(ω)e jωτ dω π C(τ) S(ω) F 43/97
Wiener-Khintchine C(τ) = π S(ω) = S(ω)e jωτ dω (0) C(τ)e jωτ dτ () x(t) F X(ω) x(t)x(t+τ) t X(ω)X (ω) t C(τ) F S(ω) t : 44/97
.5 C xy (τ) = π S xy (ω) = S xy (ω)e jωτ dω () C xy (τ)e jωτ dτ (3) X (ω)y(ω) S xy (ω) = lim = X (ω)y(ω) t (4) C xy (τ) = lim x(t)y(t+τ)dt = lim x (t)y(t+τ)dt ( x(t),y(t) R) ( )( ) = lim X (ω)e jωt dω Y(ω)e jω (t+τ) dω dt π ω π ω ( ) = lim X (ω)e jωτ Y(ω π ω ω ) e j(ω ω)t dt dω dω π = lim X (ω)e jωτ Y(ω π ω ω )δ(ω ω)dω dω = lim X (ω)y(ω)e jωτ dω π ω = X (ω)y(ω) lim e jωτ dω = S xy (ω)e jωτ dω π π ω ω 45/97
C xy (τ) = lim S xy (ω) = F {C xy (τ)} x(t)y(t+τ)dt = lim C xy (τ) R ( x(t),y(t) R) y(t)x(t τ)dt = C yx ( τ) C xy ( τ) = C yx (τ) (5) S xy ( ω) = Sxy(ω) (6) S yx (ω) = Sxy(ω) (7) S xy ( ω) = S yx (ω) (8) 46/97
g(r) = f(r + ), = ( 8,6) (Image size:8x8) C f(r) g(r) fg (r) Ĉ fg (r) C fg = F { F G t } (w/o POC) (w/ POC) } Ĉ fg = F { F Ĝ t F = F F, Ĝ = G G F Ĝ = e i(φ G φ F ) ( 8, 6) ( 8.0, 6.0) POC:Phase Only cross-correlation g(r) = f(r + ) G(k) = e ik F(k) F (k)ĝ(k) = eik (φ F (k) is canceled.) Ĉ fg (r) = F { e ik r } = lim L L (π) e ik ( r) dk = lim L Lδ(r ) 47/97
g(r) = f(αθ r + ), = ( 8,6), α =, Θ = ( cosθ sinθ sinθ cosθ ), θ = 30deg (Image size:8x8) Cartesian log-polar Hanning-Win. (x,y) (k x,k y ) (k θ,log k ) (k θ,log k ) f(r) I F (k) = F(k) H{I F (k)} C IF I G = F {(F {H{I F }}) F {H{I G }}} Cross-corr. of I F and I G (k θ,log k ) C IF I G (k) w/o POC w/ POC g(r) I G (k) = G(k) H{I G (k)} ( ) ( ) log scale log scale [0.05, 500] [0.05, 500] k θ [deg] 9.6 9.4 log k 0.343 0.68 [times] k 0.70 0.505 48/97