公式推导

(dn−1−w⃗ ′n+1x⃗ n−1)2======(dn−1−(w⃗ n+2μnenx⃗ n)′x⃗ n−1)2d2n−1−2R[d′n−1(w⃗ n+2μnenx⃗ n)′x⃗ n−1]+[(w⃗ n+2μnenx⃗ n)′x⃗ n−1]2d2n−1−2R[d′n−1w⃗ ′nx⃗ n−1]−4R[d′n−1μne′nx⃗ ′nx⃗ n−1]+(w⃗ ′nx⃗ n−1+2μne′nx⃗ ′nx⃗ n−1)2d2n−1−2R[d′n−1w⃗ ′nx⃗ n−1]−4R[d′n−1μne′nx⃗ ′nx⃗ n−1]+(w⃗ ′nx⃗ n−1)2+4R[x⃗ ′n−1w⃗ nμne′nx⃗ ′nx⃗ n−1]+4(μne′nx⃗ ′nx⃗ n−1)24(e′nx⃗ ′nx⃗ n−1)2μ2n+4R[x⃗ ′n−1w⃗ ne′nx⃗ ′nx⃗ n−1−d′n−1e′nx⃗ ′nx⃗ n−1]μn+d2n−1−2R[d′n−1w⃗ ′nx⃗ n−1]+(w⃗ ′nx⃗ n−1)24e2nx2n,n−1μ2n+4R[((wx)′n,n−1−d′n−1)e′nxn,n−1]μn+d2n−1−2R[d′n−1wxn,n−1]+(wxn,n−1)2(1)(2)(3)(4)(5)(6)(7)(8)(9)(10)(11) (1) ( d n − 1 − w → n + 1 ′ x → n − 1 ) 2 = ( d n − 1 − ( w → n + 2 μ n e n x → n ) ′ x → n − 1 ) 2 (2) = d n − 1 2 − 2 R [ d n − 1 ′ ( w → n + 2 μ n e n x → n ) ′ x → n − 1 ] (3) + [ ( w → n + 2 μ n e n x → n ) ′ x → n − 1 ] 2 (4) = d n − 1 2 − 2 R [ d n − 1 ′ w → n ′ x → n − 1 ] − 4 R [ d n − 1 ′ μ n e n ′ x → n ′ x → n − 1 ] + (5) ( w → n ′ x → n − 1 + 2 μ n e n ′ x → n ′ x → n − 1 ) 2 (6) = d n − 1 2 − 2 R [ d n − 1 ′ w → n ′ x → n − 1 ] − 4 R [ d n − 1 ′ μ n e n ′ x → n ′ x → n − 1 ] + ( w → n ′ x → n − 1 ) 2 (7) + 4 R [ x → n − 1 ′ w → n μ n e n ′ x → n ′ x → n − 1 ] + 4 ( μ n e n ′ x → n ′ x → n − 1 ) 2 (8) = 4 ( e n ′ x → n ′ x → n − 1 ) 2 μ n 2 + 4 R [ x → n − 1 ′ w → n e n ′ x → n ′ x → n − 1 − d n − 1 ′ e n ′ x → n ′ x → n − 1 ] μ n (9) + d n − 1 2 − 2 R [ d n − 1 ′ w → n ′ x → n − 1 ] + ( w → n ′ x → n − 1 ) 2 (10) = 4 e n 2 x n , n − 1 2 μ n 2 + 4 R [ ( ( w x ) n , n − 1 ′ − d n − 1 ′ ) e n ′ x n , n − 1 ] μ n + d n − 1 2 (11) − 2 R [ d n − 1 ′ w x n , n − 1 ] + ( w x n , n − 1 ) 2

xn,n−1=x⃗ ′nx⃗ n−1 x n , n − 1 = x → n ′ x → n − 1

wxn,n−1=w⃗ ′nx⃗ n−1 w x n , n − 1 = w → n ′ x → n − 1 ′==conj ′ == c o n j

∑Pl=0=S ∑ l = 0 P = S

[df(n−1)−∑l=0Pwf,l(n+1)⨀xf,l(n−1)]2=[df(n−1)−∑l=0P(wf,l(n)+2μf(n)x′f,l(n)⨀ef(n)))⨀xf,l(n−1)]2(34)(35) (34) [ d f ( n − 1 ) − ∑ l = 0 P w f , l ( n + 1 ) ⨀ x f , l ( n − 1 ) ] 2 (35) = [ d f ( n − 1 ) − ∑ l = 0 P ( w f , l ( n ) + 2 μ f ( n ) x f , l ′ ( n ) ⨀ e f ( n ) ) ) ⨀ x f , l ( n − 1 ) ] 2 如下以一个子带为例： [df(n=...===−1)−∑l=0P(wf,l(n)+2μf(n)x′f,l(n)ef(n))xf,l(n−1)]2d2f(n−1)−2R{d′f(n−1)∑l=0P[wf,l(n)xf,l(n−1)+2μf(n)x′f,l(n)ef(n)xf,l(n−1)]}+{∑l=0P[wf,l(n)xf,l(n−1)+2μf(n)x′f,l(n)ef(n)xf,l(n−1)]}2d2f(n−1)−2R{d′f(n−1)∑l=0P[wf,l(n)xf,l(n−1)]}−4R{d′f(n−1)∑l=0P[x′f,l(n)ef(n)xf,l(n−1)]}μf(n)+[∑l=0Pwf,l(n)xf,l(n−1)]2+4[∑l=0Px′f,l(n)ef(n)xf,l(n−1)]2μ2f(n)+4R{[∑l=0Pwf,l(n)xf,l(n−1)][∑l=0Px′f,l(n)ef(n)xf,l(n−1)]}μf(n)4[∑l=0Px′f,l(n)ef(n)xf,l(n−1)]2μ2f(n)+4R{[∑l=0Pwf,l(n)xf,l(n−1)][∑l=0Px′f,l(n)ef(n)xf,l(n−1)]−d′f(n−1)∑l=0P[x′f,l(n)ef(n)xf,l(n−1)]}μf(n)+d2f(n−1)−2R{d′f(n−1)∑l=0P[wf,l(n)xf,l(n−1)]}+[∑l=0Pwf,l(n)xf,l(n−1)]24[∑l=0Px′f,l(n)xf,l(n−1)]2e2f(n)μ2f(n)+4R{∑l=0P[wf,l(n)xf,l(n−1)−d′f(n−1)]∑l=0P[x′f,l(n)xf,l(n−1)]ef(n)}μf(n)+d2f(n−1)−2R{d′f(n−1)∑l=0P[wf,l(n)xf,l(n−1)]}+[∑l=0Pwf,l(n)xf,l(n−1)]2(36)(37)(38)(39)(40)(41)(42)(43)(44)(45)(46)(47)(48)(49) (36) [ d f ( n − 1 ) − ∑ l = 0 P ( w f , l ( n ) + 2 μ f ( n ) x f , l ′ ( n ) e f ( n ) ) x f , l ( n − 1 ) ] 2 (37) = d f 2 ( n − 1 ) − 2 R { d f ′ ( n − 1 ) ∑ l = 0 P [ w f , l ( n ) x f , l ( n − 1 ) + 2 μ f ( n ) x f , l ′ ( n ) e f ( n ) x f , l ( n − 1 ) ] } (38) + { ∑ l = 0 P [ w f , l ( n ) x f , l ( n − 1 ) + 2 μ f ( n ) x f , l ′ ( n ) e f ( n ) x f , l ( n − 1 ) ] } 2 (39) . . . (40) = d f 2 ( n − 1 ) − 2 R { d f ′ ( n − 1 ) ∑ l = 0 P [ w f , l ( n ) x f , l ( n − 1 ) ] } (41) − 4 R { d f ′ ( n − 1 ) ∑ l = 0 P [ x f , l ′ ( n ) e f ( n ) x f , l ( n − 1 ) ] } μ f ( n ) (42) + [ ∑ l = 0 P w f , l ( n ) x f , l ( n − 1 ) ] 2 + 4 [ ∑ l = 0 P x f , l ′ ( n ) e f ( n ) x f , l ( n − 1 ) ] 2 μ f 2 ( n ) + (43) 4 R { [ ∑ l = 0 P w f , l ( n ) x f , l ( n − 1 ) ] [ ∑ l = 0 P x f , l ′ ( n ) e f ( n ) x f , l ( n − 1 ) ] } μ f ( n ) (44) = 4 [ ∑ l = 0 P x f , l ′ ( n ) e f ( n ) x f , l ( n − 1 ) ] 2 μ f 2 ( n ) + (45) 4 R { [ ∑ l = 0 P w f , l ( n ) x f , l ( n − 1 ) ] [ ∑ l = 0 P x f , l ′ ( n ) e f ( n ) x f , l ( n − 1 ) ] − d f ′ ( n − 1 ) ∑ l = 0 P [ x f , l ′ ( n ) e f ( n ) x f , l ( n − 1 ) ] } μ f ( n ) (46) + d f 2 ( n − 1 ) − 2 R { d f ′ ( n − 1 ) ∑ l = 0 P [ w f , l ( n ) x f , l ( n − 1 ) ] } + [ ∑ l = 0 P w f , l ( n ) x f , l ( n − 1 ) ] 2 (47) = 4 [ ∑ l = 0 P x f , l ′ ( n ) x f , l ( n − 1 ) ] 2 e f 2 ( n ) μ f 2 ( n ) + (48) 4 R { ∑ l = 0 P [ w f , l ( n ) x f , l ( n − 1 ) − d f ′ ( n − 1 ) ] ∑ l = 0 P [ x f , l ′ ( n ) x f , l ( n − 1 ) ] e f ( n ) } μ f ( n ) (49) + d f 2 ( n − 1 ) − 2 R { d f ′ ( n − 1 ) ∑ l = 0 P [ w f , l ( n ) x f , l ( n − 1 ) ] } + [ ∑ l = 0 P w f , l ( n ) x f , l ( n − 1 ) ] 2 $$