国家重点实验室 Size 4 and 8 constructions U X1 W U2 X2 W U X3 W U ⊕ Ya X5 Y X6 1000 Yo 1100 Y7 G4= W 1 0 1 0 1111 X=UG 16
Size 4 and 8 constructions 16 4 1000 1100 1010 1111 G = 4 4 X1 14 =U G
国家重点实验室 Size4 construction:奇偶分离 「100 0 110 0 G ⊕ 10 1 0 1111 x4=44G4 W4(y|G)=W(I4©，4®44)W(|2,4)
Size 4 construction ：奇偶分离 17 4 1000 1100 1010 1111 G = 44 2 4 41 1 21 1 23 4 23 24 Wy u Wy u uu uWy uu ( |) ( | , )( |,) = ⊕⊕ 4 4 1 14 x uG =
国家重点实验室 Recursive construction of W XN=uNG ,N=2” 41 V2 Gw=(I.⑧F)Rw(L2⑧Gw) uN2-1 ⊕'w2 WN2 VN/21 =By FOn uNn-H SNI2 'N2 -yN2-1 yNn2 uN12+1 SN12+1 RN VNI2L -yN21 uN2+2 VNI22 o here -yw/242 1 0 SN-1 WNI2 11 uN-+⊕ VN1 YN- VN yN ⑧Kronecker Product R:reverse shuffle operation WN By bit-reversal operator if yN=uN By,then va.b=u 18
Recursive construction of WN here 18 : reverse shuffle operation : bit-reversal operator R N 1 1 N N N x uG = n BN F ⊗ = 2 2 2 ( )( ) N NN N G I FR I G =⊗ ⊗ 1 0 1 1 F = BN 2 n N = 1 1 . . if , then n n N N N bb b b v uB v u = = ⊗ Kronecker Product
国家重点实验室 (奇偶)信道的转移概率 u W0:U→YwxU w2:42 y矿旷1)2思0 g0,4)=∑90,2⊕214⊕4，) W9(0Y,214) 0r,4%)=790,。2Φ214⑧4，) W9(0Y,214） 9
（奇偶）信道的转移概率 19 1 () 1 11 i 1 1 1 1 ( , |) ( | ) N Ni 2 i i Ni N N N N N u X W yu u Wy u − + − − ∈ = ( ) ( ) 2 (2 1) 2 2 1 2 2 2 2 2 1 1 2 1 1 1, 1, 2 1 2 2 22 1 1, 2 1 ( , | ) (, | ) 2 ( , | ) i i Ni Ni i i N i N o ei i u i N i NN e i W y u u W yu u u u Wy u u − − −− − − − + = ⊕ ⊕⋅ ( ) ( ) (2 ) 2 2 1 2 2 2 2 2 1 1 2 1 1, 1, 2 1 2 2 22 1 1, 2 1 ( , |) (, | ) 2 ( , | ) i Ni Ni i i N i N o ei i i N i NN e i W y u u W yu u u u Wy u u − −− − − + = ⊕ ⊕⋅ () 1 : i Ni WUYU N → × − W2 1 u W u 2 W 1 y 2 y 1 x 2 ( ) 2 x 2 W :
一 般表示：Butterfly结构 w( wo N F朗 9 2 e 点 g 点一 Gy =By N B:bit-reversal operator if y =uNBy,then va.b=u
一般表示：Butterfly结构 1 0 1 1 F = n G BF N N ⊗ = ( ) 1 W8 ( ) 2 W8 ( ) 3 W8 ( ) 4 W8 ( ) 5 W8 ( ) 6 W8 ( ) 7 W8 ( ) 8 W8 ( ) 1 W4 ( ) 2 W4 ( ) 3 W4 ( ) 4 W4 ( ) 1 W4 ( ) 2 W4 ( ) 3 W4 ( ) 4 W4 ( ) 1 W2 ( ) 2 W2 ( ) 1 W2 ( ) 2 W2 ( ) 1 W2 ( ) 2 W2 ( ) 1 W2 ( ) 2 W2 W W W W W W W W 1 1 . . if , then n n N N N bb b b v uB v u = = BN: bit-reversal operator