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多元LDPC码 FFT-QSPA 算法的置换举例

2022-10-15 13:36 作者:乐吧的数学  | 我要投稿

(录制的视频:https://www.bilibili.com/video/BV1EG4y1n7TQ/

假设我们有个校验方程:
h_1%20c_1%20%2B%20h_2%20c_2%20%2B%20h_3c_3%20%3D%200

我们令 h_3%20%3D%201,然后,考虑给 c_3  计算概率,假如 h_1%20%3D%203%2C%20h_2%20%3D%202%20 , 则
c_1%20%3D%200%20%3D%3D%3D%3E%20c_1%5Ep%20%3D%200%20%5C%5C%0Ac_1%20%3D%201%20%3D%3D%3D%3E%20c_1%5Ep%20%3D%203%20%20%5C%5C%0Ac_1%20%3D%202%20%3D%3D%3D%3E%20c_1%5Ep%20%3D%202%20%5C%5C%0Ac_1%20%3D%203%20%3D%3D%3D%3E%20c_1%5Ep%20%3D%201且:
c_2%20%3D%200%20%3D%3D%3D%3E%20c_2%5Ep%20%3D%200%20%5C%5C%0Ac_2%20%3D%201%20%3D%3D%3D%3E%20c_2%5Ep%20%3D%202%20%5C%5C%0Ac_2%20%3D%202%20%3D%3D%3D%3E%20c_2%5Ep%20%3D%200%20%5C%5C%0Ac_2%20%3D%203%20%3D%3D%3D%3E%20c_2%5Ep%20%3D%202%20

所以,我们要计算的概率有以下四种情况:

%5Csum_%7Bh_1c_1%2Bh_2c_2%3D0%7D%20p(c_1)p(c_2)%20%20%3D%20p_1%5E0(p_2%5E0%2Bp_2%5E2)%20%2B%20p_1%5E2%20(%20p_2%5E1%2Bp_2%5E3)%20%20%5C%5C%0A%5Csum_%7Bh_1c_1%2Bh_2c_2%3D1%7D%20p(c_1)p(c_2)%20%20%3D%20p_1%5E1(p_2%5E1%2Bp_2%5E3)%20%2B%20p_1%5E3%20(%20p_2%5E0%2Bp_2%5E2)%20%20%5C%5C%0A%5Csum_%7Bh_1c_1%2Bh_2c_2%3D2%7D%20p(c_1)p(c_2)%20%20%3D%20p_1%5E0(p_2%5E1%2Bp_2%5E3)%20%2B%20p_1%5E2%20(%20p_2%5E0%2Bp_2%5E2)%20%20%5C%5C%0A%5Csum_%7Bh_1c_1%2Bh_2c_2%3D3%7D%20p(c_1)p(c_2)%20%20%3D%20p_1%5E1(p_2%5E0%2Bp_2%5E2)%20%2B%20p_1%5E3%20(%20p_2%5E1%2Bp_2%5E3)%20%20%5C%5C


所以我们可以构建两个向量:
%5Cbegin%7Bbmatrix%7D%0Ap(c_1%5Ep%20%3D%200)%20%3D%20p(c_1%3D0)%20%3D%20p_1%5E0%20%5C%5C%0Ap(c_1%5Ep%20%3D%201)%20%3D%20p(c_1%3D3)%20%3D%20p_1%5E3%20%5C%5C%0Ap(c_1%5Ep%20%3D%202)%20%3D%20p(c_1%3D2)%20%3D%20p_1%5E2%20%5C%5C%0Ap(c_1%5Ep%20%3D%203)%20%3D%20p(c_1%3D1)%20%3D%20p_1%5E1%0A%5Cend%7Bbmatrix%7D

以及:
%5Cbegin%7Bbmatrix%7D%0Ap(c_2%5Ep%3D0)%20%3Dp(c_2%3D0)%2Bp(c_2%3D2)%20%3D%20p_2%5E0%2Bp_2%5E2%20%20%5C%5C%0Ap(c_2%5Ep%3D1)%20%3D0%20%20%5C%5C%0Ap(c_2%5Ep%3D2)%20%3Dp(c_2%3D1)%2Bp(c_2%3D3)%20%3D%20p_2%5E1%2Bp_2%5E3%20%20%5C%5C%0A0%20%20%0A%5Cend%7Bbmatrix%7D


那么两个向量转到变换域后:
H_2%0A%5Cbegin%7Bbmatrix%7D%0Ap_1%5E0%20%5C%5C%0Ap_1%5E3%20%5C%5C%0Ap_1%5E2%20%5C%5C%0Ap_1%5E1%0A%5Cend%7Bbmatrix%7D%0A%3D%5Cbegin%7Bbmatrix%7D%0Ap_1%5E0%2Bp_1%5E1%2Bp_1%5E2%2Bp_1%5E3%20%5C%5C%0Ap_1%5E0-p_1%5E1%2Bp_1%5E2-p_1%5E3%20%5C%5C%0Ap_1%5E0-p_1%5E1-p_1%5E2%2Bp_1%5E3%20%5C%5C%0Ap_1%5E0%2Bp_1%5E1-p_1%5E2-p_1%5E3%0A%5Cend%7Bbmatrix%7D%20%20%5C%5C%0A%5Cquad%20%5C%5C%0A%0AH_2%0A%5Cbegin%7Bbmatrix%7D%0Ap_2%5E0%2Bp_2%5E2%20%5C%5C%0A0%20%5C%5C%0Ap_2%5E1%2Bp_2%5E3%20%5C%5C%0A0%0A%5Cend%7Bbmatrix%7D%0A%3D%5Cbegin%7Bbmatrix%7D%0Ap_2%5E0%2Bp_2%5E1%2Bp_2%5E2%2Bp_2%5E3%20%5C%5C%0Ap_2%5E0%2Bp_2%5E1%2Bp_2%5E2%2Bp_2%5E3%20%5C%5C%0Ap_2%5E0-p_2%5E1%2Bp_2%5E2-p_2%5E3%20%5C%5C%0Ap_2%5E0-p_2%5E1%2Bp_2%5E2-p_2%5E3%0A%0A%5Cend%7Bbmatrix%7D%20%20%5C%5C%0A%5Cquad%20%5C%5C

然后对上面两个向量,做对应元素相乘有:
V%20%3D%20%5Cbegin%7Bbmatrix%7D%0A(p_1%5E0%2Bp_1%5E1%2Bp_1%5E2%2Bp_1%5E3)(p_2%5E0%2Bp_2%5E1%2Bp_2%5E2%2Bp_2%5E3)%20%5C%5C%0A(p_1%5E0-p_1%5E1%2Bp_1%5E2-p_1%5E3)(p_2%5E0%2Bp_2%5E1%2Bp_2%5E2%2Bp_2%5E3)%20%5C%5C%0A(p_1%5E0-p_1%5E1-p_1%5E2%2Bp_1%5E3)(p_2%5E0-p_2%5E1%2Bp_2%5E2-p_2%5E3)%20%5C%5C%0A(p_1%5E0%2Bp_1%5E1-p_1%5E2-p_1%5E3)(p_2%5E0-p_2%5E1%2Bp_2%5E2-p_2%5E3)%0A%5Cend%7Bbmatrix%7D%20%20%5C%5C

则:
H_2%5E%7B-1%7DV%20%3D%20%5Cfrac%7B1%7D%7B4%7D%20H_2%20V%20%20%3D%20%5C%5C%0A%5Cbegin%7Bbmatrix%7D%0Ap_1%5E0(p_2%5E0%2Bp_2%5E2)%20%2B%20p_1%5E2%20(%20p_2%5E1%2Bp_2%5E3)%20%20%5C%5C%0Ap_1%5E1(p_2%5E1%2Bp_2%5E3)%20%2B%20p_1%5E3%20(%20p_2%5E0%2Bp_2%5E2)%20%5C%5C%0Ap_1%5E0(p_2%5E1%2Bp_2%5E3)%20%2B%20p_1%5E2%20(%20p_2%5E0%2Bp_2%5E2)%20%5C%5C%0Ap_1%5E1(p_2%5E0%2Bp_2%5E2)%20%2B%20p_1%5E3%20(%20p_2%5E1%2Bp_2%5E3)%0A%5Cend%7Bbmatrix%7D

多元LDPC码 FFT-QSPA 算法的置换举例的评论 (共 条)

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