LTE-uplink迭代检测（二）

算法描述：

1）软符号值计算（soft-symbol）

        根据译码器反馈回来的每比特的外信息计算每比特的概率

            \[\Pr \left( {b_{j,k}^{\left( i \right)} =  + 1} \right) = \frac{1}{2} + \frac{1}{2}\tanh \left( {\frac{1}{2}L_{j,k}^{\left( i \right)}} \right)\]

        根据每比特概率计算每符号概率

            \[\Pr \left( {x_j^{\left( i \right)} = a} \right) = \prod\nolimits_k {\Pr \left( {b_{j,k}^{\left( i \right)} = {z_k}} \right)} \]

        计算软符号值

            \[\tilde x_j^{\left( i \right)} = \sum\nolimits_{a \in {\bf{\theta }}} {\Pr \left( {x_j^{\left( i \right)} = a} \right) \cdot a} \]

        DFT变换

            \[{{\bf{\tilde s}}^{\left( i \right)}} = {\left[ {\tilde s_1^{\left( i \right)}\;\tilde s_2^{\left( i \right)}\; \cdots \tilde s_N^{\left( i \right)}} \right]^T} = {F_N}{{\bf{\tilde x}}^{\left( i \right)}}\]

2）并行干扰消除

        第w个子载波消除第i个用户后的接收信号为

             \[{{\bf{\hat y}}_{\left. w \right|i}} = {{\bf{y}}_w} - \sum\nolimits_{j \ne i} {{{\bf{h}}_{j,w}}\tilde s_w^{\left( j \right)}}  = {{\bf{H}}_w}{{\bf{z}}_{\left. w \right|i}} + {\bf{n}}\]

        其中，${{\bf{h}}_{j,w}}$为矩阵${{\bf{H}}_w}$的第j列，${{\bf{z}}_{\left. w \right|i}} = {\left[ {{\bf{z}}_{\left. w \right|i}^{\left( 1 \right)}\;{\bf{z}}_{\left. w \right|i}^{\left( 2 \right)}\; \cdots {\bf{z}}_{\left. w \right|i}^{\left( U \right)}} \right]^T}$定义为

             \[{\bf{z}}_{\left. w \right|i}^{\left( j \right)} = \left\{ \begin{array}{l}
s_w^{\left( i \right)}\;\;,\;\;\;\;\;\;\;\;\;\;\;\;if\;j = i\\
s_w^{\left( j \right)} - \tilde s_w^{\left( j \right)}\;\;,\;\;\;\;if\;j \ne i
\end{array} \right.\]

3）MMSE检测器

          MMSE信道均衡

              \[\hat s_w^{\left( i \right)} = {\bf{w}}_{\left. w \right|i}^H{{\bf{\hat y}}_{\left. w \right|i}} = {E_s}{\left( {{{\bf{h}}_{i,w}}} \right)^H}{\bf{A}}_{\left. w \right|i}^{ - 1}{{\bf{\hat y}}_{\left. w \right|i}}\]

          其中，${\bf{A}}_{\left. w \right|i}^{ - 1} = {{\bf{H}}_w}{{\bf{\Lambda }}_{\left. w \right|i}}{\bf{H}}_w^H + {N_0}{{\bf{I}}_{B \times B}}$，${{\bf{\Lambda }}_{\left. w \right|i}} = E\left[ {{{\bf{z}}_{\left. w \right|i}}{{\left( {{{\bf{z}}_{\left. w \right|i}}} \right)}^H}} \right]\]$是对角矩阵，对角元

             \[\lambda _{\left. w \right|i}^{\left( {j,j} \right)} = \left\{ \begin{array}{l}
{E_s}\;\;\;,\;\;\;\;\;\;\;\;if\;j = i\\
{\mathop{\rm var}} \left[ {s_w^{\left( j \right)}} \right],\;\;if\;j \ne i
\end{array} \right.\]

           其中，${\mathop{\rm var}} \left[ {s_w^{\left( j \right)}} \right] = {N^{ - 1}}\sum\nolimits_{k = 1}^N {\left( {E\left[ {{{\left( {x_k^{\left( j \right)}} \right)}^2}} \right] - {{\left( {\tilde x_k^{\left( j \right)}} \right)}^2}} \right)}$以及

$E\left[ {{{\left( {x_k^{\left( j \right)}} \right)}^2}} \right] = \sum\nolimits_{a \in {\bf{\theta }}} {\Pr \left( {x_j^{\left( i \right)} = a} \right) \cdot {{\left| a \right|}^2}}$ 

       DFT逆变换

          {{\bf{\hat x}}^{\left( i \right)}} = F_N^H{{\bf{\hat s}}^{\left( i \right)}} = {\left[ {\hat x_1^{\left( i \right)}\;\hat x_2^{\left( i \right)} \cdots \hat x_N^{\left( i \right)}} \right]^T}

4）LLR计算

        \hat x_t^{\left( i \right)} = {\mu _i}x_t^{\left( i \right)} + e_t^{\left( i \right)}

     其中，${\mu _i}$为等效信道增益，$e_t^{\left( i \right)}$为均衡后噪声叠加干扰项（NPI），其方差为$\upsilon _i^2$

        \[{\mu _i} = {N^{ - 1}}\sum\nolimits_{k = 1}^N {{\bf{w}}_{\left. w \right|i}^H{{\bf{h}}_{i,w}}} \]

        \[\upsilon _i^2 = {E_s}{\mu _i} - {E_s}{\left| {{\mu _i}} \right|^2}\] 

     逐比特LLR为

        \[\tilde L_{t,k}^{\left( i \right)} = \frac{1}{{\upsilon _i^2}}\left( {\mathop {\min }\limits_{a \in {\bf{\theta }}_k^0} \left| {\hat x_t^{\left( i \right)} - {\mu _i}a} \right| - \mathop {\min }\limits_{a \in {\bf{\theta }}_k^1} \left| {\hat x_t^{\left( i \right)} - {\mu _i}a} \right|} \right)\]