2 x 2 STBC 空時分組碼 解碼 與 均衡

• 本文來源於同門（Jw Lou）總結
• 主要參考文獻：MIMO系統中空時譯碼與頻域均衡研究與實現

主要流程介紹（一組數據塊示例【即兩個OFDM數據塊】）

1、四個SISO分析
$Y_{11}=[X_1H_{11},-X^*_2H_{11}]+Z_{11}$
$Y_{12}=[X_1H_{12},-X^*_2H_{12}]+Z_{12}$
$Y_{21}=[X_2H_{21},X^*_1H_{21}]+Z_{21}$
$Y_{22}=[X_2H_{22},X^*_1H_{22}]+Z_{22}$

1、$Y_{ij}$表示發射天線$i$至接收天線$j$的信號（兩個相關時隙）；
2、$X_i$爲發送信號（頻域）【未STBC編碼之前】；
3、$H_{ij}$表示發射天線$i$至接收天線$j$之間的多徑信道頻域矩陣；
4、$Z_{ij}$表示發射天線$i$至接收天線$j$之間的高斯白噪聲，功率爲$\sigma_{ij}^2$;
5、$(\cdot)^*$表示共軛。

2、MISO接收信號
$R_1=Y_{11}+Y_{21}=[X_1H_{11}+X_2H_{21},-X_2^*H_{11}+X_1^*H_{21}]+Z_{11}+Z_{21}$
$R_2=Y_{12}+Y_{22}=[X_1H_{12}+X_2H_{22},-X_2^*H_{12}+X_1^*H_{22}]+Z_{12}+Z_{22}$

$R_i$表示第$i$個接收天線接收到的信號

3、MISO分時隙表示
$R_{11}=X_1H_{11}+X_2H_{21}+Z_{11}+Z_{21}$
$R_{12}=-X_2^*H_{11}+X_1^*H_{21}+Z_{11}+Z_{21}$
$R_{21}=X_1H_{12}+X_2H_{22}+Z_{12}+Z_{22}$
$R_{22}=-X_2^*H_{12}+X_1^*H_{22}+Z_{12}+Z_{22}$

$R_{ij}$表示第$i$個天線的第$j$個時隙的接收信號。

4、MIMO-STBC解碼

$S_1=R_{11}H_{11}^*+R_{12}^*H_{21}+R_{21}H_{12}^*+R_{22}^*H_{22}$

$S_2=R_{11}H_{21}^*-R_{12}^*H_{11}+R_{21}H_{22}^*-R_{22}^*H_{12}$

$S_i$表示解調得到的發射信號$X_i$

• 進一步可以表示爲：
  $S_1=X_1\sum\limits_{i,j}|H_{ij}|^2+(Z_{11}+Z_{21})H_{11}+(Z_{11}^*+Z_{21}^*)H_{21}+(Z_{12}+Z_{22})H_{12}+(Z_{12}^*+Z_{22}^*)H_{22}$
  $S_2=X_2\sum\limits_{i,j}|H_{ij}|^2-(Z_{11}^*+Z_{21}^*)H_{11}+(Z_{11}+Z_{21})H_{21}^*-(Z_{12}^*+Z_{22}^*)H_{12}+(Z_{12}+Z_{22})H_{22}^*$

• 進一步化簡：

$S_1=KX_1+N_1$
$S_2=KX_2+N_2$

1、$K=\sum\limits_{i,j}|H_{ij}|^2$;
2、$N_1=(Z_{11}+Z_{21})H_{11}+(Z_{11}^*+Z_{21}^*)H_{21}+(Z_{12}+Z_{22})H_{12}+(Z_{12}^*+Z_{22}^*)H_{22}$
3、$N_2=-(Z_{11}^*+Z_{21}^*)H_{11}+(Z_{11}+Z_{21})H_{21}^*-(Z_{12}^*+Z_{22}^*)H_{12}+(Z_{12}+Z_{22})H_{22}^*$
4、$N_1$與$N_2$爲高斯白噪聲，功率爲$\sigma_n^2=\sigma_{n1}^2=\sigma_{n2}^2=(\sigma_{11}^2+\sigma_{21}^2)(\overline{|H_{11}|^2}+\overline{|H_{21}|^2})+(\sigma_{12}^2+\sigma_{22}^2)(\overline{|H_{12}|^2}+\overline{|H_{22}|^2})$

$\overline{(\cdot)}$表示均值

5、MMSE均衡
$W=\frac{K^*}{K^2+\sigma_n^2}=\frac{(\sum\limits_{i,j}|H_{ij}|^2)^*}{(\sum\limits_{i,j}|H_{ij}|^2)^2+(\sigma_{11}^2+\sigma_{21}^2)(\overline{|H_{11}|^2}+\overline{|H_{21}|^2})+(\sigma_{12}^2+\sigma_{22}^2)(\overline{|H_{12}|^2}+\overline{|H_{22}|^2})}$

$W$即爲均衡係數

微信公衆號：通信隨筆XIDIAN