# Capacity of the Gaussian Two-Way Relay Channel to Within 1/2 Bit

2021-09-15 05:01:41 程序员大本营

# System model

## [Signal Flow]

Wi∈{1,2,3,...,2nRi}W_i \in \left\{1,2,3,..., 2^{nR_i}\right\}Wi{1,2,3,...,2nRi}

Xi=[Xi(1),Xi(2),...,Xi(n)]⊤\bm{X_i}=\left[X^{(1)}_i,X^{(2)}_i,...,X^{(n)}_i\right]^\topXi=[Xi(1),Xi(2),...,Xi(n)]

YR=[YR(1),YR(2),...,YR(n)]⊤\bm{Y_R}=\left[Y^{(1)}_R,Y^{(2)}_R,...,Y^{(n)}_R\right]^\topYR=[YR(1),YR(2),...,YR(n)]

Relay 从接收到的 YRY_RYR 中解出信息 XRX_RXR 并把它广播出去
XR=[XR(1),XR(2),...,XR(n)]⊤\bm{X_R}=\left[X^{(1)}_R,X^{(2)}_R,...,X^{(n)}_R\right]^\topXR=[XR(1),XR(2),...,XR(n)]

Yi=[Yi(1),Yi(2),...,Yi(n)]⊤\bm{Y_i}=\left[Y^{(1)}_i,Y^{(2)}_i,...,Y^{(n)}_i\right]^\topYi=[Yi(1),Yi(2),...,Yi(n)]

W^i∈{1,2,3,...,2nRi}\hat{W}_i \in \left\{1,2,3,..., 2^{nR_i}\right\}W^i{1,2,3,...,2nRi}

## [Channels]

1. YR(t)=X1(t)+X2(t)+ZR(t)Y^{(t)}_R=X^{(t)}_1+X^{(t)}_2+Z^{(t)}_RYR(t)=X1(t)+X2(t)+ZR(t), 其中 ZR(t)∼CN(0,σR2)Z^{(t)}_R\sim\mathcal{CN}(0,\sigma^2_R)ZR(t)CN(0,σR2).
2. Xi(t)=fi(t)(Wi,Yi(t−1))X^{(t)}_i=f^{(t)}_i(W_i,\bm{Y}^{(t-1)}_i)Xi(t)=fi(t)(Wi,Yi(t1)) 即每一时刻传输的信号 Xi(t)X^{(t)}_iXi(t) 不仅是整个待传信息 WiW_iWi 的函数, 而且还是之前所有下行接收信号的函数 (Yi(t−1)\bm{Y}^{(t-1)}_iYi(t1) 是vector)。
3. 上行每个人都有功率限制 PiP_iPi: 1n∑t=1n(Xi(t))2≤Pi\frac{1}{n}\sum_{t=1}^{n}\left(X^{(t)}_i \right)^2\leq P_in1t=1n(Xi(t))2Pi.

1. Yi(t)=XR(t)+Zi(t)Y^{(t)}_i=X^{(t)}_R+Z^{(t)}_iYi(t)=XR(t)+Zi(t), 其中 Zi(t)∼CN(0,σi2)Z^{(t)}_i\sim\mathcal{CN}(0,\sigma^2_i)Zi(t)CN(0,σi2).
2. XR(t)=fR(t)(YR(t−1))X^{(t)}_R=f^{(t)}_R\left(\bm{Y}^{(t-1)}_R\right)XR(t)=fR(t)(YR(t1)), 即relay 没有自己想传输的信息, XR(t)X^{(t)}_RXR(t) 由它过去收到的所有信息决定。
3. Relay的功率限制: 1n∑t=1n(XR(t))2≤PR\frac{1}{n}\sum_{t=1}^{n}\left(X^{(t)}_R \right)^2\leq P_Rn1t=1n(XR(t))2PR.

## [Decoding]

W^2=g1(W1,Y1)\hat{W}_2=g_1(W_1,\bm{Y}_1)W^2=g1(W1,Y1)

W^1=g2(W2,Y2)\hat{W}_1=g_2(W_2,\bm{Y}_2)W^1=g2(W2,Y2)

Average probability of error:
Pe=Pr{W^1≠W1 or W^2≠W2}P_e=\text{Pr}\{\hat{W}_1\neq {W}_1~\text{or}~\hat{W}_2\neq {W}_2\}Pe=Pr{W^1=W1 or W^2=W2}

A rate pair (R1,R2)(R_1,R_2)(R1,R2) is achievable if 存在一组编码解码函数 fff and ggg 使得 n→∞n\to\inftynPe→0P_e\to 0Pe0.

Capacity region 是所有可达的 rate pair 的闭集。

https://www.pianshen.com/article/79032093600/