We consider a two-user multiple access channel $p(y \cond x_1,x_2)$ with binary input alphabets $\cX_1 = \cX_2 = \{0,1\}$ and an arbitrary output alphabet $\cY$ <cit.>. An $(R_1,R_2,n)$ code for this channel consists of encoders $g_j: [2^{nR_j}] \to \cX_j^n$ that map each message $M_j$ to an channel input sequence $X_j^n$ for $j=1,2$, and a decoder $\psi: \cY^n \to [2^{nR_1}] \times [2^{nR_2}]$ that assigns message estimates $(\hat{M}_1,\hat{M}_2)$ to each received sequence $Y^n$. The average probability of error of the code is $\eps = \P\left\{\{\widehat{M}_1 \neq M_1\} \cup \{\widehat{M}_2 \neq M_2\}\right\}$. It is well-known that a rate pair $(R_1,R_2)$ is achievable if

\begin{equation} \label{eqn:mac_region}
\begin{aligned}
R_1 &< I(X_1;Y \cond X_2),\\
R_2 &< I(X_2;Y \cond X_1),\\
R_1+R_2 &< I(X_1,X_2;Y),
\end{aligned}
\end{equation}

for some pmf $p(x_1)p(x_2)$ [47, 64, 65].

§.§ Coding Scheme

Let $p(x_1)$ and $p(x_2)$ be some desired input pmf's. The coding scheme for the multiple access channel uses the following two Lego bricks.

[Asym $\to$ MAC]
an $(R_1,n)$ asymmetric channel code $(g_1,\psi_1)$ for the channel

\[
p(y\cond x_1) = \sum_{x_2}p(x_2) p(y\cond  x_1,x_2),
\]

which targets an input distribution $p(x_1)$, such that, for $M_1 \sim \Unif([2^{nR_1}])$, the channel input $X_1^n = g_1(M_1)$ satisfies

\begin{equation} \label{eqn:p2p_mac_1}
\frac{1}{2}\sum_{x_1^n}\left| \P\{X_1^n=x_1^n\} - \prod_{i=1}^n p(x_{1i}) \right| \leq \delta_1,
\end{equation}

for some $\delta_1 > 0$. Let $\eps_1$ be the average probability of error of the asymmetric channel code when the channel input distribution is i.i.d. according to $p(x_1)$.

[Asym $\to$ MAC]
an $(R_2,n)$ asymmetric channel code $(g_2,\psi_2)$ for the channel

\[
p(y,x_1\cond x_2) = p(x_1) p(y\cond  x_1,x_2),
\]

which targets an input distribution $p(x_2)$, such that, for $M_2 \sim \Unif([2^{nR_2}])$, the channel input $X_2^n = g_2(M_2)$ satisfies

\begin{equation} \label{eqn:p2p_mac_2}
\frac{1}{2}\sum_{x_2^n}\left| \P\{X_2^n=x_2^n\} - \prod_{i=1}^n p(x_{2i}) \right| \leq \delta_2,
\end{equation}

for some $\delta_2 > 0$. Let $\eps_2$ be the average probability of error of the asymmetric channel code when the channel input distribution is i.i.d. according to $p(x_2)$.

Each of the asymmetric channel codes can be constructed starting from a point-to-point channel code and a Slepian–Wolf code, as described in Section <ref>.

Fig. <ref> shows the block diagram of the coding scheme. Inspired by Slepian and Wolf's random coding scheme for the multiple access channel [65], the two asymmetric channel encoders are used to target the desired joint pmf's $p(x_1)$ and $p(x_2)$ at the encoder side, and the decoder recovers estimates of the messages sequentially, as per successive cancellation decoding. It can be shown that the average probability of error of the coding scheme can be bounded as

\[
\P\left\{\{\widehat{M}_1 \neq M_1\} \cup \{\widehat{M}_2 \neq M_2\}\right\} \leq \eps_1 + \eps_2 + \delta_1 + \delta_2.
\]

Coding scheme for multiple access channel using two asymmetric channel codes.

If the rate of the first asymmetric channel code is $R_1 = I(X_1;Y) - \gamma_1$ for some $\gamma_1 > 0$, and the rate of the second asymmetric channel code is   $R_2 = I(X_2;X_1,Y) - \gamma_2 = I(X_2;Y \cond X_1) - \gamma_2$ for some $\gamma_2 > 0$, then the coding scheme attains the rate pair

\[
(R_1,R_2) = \left(I(X_1;Y) - \gamma_1,\: I(X_2;Y \cond X_1) - \gamma_2   \right).
\]

Note that the rate pair $\left(I(X_1;Y),\: I(X_2;Y \cond X_1)\right)$ is a corner point of the optimal rate region (<ref>) of the multiple access channel. By reversing the decoding order and appropriately modifying the channels that the two asymmetric channel codes are designed for, another corner point of the rate region can be achieved.

A similar coding scheme can be constructed for the $K$-user multiple access channel using $K$ asymmetric channel codes, where the decoder successively decodes the user messages.

§ MARTON CODING FOR BROADCAST CHANNELS

In this part, we construct a Marton coding scheme for a discrete memoryless broadcast channel starting from an asymmetric channel code and a Gelfand–Pinsker code. When the constituent codes are rate-optimal, the coding scheme can achieve a corner point in the achievable rate region of Marton coding.

§.§ Problem Statement

We consider a broadcast channel $p(y_1^K \cond x_1^K)$ with $K$ users and $K$ transmit antennas with a binary input alphabet $\cX=\{0,1\}$ and output alphabets $\cY_1, \ldots, \cY_K$ <cit.>. An $(R_1,\ldots,R_K,n)$ code $(g,\psi_1, \ldots,\psi_K)$ for this channel consists of an encoder $g:[2^{nR_1}] \times \cdots \times [2^{nR_K}] \to \cX^{Kn}$ that maps each message tuple $(M_1,\ldots,M_K)$ to $K$ input sequences $(X_1^n,\ldots,X_K^n)$, and decoders $\psi_j: \cY_j^n \to [2^{nR_j}]$, for $j\in [K]$, that assign message estimates $\hat{M}_j$ to each received sequence $Y_j^n$. The average probability of error of the code is $\eps = \P\left\{ \widehat{M}_j \neq M_j \text{ for some } j \in [K] \right\}$. For an input distribution $p(x_1,\ldots,x_K)$, the achievable rate region of Marton's coding scheme is the set of rate tuples $(R_1,\ldots,R_K)$ such that

\[
R(\cS) < \sum_{j \in \cS}I(X_j;Y_j) - I^{*}(X_\cS),
\]

for all $\cS \subseteq [K]$, where $R(\cS) = \sum_{j \in \cS} R_j$, $X_\cS = (X_j: j\in\cS)$, and

\begin{equation} \label{eqn:mutual_info_star}
I^{*}(X_\cS) = \sum_{j\in\cS}I(X_j;X_{[j-1]\cap \cS}).
\end{equation}

For example, for $K=2$ and an input distribution $p(x_1,x_2)$, the achievable rate region is the set of rate pairs $(R_1,R_2)$ such that

\begin{equation} \label{eqn:marton_rate}
\begin{aligned}
R_1 &< I(X_1;Y_1),\\
R_2 &< I(X_2;Y_2),\\
R_1 + R_2 &< I(X_1;Y_1) + I(X_2;Y_2) - I(X_1;X_2).
\end{aligned}
\end{equation}

§.§ Coding Scheme

We consider a two-user binary-input broadcast channel $p(y_1,y_2\cond x_1,x_2)$ with two transmit antennas. The proposed Marton coding scheme targets an input distribution $p(x_1,x_2)$ and uses the following asymmetric channel code and Gelfand–Pinsker code as its constituent Lego bricks.

[Asym $\to$ Marton]
an $(R_1,n)$ asymmetric channel code $(g_1,\psi_1)$ for the channel

\[
p(y_1\cond x_1) = \sum_{y_2,x_2}p(x_2\cond x_1) p(y_1,y_2\cond  x_1,x_2),
\]

which targets an input distribution $p(x_1)$, such that, for $M_1 \sim \Unif([2^{nR_1}])$, the channel input $X_1^n = g_1(M_1)$ satisfies

\begin{equation} \label{eqn:tv_marton_asym}
\frac{1}{2}\sum_{x_1^n}\left| \P\{X_1^n=x_1^n\} - \prod_{i=1}^n p(x_{1i}) \right| \leq \delta_1,
\end{equation}

for some $\delta_1 > 0$. Let $\eps_1$ be the average probability of error of the asymmetric channel code when the channel input distribution is i.i.d. according to $p(x_1)$.

[GP $\to$ Marton]
an $(R_2,n)$ code $(g_2, \psi_2)$ for the Gelfand–Pinsker problem defined by

\[
p(x_1)p(y_2\cond  x_2,x_1) = p(x_1)\sum_{y_1}p(y_1,y_2\cond  x_1,x_2),
\]

which targets a conditional distribution $p(x_2\cond x_1)$, such that, when $M_2 \sim \Unif([2^{nR_2}])$ and $\wtilde{X}_1^n$ is i.i.d. $p(x_1)$ sequence, the channel input $\wtilde{X}_2^n = g_2(M_2, \wtilde{X}_1^n)$ satisfies

\begin{equation} \label{eqn:tv_marton_gelfand}
\frac{1}{2}\sum_{x_1^n,x_2^n}\left| \P\{\wtilde{X}_1^n=x_1^n,\wtilde{X}_2^n=x_2^n\} - \prod_{i=1}^n p(x_{1i},x_{2i}) \right| \leq \delta_2,
\end{equation}

for some $\delta_2 > 0$. Let $\eps_2$ be the average probability of error of the Gelfand–Pinsker code when the conditional distribution of the channel input given the channel state is i.i.d. according to $p(x_2 \cond x_1)$.

Marton coding for the two-user broadcast channel using an asymmetric channel code and a Gelfand–Pinsker code.

Fig. <ref> shows the block diagram of the Marton coding scheme for the broadcast channel. The main idea is to view the sequence that is encoded by the asymmetric channel code as a state sequence to the Gelfand–Pinsker encoder. From conditions (<ref>) and (<ref>), it follows from standard arguments that

\[
\frac{1}{2}\sum_{x_1^n,x_2^n}\left| \P\{X_1^n=x_1^n,X_2^n=x_2^n\} - \prod_{i=1}^n p(x_{1i},x_{2i}) \right| \leq \delta_1+\delta_2,
\]

which says that the channel input distribution is $(\delta_1+\delta_2)$-away in total variation distance from the desired target distribution. The average probability of error of the coding scheme can be bounded as

\begin{equation*}
\P\left\{ \{\widehat{M}_1 \neq M_1\} \cup \{\widehat{M}_2 \neq M_2\} \right\} \, \leq \, \delta_1 + \delta_2 + \epsilon_1 + \epsilon_2,
\end{equation*}

which follows by a similar analysis as in Section <ref>.

If the rate of the asymmetric channel code is $R_1 = I(X_1;Y_1) - \gamma_1$ for some $\gamma_1 > 0$, and the rate of the Gelfand–Pinsker code is $R_2 = I(X_2;Y_2) - I(X_1;X_2) -\gamma_2$ for some $\gamma_2 > 0$, it follows that the Marton coding scheme achieves the rate pair

\[
(R_1,R_2) = \left(I(X_1;Y_1)- \gamma_1, \, I(X_2;Y_2) - I(X_1;X_2)- \gamma_2\right).
\]

Note that the rate pair $\left(I(X_1;Y_1), \, I(X_2;Y_2) - I(X_1;X_2)\right)$ is a corner point of Marton's rate region for the broadcast channel. If the encoding order is reversed (i.e., $X_2^n$ is encoded using an asymmetric channel code and used as a state sequence to encode $M_1$), another corner point of the rate region can be achieved.

A similar coding scheme can be constructed for the $K$-user broadcast channel with $K$ transmit antennas using one asymmetric channel code and $K-1$ Gelfand–Pinsker codes, where the encoder successively encodes the user messages.

§.§ Simulation Results

Now, we verify the practicality of the Marton coding scheme through simulations. We use polar codes with successive cancellation decoding as the constituent point-to-point channel codes. We consider a two-user Gaussian broadcast channel with two transmit antennas and a single receive antenna for each user. The channel can be modeled by the following input-output relations:

\begin{equation} \label{eqn:gaussian_channel}
\begin{bmatrix}
Y_1 \\
\end{bmatrix} = H_{\mathrm{ch}}W\begin{bmatrix}
X_1 \\
\end{bmatrix} + \begin{bmatrix}
Z_1 \\
\end{bmatrix},
% 	{\bf y} = {\bf H_{ch}}{\bf W} {\bf x} + {\bf z},
\end{equation}

where $H_{\mathrm{ch}} = \begin{bmatrix}
1 & g \\
g & 1
\end{bmatrix}$ is the channel gain matrix with $g=0.9$, $W$ is a $2 \times 2$ precoding matrix used by the transmitter (if needed), $X_1 \in \{\pm 1\}$ and $X_2 \in \{\pm 1\}$ are BPSK-modulated signals corresponding to the channel input codewords (i.e., bit 0 is mapped to +1 and bit 1 is mapped to $-1$), $\begin{bmatrix}
\end{bmatrix} \sim \mathcal{N}\big({\bf 0}, I\big)$ is a vector of independent Gaussian noise components ($I$ is the $2\times 2$ identity matrix), and $Y_1$ and $Y_2$ are the channel outputs at the users' side. The transmitter wishes to send messages to the users at an overall sum-rate $R_{\mathrm{sum}}$, while being subject to a sum-power constraint $P$ such that

\begin{equation} \label{eqn:marton_power_constraint}
\E\left[\left\|W\begin{bmatrix}
X_1 \\
\end{bmatrix}\right\|^2\right] \leq P.
\end{equation}

For a given input distribution $p(x_1,x_2)$ and a precoding matrix $W$ satisfying the power constraint (<ref>), the maximum achievable sum-rate of the Marton coding scheme can be expressed as

\begin{equation}\label{eqn:marton_sum_rate}
C_{\mathrm{sum}}\big(p(x_1,x_2), W\big) \triangleq I(X_1;Y_1) + I(X_2;Y_2) - I(X_1;X_2).
\end{equation}

The following coding strategies for the broadcast channel will be compared.

* “Marton coding with optimal precoding”[Marton coding over the Gaussian broadcast channel is often seen as an instance of “dirty paper coding” [27]. However, note that the channel input is binary in our case.]: corresponds to the proposed coding scheme that targets the channel input distribution $p^{*}_{\mathrm{marton,opt}}(x_1,x_2)$ and precoding matrix $W^{*}_{\mathrm{marton,\, opt}}$ that maximize $C_{\mathrm{sum}}$, while satisfying the power constraint (<ref>).

* “Marton coding without precoding”: corresponds to the proposed coding scheme that targets the channel input distribution $p^{*}_{\mathrm{marton}}(x_1,x_2)$ that maximizes $C_{\mathrm{sum}}$, while setting the precoding matrix to $W = \sqrt{\frac{P}{2}}I$.

* “Symmetric coding with optimal precoding”: corresponds to the strategy that sets the precoding matrix to $W^{*}_{\mathrm{sym,opt}}$ that maximizes $C_{\mathrm{sum}}$, while setting the input distribution to the i.i.d. $\Bern(1/2)$ distribution (i.e., the two messages are encoded independently using two separate point-to-point channel codes, and the correlation between the two transmitted signals is attributed only to the linear precoder).

* “Symmetric coding without precoding”: corresponds to the strategy that encodes the two messages independently using two separate point-to-point channel codes, while setting the precoding matrix to $W = \sqrt{\frac{P}{2}}I$.

* “Minimum mean-square error (MMSE) precoding”: corresponds to the strategy that encodes the two messages independently using two separate point-to-point channel codes, while applying the MMSE precoding matrix $W_{\mathrm{MMSE}} = \lambda(H_{\mathrm{ch}}^TH_{\mathrm{ch}} + \frac{K}{P}I)^{-1}H_{\mathrm{ch}}^T$ at the transmitter side, where $K=2$ is the number of users and $\lambda$ is a constant to satisfy the power constraint (<ref>). The MMSE precoding matrix is sometimes referred to as the “transmit Wiener filter” [66].

* “Zero-forcing precoding”: corresponds to the strategy that encodes the two messages independently using two separate point-to-point channel codes, while applying the zero-forcing precoding matrix $W_{\mathrm{ZF}} = \lambda H_{\mathrm{ch}}^{-1}$ at the transmitter side, where $\lambda$ is a constant to satisfy the power constraint (<ref>). Such a strategy suppresses the interference in the channel.

* “Time division”: corresponds to the strategy that serves only a single user of the channel. That is, a single message is encoded using a point-to-point channel code, and the same codeword is transmitted across both antennas. The power allocation between the two transmit antennas is done so as to optimize the received signal-to-noise ratio. This is done by setting the precoding matrix to $W_{\mathrm{time-division}} = \begin{bmatrix}
\sqrt{\lambda_1} & 0\\
0 & \sqrt{\lambda_2}
\end{bmatrix}$, where $(\lambda_1,\lambda_2)$ is the solution of

\[
\begin{cases}
\text{maximize}\qquad & (\sqrt{\lambda_1} + g\sqrt{\lambda_2})^2 \\
\text{subject to}\qquad & \lambda_1+\lambda_2=P,\\
& \lambda_1\geq 0, \lambda_2 \geq 0.
\end{cases}
\]

Before simulating the proposed Marton coding scheme, we first plot the maximum achievable sum-rate $C_{\mathrm{sum}}$ given in equation (<ref>) for the different coding strategies. For the coding strategies that optimize over the channel input distribution and/or the precoding matrix, we use particle swarm optimization [67, 68] as an efficient heuristic method to perform the optimization. Fig. <ref> shows the plot of $C_{\mathrm{sum}}$ as a function of the sum-power constraint $P$ for the different coding strategies over the two-user Gaussian broadcast channel model given by (<ref>). Clearly, Marton coding with optimal precoding can achieve strictly larger sum-rates asymptotically compared to the other coding strategies. When no precoding is employed, Marton coding can still achieve significantly larger sum-rates compared to common linear precoding strategies used in practice such as MMSE precoding and zero-forcing precoding. This demonstrates the significance of stochastically shaping the channel input signals over the broadcast channel and motivates our Lego-brick design of a realizable Marton coding scheme using commercial off-the-shelf codes. Such an observation has been previously noted in the literature (e.g., [69, 70]).

The maximum achievable sum-rate $C_{\mathrm{sum}}$ for the different coding strategies over the broadcast channel.

Next, we simulate the different coding strategies over the Gaussian broadcast channel model using polar codes with successive cancellation decoding as the constituent point-to-point channel codes. The coding strategies are compared for the same block length $n=1024$ and sum-rate $R_{\mathrm{sum}} = 1$. Recall that each of the asymmetric channel code and the Gelfand–Pinkser code can be implemented using a pair of point-to-point channel codes, as described in Section <ref>. It follows that four polar codes are needed to implement the Marton coding scheme. Let $(R_{11},R_{12},R_{21},R_{22})$ be the rates of the polar codes, where $(R_{11},R_{12})$ are the rates of the two polar codes needed to construct the asymmetric channel code and $(R_{21},R_{22})$ are the rates of the two polar codes needed to construct the Gelfand–Pinsker code. The rates $(R_{11},R_{12},R_{21},R_{22})$ are chosen “close” to their theoretical limits (see Remarks <ref> and <ref>). More precisely, for the Marton coding scheme, we set

\begin{equation} \label{eqn:marton_rates}
\begin{aligned}
R_{11} &= 1-H(X_1\cond Y_1)-\gamma,\\
R_{12} &= 1-H(X_1), \\
R_{21} &= 1-H(X_2\cond Y_2)-\gamma,\\
R_{22} &= 1-H(X_2 \cond X_1),
\end{aligned}
\end{equation}

where $\gamma > 0$ is a “back-off” parameter from the theoretical limit, which allows for a reasonable error probability[Note that the back-off parameter is only used for $R_{11}$ and $R_{21}$ since these rates correspond to the polar codes that are used for error correction. In contrast, our simulations show that the polar codes used for shaping perform pretty well, even at rates that are extremely close to the theoretical limit; hence, no back-off is needed for $R_{12}$ and $R_{22}$.]. Note that the sum-rate attained by the coding scheme is equal to

\[
R_{11}-R_{12}+R_{21}-R_{22} = I(X_1;Y_1)+I(X_2;Y_2)-I(X_1;X_2)-2\gamma.
\]

Hence, in order to guarantee that this sum-rate is equal to $R_{\mathrm{sum}}$, we consider the power level $P^{*}$ at which $C_{\mathrm{sum}} = R_{\mathrm{sum}} + 2\gamma$.[For each coding strategy, the power level $P^{*}$ can be estimated from the plot of Fig. <ref>.] For example, for the Marton coding strategy with optimal precoding, the target distribution $p^{*}_{\mathrm{marton,opt}}(x_1,x_2)$ and the precoding matrix $W^{*}_{\mathrm{marton,opt}}$ are chosen to be the maximizers of $C_{\mathrm{sum}}$, when the sum-power constraint is $P^{*}$. A similar approach is taken to find the target distribution $p^{*}_{\mathrm{marton}}(x_1,x_2)$ and the precoding matrix $W^{*}_{\mathrm{sym,opt}}$. Given the target distribution and the precoding matrix, the rates $(R_{11},R_{12},R_{21},R_{22})$ are set according to (<ref>). This allows us to compare the different coding strategies for the same sum-rate $R_{\mathrm{sum}}$. In our simulations over the broadcast channel, we used $\gamma = 1/16$. For more details about the simulation setup, our code is available on GitHub [63].

The simulation results are shown in Fig. <ref>. Clearly, the Marton coding scheme with optimal precoding can achieve improved block error rate performance compared to the other coding strategies. Even when no precoding is used, the Marton coding scheme can achieve better performance compared to common coding strategies often employed in practice, such as time division, MMSE precoding and zero-forcing precoding.

Simulation results for the different coding strategies over a two-user Gaussian broadcast channel for the same block length $n=1024$ and sum-rate $R_\mathrm{sum} = 1$.

Note that all the coding strategies that are considered in this part can be implemented as instances of the proposed Marton coding scheme (Fig. <ref>) for particular choices of the rates $(R_{11},R_{12},R_{21},R_{22})$, the target channel input distribution $p(x_1,x_2)$ and the precoding matrix $W$.

§ BERGER–TUNG CODING FOR DISTRIBUTED LOSSY COMPRESSION

In this section, we describe a Berger–Tung coding scheme for distributed lossy compression starting from a lossy source code and a Wyner–Ziv code. When the constituent codes are rate-optimal, the coding scheme can achieve a corner point in the Berger–Tung rate region.

§.§ Problem Statement

We consider the problem of distributed lossy compression of two binary sources that generate two jointly i.i.d. binary source sequences $X_1^n$ and $X_2^n$ such that $(X_{1i},X_{2i}) \sim p(x_1,x_2)$ <cit.>. The goal is to efficiently represent the two sequences using two separate encoders to a decoder that wishes to reconstruct the sequences with some distortion levels $D_1$ and $D_2$. More specifically, an $(R_1,R_2,n)$ code for the distributed lossy compression problem consists of encoders $g_j: \{0,1\}^n \to [2^{nR_j}]$ that assign an index $M_j$ to the source sequence $X_j^n$ for $j=1,2$, and a decoder $\psi: [2^{nR_1}] \times [2^{nR_2}] \to \{0,1\}^{2n}$ that assigns a pair of estimates $(\hat{X}_1^n,\hat{X}_2^n)$ to each index pair $(M_1,M_2)$. A rate-distortion quadruple $(R_1,R_2,D_1,D_2)$ is said to be achievable if there exists a sequence of $(R_1,R_2,n)$ codes such that

\[
\limsup_{n\to \infty}\tfrac{1}{n}\E[d_H(X_j^n, \widehat{X}_j^n)] \leq D_j, \qquad j=1,2,
\]

where $d_H(.,.)$ denotes the Hamming distance metric. Berger [48] and Tung [71] showed that a rate pair $(R_1,R_2)$ is achievable for the distributed lossy compression problem with distortion pair $(D_1,D_2)$ if

\begin{equation} \label{eqn:berger_region}
\begin{aligned}
R_1 &> I(X_1;\widehat{X}_1 \cond \widehat{X}_2),\\
R_2 &> I(X_2;\widehat{X}_2 \cond \widehat{X}_1),\\
R_1+R_2 &> I(X_1,X_2;\widehat{X}_1,\widehat{X}_2)
\end{aligned}
\end{equation}

for some conditional pmf $p(\hat{x}_1 \cond x_1)p(\hat{x}_2 \cond x_2)$ such that $\E[d_H(X_j,\widehat{X}_j)] \leq D_j$, $j=1,2$.

§.§ Coding Scheme

Consider two conditional pmf's $p(\hat{x}_1 \cond x_1)$ and $p(\hat{x}_2 \cond x_2)$ such that $\E[d_H(X,\widehat{X}_j)] \leq D_j$, for $j=1,2$. This completely specifies the source-reconstruction joint distribution as

\[
p(x_1,x_2,\hat{x}_1,\hat{x}_2) = p(x_1,x_2)p(\hat{x}_1 \cond x_1)p(\hat{x}_2 \cond x_2).
\]

The coding scheme for distributed lossy compression uses a lossy source code and a Wyner–Ziv code that target the desired conditional pmf's, as described in the following Lego-brick definitions.

[Lossy $\to$ BT]
an $(R_1,n)$ lossy source code $(g_1,\psi_1)$ for a $p(x_1)$-source that targets a conditional distribution $p(\hat{x}_1 \cond x_1)$ s.t. for $X_1^n \IID p(x_1)$ and $\widehat{X}_1^n = \psi_1\big(g_1(X_1^n)\big)$, we have

\begin{equation}\label{eqn:lossy_berger}
\frac{1}{2}\sum_{x_1^n,\hat{x}_1^n} \left| \P\{X_1^n=x_1^n,\widehat{X}_1^n = \hat{x}_1^n\} - \prod_{i=1}^n  p(x_{1i})p(\hat{x}_{1i} \cond x_{1i})\right| \leq \delta_1
\end{equation}

for some $\delta_1 > 0$.

[WZ $\to$ BT]
an $(R_2,n)$ Wyner–Ziv code $(g_2,\psi_2)$ for a $p(x_2,\hat{x}_1)$-source that targets a conditional distribution $p(\hat{x}_2 \cond x_2)$ s.t. for $(X_2^n,\widehat{X}_1^n) \IID p(x_2,\hat{x}_1)$ and $\widehat{X}_2^n = \psi_2\big(g_2(X_2^n), \widehat{X}_1^n\big)$, we have

\begin{equation} \label{eqn:wz_berger}
\frac{1}{2}\sum_{x_2^n,\hat{x}_2^n} \left| \P\{X_2^n=x_2^n,\widehat{X}_2^n = \hat{x}_2^n\} - \prod_{i=1}^n  p(x_{2i})p(\hat{x}_{2i} \cond x_{2i})\right| \leq \delta_2
\end{equation}

for some $\delta_2 > 0$.

Note that a lossy source code satisfying condition (<ref>) can be constructed starting from a point-to-point channel code and a lossless source code, as described in Section <ref>. Also, a Wyner–Ziv code satisfying condition (<ref>) can be constructed starting from a point-to-point channel code and a Slepian–Wolf code, as described in Section <ref>.

Fig. <ref> shows the block diagram of a distributed lossy compression code that uses the aforementioned Lego bricks. Similar to the distortion analysis in Section <ref>, it can be shown using conditions (<ref>) and (<ref>) that the average distortions of the Berger–Tung coding scheme can be bounded as

\begin{align*}
\tfrac{1}{n}\E[d_H(X_1^n,\widehat{X}_1^n)] &\leq D_1 + \delta_1,\\
\tfrac{1}{n}\E[d_H(X_2^n,\widehat{X}_2^n)] &\leq D_2  +\delta_1 + \delta_2.
\end{align*}

Coding scheme for distributed lossy compression using a lossy source code and a Wyner–Ziv code.

If the rate of the lossy source code is $R_1 = I(X_1;\widehat{X}_1) + \gamma_1$ for some $\gamma_1 > 0$, and the rate of the Wyner–Ziv code is $R_2 = I(X_2;\widehat{X}_2 \cond \widehat{X}_1) + \gamma_2$ for some $\gamma_2>0$, then the coding scheme can achieve the rate pair

\[
(R_1,R_2) = \left(I(X_1;\widehat{X}_1) + \gamma_1, I(X_2;\widehat{X}_2 \cond \widehat{X}_1) + \gamma_2\right).
\]

Note that $\left(I(X_1;\widehat{X}_1), I(X_2;\widehat{X}_2 \cond \widehat{X}_1)\right)$ is a corner point of the rate-distortion region given in (<ref>).

A similar coding scheme can be constructed for a distributed compression problem with $L$ sources using one lossy source code and $L-1$ Wyner–Ziv codes, where the decoder successively decodes the source sequences.

§ MULTIPLE DESCRIPTION CODING

In this appendix, we describe a coding scheme for the multiple description coding problem starting from two lossy source codes and one Wyner–Ziv code. For illustrative purposes, each of the lossy source codes and Wyner–Ziv code will be described using their constituent codes, as described in previous sections. Therefore, we will construct a multiple description coding scheme starting from three point-to-point channel codes, two lossless source codes and a Slepian–Wolf code. Provided that these constituent codes are rate-optimal, the coding scheme can achieve a corner point of the El Gamal–Cover rate-distortion region for the multiple description coding problem.

§.§ Problem Statement

Consider a binary source sequence $X^n \IID \BERN(\theta)$ to be encoded through two descriptions such that each description by itself can be used to reconstruct the source with some distortions $D_1$ and $D_2$, and the two descriptions together can be used to reconstruct the source with a lower distortion $D_0$ <cit.>, as depicted in Fig. <ref>. The goal is to characterize the optimal tradeoff between the description rate pair $(R_1,R_2)$ and the distortion triple $(D_0,D_1,D_2)$. An $(R_1,R_2,n)$ multiple description code consists of an encoder $g:\{0,1\}^n \to [2^{nR_1}] \times [2^{nR_2}]$ that assigns two indices $M_1$ and $M_2$ to the source sequence $X^n$, and three decoders $\psi_j$, for $j=0,1,2$, such that $\psi_1$ assigns an estimate $\hat{X}_1^n$ to the index $M_1$, $\psi_2$ assigns an estimate $\hat{X}_2^n$ to the index $M_2$, and $\psi_0$ assigns an estimate $\hat{X}_0^n$ to the index pair $(M_1,M_2)$. A rate-distortion quintuple $(R_1,R_2,D_0,D_1,D_2)$ is achievable if there exists a sequence of $(R_1,R_2,n)$ codes such that

\[
\limsup_{n\to \infty}\frac{1}{n}\E[d_H(X^n, \widehat{X}_j^n)] \leq D_j\qquad j=0,1,2,
\]

where $d_H(.,.)$ denotes the Hamming distance metric. El Gamal and Cover [72] showed that a rate pair $(R_1,R_2)$ is achievable with distortions $(D_0,D_1,D_2)$ if

\begin{equation} \label{eqn:mdc_region}
\begin{aligned}
R_1 &> I(X;\widehat{X}_1)\\
R_2 &> I(X;\widehat{X}_2)\\
R_1+R_2 &> I(X;\widehat{X}_0,\widehat{X}_1,\widehat{X}_2)+I(\widehat{X}_1;\widehat{X}_2)
\end{aligned}
\end{equation}

for some conditional pmf $p(\hat{x}_0,\hat{x}_1,\hat{x}_2 \cond x)$ such that $\E[d_H(X,\widehat{X}_j)] \leq D_j$, $j=0,1,2$.

A multiple description code.

§.§ Coding Scheme

Let $p(\hat{x}_0,\hat{x}_1,\hat{x}_2 \cond x)$ be some desired conditional pmf that satisfies $\E[d_H(X,\widehat{X}_j)] \leq D_j$, for each $j=0,1,2$. Therefore, the source-reconstruction joint distribution can be written as

\[
p(x,\hat{x}_0,\hat{x}_1,\hat{x}_2) = p(x)p(\hat{x}_0,\hat{x}_1,\hat{x}_2 \cond x).
\]

The coding scheme for the multiple description coding problem can be constructed starting from the following Lego bricks.

[P2P $\to$ MDC]
a $(k_{11},n)$ linear point-to-point channel code $(H_{11},\phi_{11})$ with codebook $\cC_{11}$ for the channel

\begin{equation} \label{eqn:p2p_lossy_1}
\bar{p}_1(x, v\cond\hat{x}_1) = p_{X,\widehat{X}_1}(x,\hat{x}_1\oplus v).
\end{equation}

Let $\delta_1$ denote the shaping distance of the code $(H_{11},\phi_{11})$ with respect to the channel $\bar{p}_1$.

[Lossless $\to$ MDC]
an $(n-k_{12},n)$ lossless source code $(H_{12},\phi_{12})$ for a $\BERN(p_{\widehat{X}_1}(1))$ source with codebook $\cC_{12}$ and average probability of error $\eps_1$. We further assume that $\cC_{12} \subseteq \cC_{11}$.

[P2P $\to$ MDC]
a $(k_{21},n)$ linear point-to-point channel code $(H_{21},\phi_{21})$ with codebook $\cC_{21}$ for the channel

\begin{equation} \label{eqn:p2p_lossy_2}
\bar{p}_2(x, \hat{x}_1, v\cond\hat{x}_2) = p_{X,\widehat{X}_1,\widehat{X}_2}(x,\hat{x}_1, \hat{x}_2\oplus v).
\end{equation}

Let $\delta_2$ denote the shaping distance of the code $(H_{21},\phi_{21})$ with respect to the channel $\bar{p}_2$.

[Lossless $\to$ MDC]
an $(n-k_{22},n)$ lossless source code $(H_{22},\phi_{22})$ for a $\BERN(p_{\widehat{X}_2}(1))$ source with codebook $\cC_{22}$ and average probability of error $\eps_2$. We further assume that $\cC_{22} \subseteq \cC_{21}$.

[P2P $\to$ MDC]
a $(k_{01},n)$ linear point-to-point channel code $(H_{01},\phi_{31})$ with codebook $\cC_{01}$ for the channel

\begin{equation} \label{eqn:p2p_lossy_3}
\bar{p}_0(x, \hat{x}_1, \hat{x}_2, v\cond\hat{x}_0) = p_{X,\widehat{X}_1,\widehat{X}_2,\widehat{X}_0}(x,\hat{x}_1, \hat{x}_2, \hat{x}_0\oplus v).
\end{equation}

Let $\delta_0$ denote the shaping distance of the code $(H_{01},\phi_{01})$ with respect to the channel $\bar{p}_0$.

[SW $\to$ MDC]
an $(n-k_{02},n)$ Slepian–Wolf code $(H_{02},\phi_{02})$ for the problem $p\big(\hat{x}_0,(\hat{x}_1,\hat{x}_2)\big)$ with codebook $\cC_{02}$ and average probability of error $\eps_0$. We further assume that $\cC_{02} \subseteq \cC_{01}$.

Due to nestedness, we will assume, without loss of generality, that $H_{j1}$ is a submatrix of $H_{j2}$ for each $j=0,1,2$, i.e., $H_{j2} = \begin{bmatrix}
\end{bmatrix}$ for some $(k_{j1}-k_{j2})\times n$ matrix $Q_j$.

Encoder of a multiple description code using three point-to-point channel codes.

The point-to-point channels $\bar{p}_1$, $\bar{p}_2$ and $\bar{p}_0$ are the symmetrized channels corresponding to the joint distributions $p\big(\hat{x}_1,x\big)$, $p\big(\hat{x}_2,(\hat{x}_1,x\big))$ and $p\big(\hat{x}_0,(\hat{x}_1,\hat{x}_2,x)\big)$, respectively.

Fig. <ref> and Fig. <ref> show the block diagrams of the encoder and decoder of the multiple description code, respectively, where $(V_0^n,V_1^n,V_2^n)$ are i.i.d. $\BERN(1/2)$ random dithers such that $V_1^n$ is shared with decoder 1, $V_2^n$ is shared with decoder 2, and all three random dithers are shared with decoder 0. Similar to lossy source coding, the basic idea of the coding scheme is to generate three sequences $(U_0^n,U_1^n,U_2^n)$ whose distribution is “close” to the i.i.d. $p(\hat{x}_0,\hat{x}_1,\hat{x}_2)$ distribution, and then perform lossless source compression to recover estimates of the sequences at the decoders. The construction of the sequences $(U_0^n,U_1^n,U_2^n)$ is done successively at the encoder side (Fig. <ref>). That is, first, the sequence $U_1^n$ is generated using the decoding function $\phi_{11}$. Then, $U_1^n$ is inputted to the decoding function $\phi_{21}$ to construct the sequence $U_2^n$. Intuitively, this step attempts to generate $U_2^n$ according to the conditional distribution $p(\hat{x}_2 \cond x, \hat{x}_1)$. Similarly, $(U_1^n,U_2^n)$ are inputted to the decoding function $\phi_{01}$ to construct the sequence $U_0^n$ according to the conditional distribution $p(\hat{x}_0 \cond x, \hat{x}_1,\hat{x}_2)$. By repeated applications of Lemma <ref> and the definition of shaping distance, it follows that

\begin{equation*}
\begin{aligned}
&\frac{1}{2}\sum_{u_0^n,u_1^n,u_2^n}\Big| \P\{U_0^n=u_0^n,U_1^n=u_1^n,U_2^n=u_2^n\} \\
&\qquad \qquad \quad - \prod_{i=1}^{n}p_{\widehat{X}_0,\widehat{X}_1,\widehat{X}_2}(u_{0i},u_{1i},u_{2i})\Big| \\
&\leq \delta_0 + \delta_1 + \delta_2.
\end{aligned}
\end{equation*}

Moreover, for each $j=0,1,2$, we have

\[
H_{j2}U_j^n = \begin{bmatrix}
\end{bmatrix}U_j^n = \begin{bmatrix}
\end{bmatrix},
\]

where $Q_jU_j^n$, $j=0,1,2$, are indices transmitted to the decoders. Therefore, the decoder can recover the estimates of the source sequence knowing the shared indices and random dithers. The coding scheme can be summarized as follows.

Decoder of a multiple description code using two lossless source decoders and a Slepian–Wolf decoder.

Encoding: Upon observing the source sequence $x^n$, the encoder computes the sequence $u_1^n = \phi_{11}\left(x^n, v_1^n\right)\oplus v_1^n$ and transmits the index $m_1^{k_{11}-k_{12}} = Q_1u_1^n$ to decoder 1 and decoder 0, where $v_1^n$ is a realization of a random dither shared with the decoders. The encoder then computes the sequences $u_2^n = \phi_{21}\left(x^n, u_1^n, v_2^n\right)\oplus v_2^n$ and $u_0^n = \phi_{01}\left(x^n, u_1^n, u_2^n, v_0^n\right)\oplus v_2^n$, and transmits the index pair

\[
(m_{22}^{k_{21}-k_{22}}, m_{02}^{k_{01}-k_{02}}) = (Q_2u_2^n, Q_0u_0^n)
\]

to decoder 2 and decoder 0, where $v_2^n$ is a random dither shared with decoder 2 and decoder 0, and $v_0^n$ is a random dither shared with the decoder 0.

Decoding: Upon observing the index $m_1^{k_{11}-k_{12}}$, decoder 1 declares the sequence $\hat{x}_1^n = \phi_{12}\left( \begin{bmatrix}\bf 0 \\ H_{11}v_1^n \\ m_1^{k_{11}-k_{12}}\end{bmatrix} \right)$ as the source estimate. Upon observing the index $m_{22}^{k_{21}-k_{22}}$, decoder 2 declares the sequence $\hat{x}_2^n = \phi_{22}\left( \begin{bmatrix}\bf 0 \\ H_{21}v_2^n \\ m_{22}^{k_{21}-k_{22}}\end{bmatrix} \right)$ as the source estimate. And upon observing the index triplet $(m_1^{k_{11}-k_{12}},m_{22}^{k_{21}-k_{22}},m_{02}^{k_{01}-k_{02}})$, decoder 0 first computes the sequences $\hat{x}_1^n$ and $\hat{x}_2^n$, and declares the sequence $\hat{x}_0^n = \phi_{02}\left( \begin{bmatrix}\bf 0 \\ H_{01}v_0^n \\ m_{02}^{k_{01}-k_{02}}\end{bmatrix}, \hat{x}_1^n, \hat{x}_2^n\right)$ as the source estimate (recall that $\phi_{02}(.)$ is a Slepian–Wolf decoder with side information sequences $\hat{x}_1^n$ and $\hat{x}_2^n$).

Analysis of the average distortion: Similar to the analysis in lossy source coding, it can be shown that the average distortions can be bounded as

\begin{align*}
\tfrac{1}{n}\E[d_H(X^n,\widehat{X}_1^n)] &\leq D_1 + \delta_1 + \eps_1\\
\tfrac{1}{n}\E[d_H(X^n,\widehat{X}_2^n)] &\leq D_2  +\delta_1 + \delta_2 + \eps_2\\
\tfrac{1}{n}\E[d_H(X^n,\widehat{X}_0^n)] &\leq D_0 +\delta_0 + \delta_1 + \delta_2 + \eps_0 + \eps_1 + \eps_2
\end{align*}

Rate: The coding scheme attains the rate pair $(R_1,R_2) = \left(\frac{k_{11}-k_{12}}{n}, \frac{k_{21}-k_{22}+k_{01}-k_{02}}{n}\right)$.

Following the discussion in Section <ref> and the properties of a symmetrized channel (Remark <ref>), it follows that if the rates of point-to-point channel codes are

\begin{equation*}
\begin{aligned}
\tfrac{k_{11}}{n}&=1-H(\widehat{X}_1 \cond X)+\gamma_{11},\\
\tfrac{k_{21}}{n}&=1-H(\widehat{X}_2 \cond X, \widehat{X}_1)+\gamma_{21},\\
\tfrac{k_{01}}{n}&=1-H(\widehat{X}_0 \cond X, \widehat{X}_1, \widehat{X}_2)+\gamma_{01},
\end{aligned}
\end{equation*}

for some $\gamma_{11}$, $\gamma_{21}$, $\gamma_{01}>0$, and the rates of the two lossless source codes and Slepian–Wolf code are, respectively,

\begin{equation*}
\begin{aligned}
\tfrac{n-k_{12}}{n}&=H(\widehat{X}_1)+\gamma_{12},\\
\tfrac{n-k_{22}}{n}&=H(\widehat{X}_2)+\gamma_{22},\\
\tfrac{n-k_{02}}{n}&=H(\widehat{X}_0 \cond \widehat{X}_1, \widehat{X}_2)+\gamma_{02},
\end{aligned}
\end{equation*}

for some $\gamma_{12}$, $\gamma_{22}$, $\gamma_{02}>0$, then the multiple description coding scheme attains the rate pair

\begin{equation*}
\begin{aligned}
&(R_1,R_2)  = \left(\tfrac{k_{11}-k_{12}}{n}, \tfrac{k_{21}-k_{22}+k_{01}-k_{02}}{n}\right) \\
&= \hspace*{-0.1em}\Big( I(X;\widehat{X}_1) \hspace*{-0.1em}+\hspace*{-0.1em} \gamma_{11}\hspace*{-0.1em}+\hspace*{-0.1em}\gamma_{12}, \: I(X,\widehat{X}_1;\widehat{X}_2) \hspace*{-0.1em}+\hspace*{-0.1em} I(X;\widehat{X}_0 \cond \widehat{X}_1,\widehat{X}_2)\\
&\qquad\qquad\qquad\qquad\qquad\quad +\hspace*{-0.1em} \gamma_{21} \hspace*{-0.1em} + \hspace*{-0.1em} \gamma_{22} \hspace*{-0.1em} + \hspace*{-0.1em} \gamma_{01}\hspace*{-0.1em}+ \hspace*{-0.1em} \gamma_{02}\Big).
\end{aligned}
\end{equation*}

Note that the rate pair $\big( I(X;\widehat{X}_1), \:I(X,\widehat{X}_1;\widehat{X}_2) + I(X;\widehat{X}_0 \cond \widehat{X}_1,\widehat{X}_2) \big)$ is a corner point of the El Gamal–Cover rate-distortion region given in (<ref>). By reversing the order of generating the sequences $U_1^n$ and $U_2^n$ at the encoder side, another corner point can be achieved.

§ CODING FOR DOWNLINK C-RAN

In this appendix, we construct a coding scheme for the downlink of a cloud radio access network (C-RAN) [50] starting from a Marton code and two lossless source codes. When the constituent codes are rate-optimal, the coding scheme can achieve a corner point of the achievable rate region of distributed decode-forward [73], which is the best known inner bound for the downlink C-RAN problem.

§.§ Problem Statement

Consider the downlink of the C-RAN model with two users and two relays, as shown in Fig. <ref>. A central processor (CP) communicates with the relays through noiseless fronthaul links of finite capacities $C_1$ and $C_2$. A memoryless channel $p(y_1,y_2 \cond x_1,x_2)$ is assumed between the relays and the users, with an input alphabet $\cX^2 = \{0,1\}^2$ and output alphabet $\cY_1 \times \cY_2$. An $(R_1,R_2,n)$ code for the downlink C-RAN problem consists of an encoder $g:[2^{nR_1}] \times [2^{nR_2}] \to [2^{nC_1}] \times [2^{nC_2}]$ at the CP that maps a message pair $(M_1,M_2)$ to a pair of indices $(S_1,S_2)$, encoders $h_j: [2^{nC_j}] \to \cX^n$ at the $j$th relay for $j=1,2$, that map the index $S_j$ to a channel input sequence $X_j^n$, and decoders $\psi_j: \cY_j^n \to [2^{nR_j}]$ at the $j$th user for $j=1,2$, that assign message estimates $\hat{M}_j$ to the received sequence $Y_j^n$. The average probability of error of the code is $\eps = \P\big\{ \{\widehat{M}_1 \neq M_1\} \cup \{\widehat{M}_2 \neq M_2\} \big\}$. A rate pair $(R_1, R_2)$ is said to be achievable for the downlink C-RAN problem if there exists a sequence of $(R_1,R_2,n)$ codes with vanishing error probability asymptotically.

The C-RAN architecture is a key part in the deployment of 5G standards [74, 75]. Coding schemes for the C-RAN model have been proposed in [76, 77, 73, 78]; in particular, [73] specializes the distributed decode-forward relaying scheme [79] to the downlink C-RAN problem under consideration and shows that the achievable rate region for the two-user two-relay problem is the set of rate tuples $(R_1,R_2)$ satisfying

\begin{equation} \label{eqn:cran_rate_region}
\sum_{k\in \cK}R_k < \sum_{k \in \cK}I(U_k;Y_k) + \sum_{l\in \cL} C_l - I(U_\cK;X_{\cL}) - I^{*}(U_\cK) - I^{*}(X_\cL),
% \sum_{k\in \cK}R_k < \sum_{k \in \cK}\big(I(U_k;Y_k)-I(U_k; U([k-1]\cap \cK))\big) + \sum_{l\in \cL} \big(C_l-I(X_l; X([l-1]\cap \cL)))\big) - I(U(\cK);X({\cL})),
\end{equation}

for all $\cK \subseteq \{1,2\}$ and $\cL \subseteq \{1,2\}$ and some joint distribution $p(u_1,u_2,x_1,x_2)$, where $I^{*}(.)$ is as defined in (<ref>).

Downlink C-RAN problem with two users, two relays and a channel $p(y_1,y_2 \cond x_1, x_2)$ between the relays and the users.

§.§ Coding Scheme

The coding scheme for the two-user, two-relay downlink C-RAN problem is constructed starting from a Marton code for a broadcast channel and two lossy source codes. The Marton code shapes the channel input distribution while reliably sending messages to the intended users, whereas the lossy source codes compress the encoded signals to accommodate the limited capacity constraints of the fronthaul links. For clarity of representation, we will describe the Marton code by its constituent asymmetric channel code and Gelfand–Pinsker code, as described in Appendix <ref>. Similarly, each lossy source code will be implemented using its constituent point-to-point channel code and lossless source code, as described in Section <ref>. Therefore, the following Lego bricks (one asymmetric channel code, one Gelfand–Pinsker code, two point-to-point channel codes and two lossless source codes) will be used in the construction, which will target some desired joint distribution $p(u_1,u_2,x_1,x_2)$.

[Asym $\to$ DL C-RAN]
an $(R_1,n)$ asymmetric channel code $(f_1,\psi_1)$ for the channel

\[
p(y_1|u_1) = \sum_{\substack{u_2,x_1,\\x_2,y_2}}p(u_2,x_1,x_2|u_1) p(y_1,y_2| x_1,x_2),
\]

which targets an input distribution $p(u_1)$, such that, for $M_1 \sim \Unif([2^{nR_1}])$, the sequence $U_1^n = g_1(M_1)$ satisfies

\begin{equation} \label{eqn:tv_cran_asym}
\frac{1}{2}\sum_{u_1^n}\left| \P\{U_1^n=u_1^n\} - \prod_{i=1}^n p(u_{1i}) \right| \leq \delta_1,
\end{equation}

for some $\delta_1 > 0$. Let $\eps_1$ be the average probability of error of the asymmetric channel code when the channel input distribution is i.i.d. according to $p(u_1)$.

[GP $\to$ DL C-RAN]
an $(R_2,n)$ code $(f_2, \psi_2)$ for the Gelfand–Pinsker problem defined by

\[
p(u_1)p(y_2| u_2,u_1) = p(u_1)\sum_{x_1,x_2,y_1}p(x_1,x_2|u_1,u_2)p(y_1,y_2| x_1,x_2),
\]

which targets a conditional distribution $p(u_2 \cond u_1)$, such that, when $M_2 \sim \Unif([2^{nR_2}])$ and $\wtilde{U}_1^n$ is an i.i.d. $p(u_1)$ sequence, the sequence $\wtilde{U}_2^n = g_2(M_2, \wtilde{U}_1^n)$ satisfies

\begin{equation} \label{eqn:tv_cran_gelfand}
\frac{1}{2}\sum_{u_1^n,u_2^n}\left| \P\{\wtilde{U}_1^n=u_1^n,\wtilde{U}_2^n=u_2^n\} - \prod_{i=1}^n p(u_{1i},u_{2i}) \right| \leq \delta_2,
\end{equation}

for some $\delta_2 > 0$. Let $\eps_2$ be the average probability of error of the Gelfand–Pinsker code when the conditional distribution of the channel input given the channel state is i.i.d. according to $p(u_2 \cond u_1)$.

[P2P $\to$ Lossy $\to$ DL C-RAN]
a $(k_{31},n)$ linear point-to-point channel code $(H_{31},\phi_{31})$ with codebook $\cC_{31}$ for the channel

\begin{equation} \label{eqn:p2p_lossy_cran1}
\bar{p}_3(u_1, u_2, v\cond x_1) = p_{U_1,U_2,X_1}(u_1,u_2,x_1\oplus v).
\end{equation}

Let $\delta_3$ denote the shaping distance of the code $(H_{31},\phi_{31})$ with respect to the channel $\bar{p}_3$.

[Lossless $\to$ Lossy $\to$ DL C-RAN]
an $(n-k_{32},n)$ lossless source code $(H_{32},\phi_{32})$ for a $\BERN(p_{X_1}(1))$ source with codebook $\cC_{32}$ and average probability of error $\eps_3$. We further assume that $\cC_{32} \subseteq \cC_{31}$ (i.e., the two codes are nested) and that $k_{31}-k_{32} < nC_1$.

[P2P $\to$ Lossy $\to$ DL C-RAN]
a $(k_{41},n)$ linear point-to-point channel code $(H_{41},\phi_{41})$ with codebook $\cC_{41}$ for the channel

\begin{equation} \label{eqn:p2p_lossy_cran2}
\bar{p}_4(u_1, u_2, x_1, v\cond x_2) = p_{U_1,U_2,X_1,X_2}(u_1,u_2,x_1,x_2\oplus v).
\end{equation}

Let $\delta_4$ denote the shaping distance of the code $(H_{41},\phi_{41})$ with respect to the channel $\bar{p}_4$.

[Lossless $\to$ Lossy $\to$ DL C-RAN]
an $(n-k_{42},n)$ lossless source code $(H_{42},\phi_{42})$ for a $\BERN(p_{X_2}(1))$ source with codebook $\cC_{42}$ and average probability of error $\eps_4$. We further assume that $\cC_{42} \subseteq \cC_{41}$ (i.e., the two codes are nested) and that $k_{41}-k_{42} < nC_2$.

The channels $\bar{p}_3$ and $\bar{p}_4$ are the symmetrized channels corresponding to the joint distributions $p\big(x_1,(u_1,u_2)\big)$ and $p\big(x_2,(u_1,u_2,x_1)\big)$, respectively.

Without loss of generality, assume that $H_{j1}$ is a submatrix of $H_{j2}$ for $j=3,4$, i.e., $H_{j2} = \begin{bmatrix}
\end{bmatrix}$ for some $(k_{j1}-k_{j2})\times n$ matrix $Q_j$.

Encoding scheme at the central processor for the two-user, two-relay downlink C-RAN problem using an asymmetric channel code, a Gelfand–Pinsker code and two point-to-point channel codes.

Fig. <ref> and Fig. <ref> show the block diagrams of the coding scheme at the central processor, relays and users. The key point at the central processor is to construct a tuple $(U_1^n,U_2^n,X_1^n,X_2^n)$ that is close in total variation distance to the i.i.d. distribution according to $p(u_1,u_2,x_1,x_2)$, and then compress $(X_1^n,X_2^n)$ through a pair of indices that are sent to the relays through the fronthaul links. The relays recover estimates of $(X_1^n,X_2^n)$, which are transmitted to the users through the channel. Using standard arguments that were used throughout this paper, it follows from conditions (<ref>), (<ref>), and Lemma <ref> that

\begin{equation*}
\begin{aligned}
&\frac{1}{2}\sum_{\substack{u_1^n,u_2^n,\\ x_1^n,x_2^n}}\Big| \P\{U_1^n=u_1^n, U_2^n=u_2^n,X_1^n=x_1^n,X_2^n=x_2^n\} \\
&\qquad \qquad - \prod_{i=1}^n p(u_{1i},u_{2i},x_{1i},x_{2i}) \Big| \\
&\leq \delta_1+\delta_2+\delta_3+\delta_4,
\end{aligned}
\end{equation*}

which says that the distribution of $(U_1^n,U_2^n,X_1^n,X_2^n)$ is $(\delta_1+\delta_2+\delta_3+\delta_4)$-away in total variation distance from the i.i.d. $p(u_1,u_2,x_1,x_2)$ distribution. Furthermore, the average probability of error of the coding scheme can be bounded as

\begin{equation*}
\begin{aligned}
&\P\left\{ \{\widehat{M}_1 \neq M_1\} \cup \{\widehat{M}_2 \neq M_2\} \right\} \\
&\leq \, \delta_1 + \delta_2 + \delta_3 + \delta_4 + \epsilon_1 + \epsilon_2 + \eps_3 + \eps_4,
\end{aligned}
\end{equation*}

following similar bounding techniques as in previous sections.

The coding scheme attains the rate pair $(R_1,R_2)$, and the compression rates are $\frac{k_{31}-k_{32}}{n}$ and $\frac{k_{41}-k_{42}}{n}$. Notice that these compression rates do not exceed the fronthaul link capacities (by assumption).

Similar to the analysis of the Marton coding scheme of Appendix <ref>, the coding rate pair $(R_1,R_2)$ can be made arbitrarily close to $(I(U_1;Y_1), \, I(U_2;Y_2) - I(U_2; U_1))$ for sufficiently large $n$. On the other hand, following the discussion on the achievable rates of the lossy source coding scheme (specifically, Remarks <ref> and <ref>), the compression rate pair $\left(\frac{k_{31}-k_{32}}{n}, \frac{k_{41}-k_{42}}{n}\right)$ can be made arbitrarily close to $\big(I(U_1,U_2;X_1), I(U_1,U_2,X_1;X_2)\big)$ for sufficiently large $n$. It can be checked that this corresponds to a corner point in the achievable rate region of the distributed decode-forward relaying scheme given in (<ref>). If the encoding order at the central processor is modified (e.g., if $X_1^n$ is generated before $U_2^n$ in Fig. <ref>), other corner points of the rate region can be achieved.

A similar approach can be taken to construct a coding scheme for the general $K$-user, $L$-relay downlink C-RAN problem using one asymmetric channel code, $K-1$ Gelfand–Pinsker codes, $L$ point-to-point channel codes and $L$ lossless source codes.

Encoding scheme at the relays and decoding scheme at the users for the downlink C-RAN problem.

§.§ Simulation Results

By implementing each of the lossless source codes, the asymmetric channel code and the Gelfand–Pinsker code using their constituent point-to-point channel codes as described in Section <ref> and Section <ref>, the coding scheme for downlink C-RAN can be simulated using point-to-point channel codes. We consider a two-user two-relay model with a Gaussian channel between the relays and the users, i.e., the channel output can be expressed as

\begin{equation} \label{eqn:gaussian_channel_cran}
% Y_1^2 = G\Lambda X_1^2 + Z_1^2
\begin{bmatrix}
Y_1 \\
\end{bmatrix} = H_{\mathrm{ch}}\Lambda \begin{bmatrix}
\end{bmatrix} + \begin{bmatrix}
\end{bmatrix}
%{\bf y} = {\bf H_{ch}}{\bf \Lambda} {\bf x} + {\bf z},
\end{equation}

where $H_{\mathrm{ch}} = \begin{bmatrix}
1 & g \\
g & 1
\end{bmatrix}$ is the channel gain matrix with $g=0.9$, $\Lambda = \begin{bmatrix}
\sqrt{\lambda_1} & 0\\
0 & \sqrt{\lambda_2}
\end{bmatrix}$ is a power allocation matrix for the relays[Note that since the two relays do not communicate and the fronthaul links are capacity-limited, we do not consider the possibility of applying a general precoding matrix as in Appendix <ref>.], $X_1 \in \{\pm 1\}$ and $X_2 \in \{\pm 1\}$ are BPSK-modulated signals, and $\begin{bmatrix}
\end{bmatrix} \sim \mathcal{N}\big({\bf 0}, I\big)$ is a vector of independent Gaussian noise components. The transmitter wishes to send messages to the users at an overall sum-rate $R_{\mathrm{sum}}$, and each relay is subject to a power constraint $P$ such that $\lambda_j \leq P$ for each $j=1,2$.

We first look at the maximum achievable sum-rate of the proposed coding scheme under different power constraints $P$ and fronthaul capacity constraints $C_1$ and $C_2$. We know from (<ref>) that the maximum sum-rate $C_{\mathrm{sum}}$ that can be achieved by our coding scheme is given by equation (<ref>) below,

\begin{equation} \label{eqn:cran_optimization_ddf}
C_{\mathrm{sum}} \triangleq \underset{\substack{p(u_1,u_2,x_1,x_2), \\ (\lambda_1, \lambda_2) \in \cK(P)}}{\max} \, \left(I(U_1;Y_1) + I(U_2;Y_2) - I(U_1;U_2) + \min\left\{\begin{matrix}
C_1+C_2-I(U_1,U_2;X_1,X_2) - I(X_1;X_2),\\
\end{matrix} \right\}\right),
\end{equation}

where $\cK(P) = \{(\lambda_1,\lambda_2): 0\leq \lambda_1 \leq P, 0 \leq \lambda_2 \leq P\}$. For the encoding order adopted in Fig. <ref> and the corresponding achievable rates (see Remark <ref>), the maximum achievable sum-rate $C_{\mathrm{sum}}$ can be alternatively expressed as the solution to a simpler optimization problem, i.e.,

\begin{equation} \label{eqn:cran_optimization}
C_{\mathrm{sum}} = \hspace*{-2.5em}\underset{\substack{p(u_1,u_2,x_1,x_2) \in \cD(C_1,C_2), \\ (\lambda_1, \lambda_2) \in \cK(P)}}{\max} \, \big(I(U_1;Y_1) + I(U_2;Y_2) - I(U_1;U_2)\big),
\end{equation}

where $\cD(C_1,C_2)$ is the set of joint distributions $p(u_1,u_2,x_1,x_2)$ that satisfy $C_1 > I(U_1,U_2; X_1)$ and $C_2 > I(U_1,U_2,X_1; X_2)$.

We use the genetic algorithm [80, 68] as an efficient heuristic method to perform the optimization in (<ref>). Fig. <ref> shows the plot of $C_{\mathrm{sum}}$ as a function of the power constraint $P$ under different fronthaul capacity constraints $C_1$ and $C_2$. As expected, larger sum-rates can be achieved by the proposed coding scheme when the fronthaul link capacities are increased.[Note that when $C_1 \geq 1$ and $C_2\geq 1$, the maximum achievable sum-rate of the downlink C-RAN coding scheme over the given Gaussian model is the same as that of the Marton coding scheme over the Gaussian broadcast channel model considered in Appendix <ref>, in the special case when the transmitter is subject to a per-antenna power constraint and the precoding matrix is restricted to be a diagonal matrix.]

Next, the proposed coding scheme for the downlink C-RAN problem is simulated for a block length $n=1024$ and a sum-rate $R_{\mathrm{sum}} = 0.75$ using polar codes with successive cancellation decoding as the constituent point-to-point channel codes. Note that each of the asymmetric channel code and the Gelfand–Pinsker code can be implemented using a pair of point-to-point channel codes, and each of the lossless source codes can be implemented using a single point-to-point channel code. Hence, the downlink C-RAN coding scheme can be constructed starting from eight polar codes. Let $(R_{11},R_{12},R_{21},R_{22}, R_{31}, R_{32}, R_{41}, R_{42})$ be the rates of the constituent polar codes, where $(R_{11},R_{12})$ are the rates of the two polar codes needed to construct the asymmetric channel code, $(R_{21},R_{22})$ are the rates of the two polar codes needed to construct the Gelfand–Pinsker code, $(R_{31}, R_{32})$ are the rates of the two polar codes needed to construct the codes $\cC_{31}$ and $\cC_{32}$, and $(R_{41}, R_{42})$ are the rates of the two polar codes needed to construct the codes $\cC_{41}$ and $\cC_{42}$. The rates are chosen “close” to their theoretical limits, i.e., we set

\begin{equation*}
\begin{aligned}
R_{11} &= 1-H(U_1\cond Y_1)-\gamma_r,\\
R_{21} &= 1-H(U_2\cond Y_2)-\gamma_r,\\
R_{31} &= 1-H(X_1 \cond U_1,U_2),\\
R_{41} &= 1-H(X_2 \cond U_1,U_2,X_1),
\end{aligned}
\end{equation*}

\begin{equation} \label{eqn:cran_rates}
\begin{aligned}
R_{12} &= 1-H(U_1), \\
R_{22} &= 1-H(U_2 \cond U_1), \\
R_{32} &= 1-H(X_1)-\gamma_c,\\
R_{42} &= 1-H(X_2)-\gamma_c,
\end{aligned}
\end{equation}

where $\gamma_r > 0$ and $\gamma_c>0$ are “back-off” parameters from the theoretical limit that are used for the polar codes involved in error correction (i.e., not shaping). Thus, the sum-rate attained by the coding scheme is equal to

\[
R_{11}-R_{12}+R_{21}-R_{22} \hspace*{-0.1em}=\hspace*{-0.1em} I(U_1;Y_1) + I(U_2;Y_2) - I(U_1;U_2) -2\gamma_r.
\]

As in Appendix <ref>, in order to guarantee that the sum-rate is equal to $R_{\mathrm{sum}}$, the coding scheme targets the joint distribution $p^{*}(u_1,u_2,x_1,x_2)$ and the power levels $(\lambda_1^{*},\lambda_2^{*})$ that maximize $C_{\mathrm{sum}}$ under the optimization problem of (<ref>), where the power constraint $P$ is set to be equal to the power level $P^{*}$ at which $C_{\mathrm{sum}} = R_{\mathrm{sum}} + 2\gamma_r$.[The power level $P^{*}$ can be estimated using the plot of Fig. <ref>.] Under the joint distribution $p^{*}(u_1,u_2,x_1,x_2)$ and the power levels $(\lambda_1^{*},\lambda_2^{*})$, the rates of the constituent polar codes are set according to (<ref>). Note that $\gamma_c$ should be chosen so that $R_{31}-R_{32} \leq C_1$ and $R_{41}-R_{42} \leq C_2$. In our simulations over the downlink C-RAN channel model, we used $\gamma_r = 1/8$ and $\gamma_c = 5/32$. For more details about the simulation setup, our code is available on GitHub [63]. The block error rate performance of the downlink C-RAN coding scheme is shown in Fig. <ref> for different fronthaul capacity constraints $C_1$ and $C_2$. The results demonstrate the practicality of the proposed coding scheme over the downlink C-RAN channel model.

The maximum achievable sum-rate of the proposed coding scheme for the downlink C-RAN problem under different fronthaul capacity constraints.

Simulation results of the downlink C-RAN coding scheme for a block length $n=1024$ and sum-rate $R_\mathrm{sum} = 0.75$ under different fronthaul capacity constraints.

§ CODING FOR UPLINK C-RAN

In this appendix, we describe a coding scheme for the uplink C-RAN model starting from a multiple access channel code and a Berger–Tung code. We show that, if the constituent codes are rate-optimal, the coding scheme can achieve a corner point of the rate region achieved by the network compress-and-forward coding scheme with successive decoding [81].

§.§ Problem Statement

Consider the uplink of a C-RAN model with two users and two relays [50], as shown in Fig. <ref>. Two users wish to communicate with a central processor (CP) through two relays that are connected to the CP through noiseless backhaul links of finite capacities $C_1$ and $C_2$. A memoryless channel $p(y_1,y_2 \cond x_1,x_2)$ is assumed between the users and the relays, with an input alphabet $\cX^2 = \{0,1\}^2$ and output alphabet $\cY_1 \times \cY_2$. An $(R_1,R_2,n)$ code for uplink C-RAN problem consists of encoders $g_j:[2^{nR_j}] \to \cX^n$ at the $j$th user for $j=1,2$ that map each message $M_j$ to a channel input sequence $X_j^n$, encoders $h_j: \cY_j^n \to [2^{nC_j}]$ at the $j$th relay for $j=1,2$ that map the received sequence $Y_j^n$ to an index $S_j$, and a decoder $\psi: [2^{nC_1}] \times [2^{nC_2}] \to [2^{nR_1}] \times [2^{nR_2}]$ at the CP that assign message estimates $(\hat{M}_1,\hat{M}_2)$ to the index pair $(S_1,S_2)$. The average probability of error of the code and achievability of a rate pair $(R_1, R_2)$ are defined as in Appendix <ref>.

Uplink C-RAN problem with two users and two relays.

Coding schemes for the uplink C-RAN problem have been proposed in [82, 83, 73] based on the network compress-and-forward relaying scheme [84], the achievable rate region of which can be expressed as the rate tuples $(R_1,R_2)$ satisfying

\begin{equation} \label{eqn:uplink_cran_rate_region}
\sum_{k\in \cK}R_k < \sum_{l\in \cL} \big(C_l - I(Y_l;\widehat{Y}_l \cond X_1,X_2)\big) + I(X_\cK;\widehat{Y}_{\cL^c} \cond X_{\cK^c}),
% \sum_{k\in \cK}R_k < \sum_{k \in \cK}\big(I(U_k;Y_k)-I(U_k; U([k-1]\cap \cK))\big) + \sum_{l\in \cL} \big(C_l-I(X_l; X([l-1]\cap \cL)))\big) - I(U(\cK);X({\cL})),
\end{equation}

for all $\cK \subseteq \{1,2\}$ and $\cL \subseteq \{1,2\}$ and some product distribution $p(x_1)p(x_2)p(\hat{y}_1 \cond y_1)p(\hat{y}_2 \cond y_2)$.

§.§ Coding Scheme

The coding scheme for the uplink C-RAN problem is constructed starting from a multiple access channel code and a Berger–Tung code. The coding scheme will target a given product distribution $p(x_1)p(x_2) p(\hat{y}_1 \cond y_1)p(\hat{y}_2 \cond y_2)$[For example, such a product distribution can be chosen to maximize the maximum achievable sum-rate $I(X_1,X_2;\widehat{Y}_1,\widehat{Y}_2)$.]. Therefore, the desired input-output joint distribution can be given by

\begin{equation*}
\begin{aligned}
&p(x_1,x_2,y_1,y_2,\hat{y}_1,\hat{y}_2) \\
&= p(x_1)p(x_2) p(y_1,y_2 \cond x_1,x_2)p(\hat{y}_1 \cond y_1)p(\hat{y}_2 \cond y_2).
\end{aligned}
\end{equation*}

The coding scheme for the uplink C-RAN problem uses the following Lego bricks.

[MAC $\to$ UL C-RAN]
an $(R_1,R_2,n)$ code $(f_1,f_2,\phi)$ for the multiple access channel $p(\hat{y}_1,\hat{y}_2 \cond x_1,x_2)$ which targets input distributions $p(x_1)p(x_2)$, such that, for $M_j \sim \Unif([2^{nR_j}])$, the channel input $X_j^n = f_j(M_j)$ satisfies

\begin{equation} \label{eqn:asym_uplink_cran_1}
\frac{1}{2}\sum_{x_j^n}\left| \P\{X_j^n=x_j^n\} - \prod_{i=1}^n p(x_{ji}) \right| \leq \delta_j^{\mathrm{MAC}},
\end{equation}

for $j = 1,2$ and some $\delta_1^{\mathrm{MAC}}$, $\delta_2^{\mathrm{MAC}} > 0$. Let $\eps$ be the average probability of error of the multiple access channel code when the channel input distributions are i.i.d. according to $p(x_1)p(x_2)$.

[BT $\to$ UL C-RAN]
an $(R_3,R_4,n)$ Berger–Tung code $(f_3,f_4,\psi)$ for a $p(y_1,y_2)$-source that targets conditional distributions $p(\hat{y}_1 \cond y_1)p(\hat{y}_2 \cond y_2)$, such that, for $(Y_1^n,Y_2^n) \IID p(y_1,y_2)$, the sequences $(\widehat{Y}_1^n,\widehat{Y}_2^n) = \psi\big(g_3(Y_1^n), g_4(Y_2^n)\big)$ satisfy

\begin{equation}\label{eqn:lossy_uplink_cran}
\frac{1}{2}\hspace*{-0.1em}\sum_{y_j^n, \, \hat{y}_j^n}\hspace*{-0.1em} \left| \P\{Y_j^n\hspace*{-0.1em}=\hspace*{-0.1em}y_j^n,\widehat{Y}_j^n \hspace*{-0.1em}= \hspace*{-0.1em}\hat{y}_j^n\} \hspace*{-0.1em}-\hspace*{-0.1em} \prod_{i=1}^n  p(y_{ji})p(\hat{y}_{ji} \cond y_{ji})\right| \leq \delta_j^{\mathrm{BT}}
\end{equation}

for $j = 1,2$ and some $\delta_1^{\mathrm{BT}}$, $\delta_2^{\mathrm{BT}} > 0$. Furthermore, we assume that $R_3 < C_1$ and $R_4 < C_2$.

The multiple access channel code can be implemented using two asymmetric channel codes, as described in Appendix <ref>, and the Berger–Tung code can be implemented using a lossy source code and a Wyner–Ziv code, as described in Appendix <ref>.

Fig. <ref> shows the block diagram of the uplink C-RAN problem. The multiple access channel code is used to encode the user messages while shaping the channel input signals, whereas the Berger–Tung code is used to compress the received signals over the backhaul links. The central processor first decodes the compressed codewords using the Beger–Tung decoder, then decodes the user messages using the multiple access channel decoder. Following similar bounding techniques seen thus far, the average probability of error of the coding scheme can be bounded as

\[
\P\big\{ \{\widehat{M}_1 \neq M_1\} \cup \{\widehat{M}_2 \neq M_2\} \big\} \leq \eps + \delta_1^{\mathrm{MAC}} +\delta_2^{\mathrm{MAC}} + \delta_1^{\mathrm{BT}} + \delta_2^{\mathrm{BT}}.
\]

The user rate pair is given by $(R_1,R_2)$. Note that the conditions $R_3< C_1$ and $R_4< C_2$ are needed so that the compression rates of the coding scheme do not exceed the backhaul link capacities.

Coding scheme for the uplink C-RAN problem using a multiple access channel code and a Berger–Tung code.

For sufficiently large $n$, the rate pair $(R_1,R_2)$ can be made arbitrarily close to $\big(I(X_1;\widehat{Y}_1,\widehat{Y}_2),I(X_2;\widehat{Y}_1,\widehat{Y}_2 \cond X_1)\big)$ (e.g., using the multiple access channel coding scheme of Appendix <ref>). Similarly, for sufficiently large $n$, the compression rate pair $(R_3,R_4)$ can be chosen to be arbitrarily close to $\big(I(Y_1;\widehat{Y}_1), I(Y_2;\widehat{Y}_2) - I(\widehat{Y}_1;\widehat{Y}_2)\big)$ (e.g., using the Berger–Tung coding scheme of Appendix <ref>). It can be checked that these coding and compression rate pairs correspond to a corner point of the compress-and-forward achievable rate region given in (<ref>). Note that the decoding at the central processor in Fig. <ref> is done in the order $\widehat{Y}_1 - \widehat{Y}_2 - X_1 - X_2$. By exploiting other decoding orders, the coding scheme can achieve other corner points of the rate region (<ref>).

A similar coding scheme can be constructed for the general $K$-user, $L$-relay uplink C-RAN problem using a code for a $K$-user multiple access channel and a Berger–Tung code for a distributed lossy compression problem with $L$ sources.

Note that the uplink C-RAN coding scheme can be implemented using point-to-point channel codes by constructing the multiple access channel and Berger–Tung codes using their constituent point-to-point channel codes, as described in Appendices <ref> and <ref> respectively. Nevertheless, note that our construction of a Berger–Tung code using point-to-point channel codes requires binary sources. Hence, the channel output alphabets $\cY_1$ and $\cY_2$ of the uplink C-RAN model should be binary in this case.

§.§ Simulation Results

By implementing the multiple access channel code and the Berger–Tung code using their constituent point-to-point channel codes, the coding scheme for uplink C-RAN can be simulated using point-to-point channel codes. We consider a two-user two-relay model with a binary-quantized Gaussian channel between the users and the relays. That is, the channel output at the relays can be expressed as

\begin{align*} \label{eqn:uplink_cran_model}
\wtilde{Y}_1 &= X_1 + gX_2 + Z_1, \qquad Y_1 = q_1(\wtilde{Y}_1), \\
\wtilde{Y}_2 &= X_2 + gX_1 + Z_2, \qquad Y_2 = q_2(\wtilde{Y}_2),
\end{align*}

where $g=0.9$, $(X_1, X_2) \in \{-\sqrt{P},\sqrt{P}\}^2$ are BPSK-modulated signals, $(Z_1, Z_2) \sim \cN({\bf 0},I)$ are independent Gaussian noise components, and $q_1(.)$ and $q_2(.)$ are binary quantizers. The use of binary quantizers is motivated by the case of severely capacity-limited backhaul links (e.g., when $C_1 \leq 1$ and $C_2\leq 1$).

We first look at the maximum achievable sum-rate of the proposed coding scheme as a function of the power constraint $P$ and the backhaul capacity constraints $C_1$ and $C_2$. For the decoding order considered in Fig. <ref> and the corresponding achievable rates (see Remark <ref>), the maximum achievable sum-rate can be expressed as

\begin{equation} \label{eqn:uplink_cran_optimization}
C_{\mathrm{sum}} \triangleq \underset{\substack{p(x_1)p(x_2)p(\hat{y}_1 \cond y_1)p(\hat{y}_2 \cond y_2) \in \cD(C_1,C_2)}}{\max} \, I(X_1,X_2;\widehat{Y}_1,\widehat{Y}_2),
\end{equation}

where $\cD(C_1,C_2)$ is the set of all product distributions $p(x_1)p(x_2)p(\hat{y}_1 \cond y_1)p(\hat{y}_2 \cond y_2)$ satisfying $C_1>I(Y_1;\widehat{Y}_1)$ and $C_2>I(Y_2;\widehat{Y}_2)-I(\widehat{Y}_1;\widehat{Y}_2)$.

We use the genetic algorithm [80, 68] as an efficient heuristic method to perform the optimization in (<ref>). The quantizers $q_1(.)$ and $q_2(.)$ are optimized using the Lloyd–Max algorithm [85, 86]. That is, for any pair of input distributions $p(x_1)$ and $p(x_2)$, the distribution of $\wtilde{Y}_1$ (resp., $\wtilde{Y}_2$) is derived and used to compute the optimal partition points and quantization levels of $q_1(.)$ (resp., $q_2(.)$) using Lloyd–Max. Fig. <ref> shows the plot of $C_{\mathrm{sum}}$ as a function of the power constraint $P$ under different backhaul capacity constraints $C_1$ and $C_2$.

The maximum achievable sum-rate of the proposed coding scheme for the uplink C-RAN problem under different backhaul capacity constraints.

Simulation results of the uplink C-RAN coding scheme for a block length $n=1024$ and sum-rate $R_\mathrm{sum} = 0.25$ under different backhaul capacity constraints.

Next, the proposed coding scheme for the uplink C-RAN problem is simulated for a block length $n=1024$ and a sum-rate $R_{\mathrm{sum}} = 0.25$ using polar codes with successive cancellation decoding as the constituent point-to-point channel codes. Note that the multiple access channel code can be implemented using two asymmetric channel codes, each of which can be implemented using a pair of point-to-point channel codes. Moreover, the Berger–Tung code can be implemented using a lossy source code and a Wyner–Ziv code, each of which can be implemented using a pair of point-to-point channel codes. Hence, the uplink C-RAN coding scheme can be constructed starting from eight polar codes. Let $(R_{11},R_{12},R_{21},R_{22}, R_{31}, R_{32}, R_{41}, R_{42})$ be the rates of the constituent polar codes, where $(R_{11},R_{12})$ and $(R_{21},R_{22})$ are the rates of the two polar codes needed to construct the first and second asymmetric channel code respectively, and $(R_{31},R_{32})$ and $(R_{41},R_{42})$ are the rates of the two polar codes needed to construct the lossy source code and Wyner–Ziv code respectively. The rates are chosen “close” to their theoretical limits, i.e., we set[These theoretical limits can be derived for the uplink C-RAN model as a consequence of Remarks <ref>, <ref>, and <ref>.]

\begin{equation}\label{eqn:uplink_cran_rates}
\begin{aligned}
R_{11} &=1-H(X_1\cond \widehat{Y}_1,\widehat{Y}_2)-\gamma_r,\\
R_{12} &= 1-H(X_1), \\
R_{21} &=1-H(X_2\cond \widehat{Y}_1,\widehat{Y}_2,X_1)-\gamma_r,\\
R_{22} &= 1-H(X_2), \\
R_{31} &= 1-H(\widehat{Y}_1 \cond Y_1),\\
R_{32} &= 1-H(\widehat{Y}_1)-\gamma_c,\\
R_{41} &= 1-H(\widehat{Y}_2 \cond Y_2),\\
R_{42} &= 1-H(\widehat{Y}_2 \cond \widehat{Y}_1)-\gamma_c,
\end{aligned}
\end{equation}

where $\gamma_r > 0$ and $\gamma_c >0$ are “back-off” parameters from the theoretical limit that are used for the polar codes involved in error correction (i.e., not shaping). Thus, the sum-rate attained by the coding scheme is equal to

\[
R_{11}-R_{12}+R_{21}-R_{22} = I(X_1,X_2;\widehat{Y}_1,\widehat{Y}_2) -2\gamma_r.
\]

Similar to the approach taken in Appendix <ref>, the coding scheme targets the distributions $p(x_1)p(x_2)p(\hat{y}_1 \cond y_1)p(\hat{y}_2 \cond y_2)$ that maximize $C_{\mathrm{sum}}$ under the optimization problem of (<ref>), where the power constraint $P$ is set to be equal to the power level $P^{*}$ at which $C_{\mathrm{sum}} = R_{\mathrm{sum}} + 2\gamma_r$.[As before, the power level $P^{*}$ can be estimated using the plot of Fig. <ref>.] This guarantees that the sum-rate of the coding scheme is equal to $R_{\mathrm{sum}}$. Under the target distributions, the rates of the constituent polar codes are set according to (<ref>). Note that $\gamma_c$ should be chosen so that $R_{31}-R_{32} \leq C_1$ and $R_{41}-R_{42} \leq C_2$. In our simulations over the uplink C-RAN channel model, we used $\gamma_r = 1/8$ and $\gamma_c = 5/32$. For more details about the simulation setup, our code is available on GitHub [63]. The block error rate performance of the uplink C-RAN coding scheme is shown in Fig. <ref> for different backhaul capacity constraints $C_1$ and $C_2$. The results demonstrate the practicality of the proposed coding scheme over the uplink C-RAN channel model. Note that the observed error floor is due to binary quantization.

§ ACKNOWLEDGEMENT

The authors would like to thank the anonymous reviewers for several valuable comments that helped improve the presentation of this paper. This work was supported in part by the Institute for Information & Communication Technology Planning & Evaluation (IITP) grant funded by the Korean government (MSIT) (No. 2018-0-01659, 5G Open Intelligence-Defined RAN (ID-RAN) Technique based on 5G New Radio), in part by the NSERC Discovery Grant No. RGPIN-2019-05448, and in part by the NSERC Collaborative Research and Development Grant CRDPJ 543676-19.

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Nadim Ghaddar (S’16)
received a B.E. degree in computer and communication engineering from the American University of Beirut, Lebanon, in 2014, an M.S. degree in communication systems from the École Polytechnique Fédérale de Lausanne (EPFL), Switzerland, in 2016, and a Ph.D. degree in Communication Theory and Systems from the University of California San Diego, La Jolla, CA, USA, in 2022. He is currently a postdoctoral research fellow in the Department of Electrical and Computer Engineering at the University of Toronto. His research interests include modern coding theory, information theory, signal processing and wireless communication systems.

Shouvik Ganguly (Member, IEEE)
received the B.Tech. degree in electrical engineering from IIT Kanpur, India in 2013 and the Ph.D. degree in electrical engineering from the University of California San Diego (UCSD) in 2020. From 2020—2022, he was with XCOM Labs, San Diego, CA, USA as a member of technical staff. In 2022, he joined Samsung Research America, Plano, TX, USA, where he is currently a staff research engineer. His research interests include network information theory and communication theory.

Lele Wang
is an Assistant Professor in the Department of Electrical and Computer Engineering at the University of British Columbia, Vancouver. Before joining UBC, she was an NSF Center for Science of Information postdoctoral fellow. From 2015 to 2017, she was a postdoctoral researcher at Stanford University and Tel Aviv University. She received her Ph.D. degree in Communication Theory and Systems at the University of California, San Diego, in 2015. She attended the Academic Talent Program and obtained a B.E. degree at Tsinghua University, China in 2009. Her research interests include information theory, coding theory, communication theory, statistical inference on graphs, high dimensional statistics, and generative models in machine learning. She is a recipient of the 2013 UCSD Shannon Memorial Fellowship, the 2013-2014 Qualcomm Innovation Fellowship, and the 2017 NSF Center for Science of Information Postdoctoral Fellowship. Her PhD thesis "Channel coding techniques for network communication" won the 2017 IEEE Information Theory Society Thomas M. Cover Dissertation Award. She serves as the Vice President of the Canadian Society of Information Theory 2022–2025, an associate editor for the IEEE Transactions on Communications 2023–2026, and a guest associate editor for the IEEE Journal on Selected Areas on Information Theory 2023.

Young-Han Kim (S’99–M’06–SM’12–F’15)
received his B.S. degree (Hons.) in electrical engineering from Seoul National University, Seoul, South Korea, in 1996, and M.S. degrees in electrical engineering and in statistics and Ph.D. degree in electrical engineering from Stanford University, Stanford, CA, USA, in 2001, 2006, and 2006, respectively. In 2006, he joined the University of California at San Diego, La Jolla, CA USA, where he is currently a Professor in the Department of Electrical and Computer Engineering. Since 2020, he has also been a founding CEO of Gauss Labs Inc., an industrial AI startup company. He has co-authored the book Network Information Theory (Cambridge University Press, 2011) and the monograph Fundamentals of Index Coding (Now Publishers, 2018). His current research interests include data science, machine learning, information theory, and their applications in manufacturing, microelectronics, communications, networking, cryptography, and bioinformatics. Prof. Kim was a recipient of the 2008 NSF Faculty Early Career Development Award, the 2009 US–Israel Binational Science Foundation Bergmann Memorial Award, the 2012 IEEE Information Theory Paper Award, and the 2015 IEEE Information Theory Society James L. Massey Research and Teaching Award for Young Scholars. He served as an Associate Editor of the IEEE Transactions on Information Theory and a Distinguished Lecturer for the IEEE Information Theory Society. He is a foreign member of the National Academy of Engineering of Korea.