IC-DMS: Interference Channels with Degraded Message Sets

• IC-DMS is a communication model where one sender has complete, non-causal knowledge of the other's message, enabling cooperative interference management.
• The scheme integrates cooperative, collaborative, and dirty paper coding to enhance rate regions and pre-cancel interference in wireless channels.
• Quantitative evaluations in both discrete memoryless and Gaussian settings demonstrate significant rate improvements, especially under high interference.

An interference channel with degraded message sets (IC-DMS) is a communication model featuring two senders and two receivers, where each sender transmits a separate message, but one sender (Sender 2) has complete, non-causal knowledge of the other's message (Sender 1). This model captures scenarios in which secondary users in wireless networks (e.g., cognitive radio) not only avoid interfering with primary users but can actively cooperate by leveraging knowledge of the primary message. The paper of such channels addresses fundamental limits on information transfer in the presence of interference when asymmetric transmitter side information is present.

1. IC-DMS Model and Structure

The IC-DMS is defined by a discrete memoryless channel (DMC)

$$(\mathcal{X}_1, \mathcal{X}_2, \mathcal{Y}_1, \mathcal{Y}_2, p(y_1, y_2 | x_1, x_2))$$

where Sender 1 transmits message $w_1$ to Receiver 1, and Sender 2 transmits message $w_2$ to Receiver 2. Uniquely, Sender 2 knows the entirety of $w_1$ prior to transmission. The schematic in the source highlights that Sender 2's encoder processes its own message, $w_2$, while exploiting its a priori knowledge of $w_1$ to support its own communication and to opportunistically assist Sender 1.

This setting generalizes both the classical interference channel and broadcast scenarios; the IC-DMS interpolates between them depending on the coding strategy.

2. Coding Strategies: Cooperation, Collaboration, and Dirty Paper Coding

The paper develops a composite coding scheme that synthesizes three classical techniques:

• Cooperative Coding: Sender 2 transmits signal components that reinforce Sender 1’s transmission at Receiver 1.
• Collaborative Coding: Sender 2 splits its message into two parts, $w_2 = (w_{21}, w_{22})$, with $w_{21}$ encoded so that both receivers can decode it, mitigating interference at Receiver 1.
• Dirty Paper Coding (DPC) (Gel’fand–Pinsker approach): Sender 2 pre-cancels the interference from Sender 1’s message (using known side information) by an encoding strategy that effectively treats $w_1$ as “state” in the classic state-dependent channel problem. The encoder computes signals as functions of $w_1$ to “write on dirty paper.”

The joint distribution of random variables under this scheme is

$$p(q, w, x_1, u, \tilde{u}, v, \tilde{v}, x_2, y_1, y_2) = p(q)p(x_1, w|q)p(u, \tilde{u}|w,q)p(v, \tilde{v}|w,q) p(x_2|\tilde{u}, \tilde{v}, w, q)p(y_1, y_2|x_1, x_2)$$

where $Q$ is a time-sharing variable and $U, V, \tilde{U}, \tilde{V}$ are auxiliaries that enable layered coding for cooperation and DPC.

The encoding procedure for Sender 2 involves generating codewords using both message parts and $w_1$, combining codeword segments to form $x_2$ as per the function $x_2 = f(\tilde{u}, \tilde{v}, w, q)$. Receiver 1 performs joint decoding on $w_1$ and $w_{21}$; Receiver 2 recovers both $w_{21}$ and $w_{22}$.

3. Achievable Rate Regions: Discrete Memoryless and Gaussian IC-DMS

The achievable rate region $\mathcal{R}$ for the discrete memoryless case is specified by

$$\begin{aligned} R_1 &\leq I(W; Y_1 U \mid Q) \ R_2 &\leq I(UV; Y_2 \mid Q) - I(U; W \mid Q) - I(V; W \mid Q) \ R_1 + R_2 &\leq I(WU; Y_1 \mid Q) + I(V; Y_2 U \mid Q) - I(U; W \mid Q) - I(V; W \mid Q) \ \text{plus:} \quad 0 &\leq I(UW; Y_1 \mid Q) - I(U; W \mid Q) \quad \text{and further non-negativity constraints} \end{aligned}$$

A structurally simpler subregion $\mathcal{R}_{\text{sim}}$ is given by: $$R_1 \leq I(W; Y_1 \mid U, Q), \quad R_2 \leq I(UV; Y_2 \mid Q) - I(V; W \mid Q), \quad \cdots$$ with similarly interpreted mutual information constraints.

In the Gaussian channel case, discrete auxiliaries are mapped to Gaussian random variables: $$\begin{align*} &\text{Let} \quad W \sim \mathcal{N}(0,1), \quad X_1 = \sqrt{P_1} W\ &\tilde{u} \sim \mathcal{N}(0, \alpha\beta P_2), \quad \tilde{v} \sim \mathcal{N}(0, \alpha(1-\beta) P_2)\ &U = \tilde{u} + \lambda_1 W, \quad V = \tilde{v} + \lambda_2 W\ &X_2 = \tilde{u} + \tilde{v} + \sqrt{(1-\alpha) P_2} W\ &Y_1 = X_1 + \sqrt{c_{21}} X_2 + Z_1\ &Y_2 = X_2 + \sqrt{c_{12}} X_1 + Z_2 \end{align*}$$ where $Z_1, Z_2 \sim \mathcal{N}(0,1)$ and $c_{21}, c_{12}$ are normalized interference gains.

A key achievable rate subregion (via a successive-decoding based strategy $\mathcal{G}_{\text{suc}}$) is: $$\begin{aligned} R_1 &\leq \frac{1}{2} \log_2 \left[ 1 + \frac{(\sqrt{P_1} + \sqrt{c_{21}(1-\alpha) P_2})^2}{c_{21} \alpha P_2 + 1} \right] \ R_2 &\leq \frac{1}{2} \log_2 [1 + \alpha(1-\beta) P_2] + \min \left\{ \begin{array}{l} \frac{1}{2} \log_2 \left[ 1 + \frac{c_{21} \alpha \beta P_2}{(\sqrt{P_1} + \sqrt{c_{21}(1-\alpha) P_2})^2 + c_{21}\alpha(1-\beta) P_2 + 1} \right] \ \frac{1}{2} \log_2 \left[ 1 + \frac{\alpha\beta P_2}{\alpha(1-\beta)P_2 + (\sqrt{(1-\alpha)P_2} + \sqrt{c_{12}P_1})^2 + 1} \right] \end{array} \right\} \end{aligned}$$ after optimizing over the splitting and DPC parameters $(\alpha, \beta, \lambda_1, \lambda_2)$.

4. Comparative Evaluation and Numerical Results

The derived achievable regions systematically include and improve upon previously known rate regions for the IC-DMS:

• Prior approaches, such as by Tarokh et al., did not exploit cooperative coding and DPC in combination; the present scheme integrates both, yielding strictly larger achievable regions in many regimes.
• Specific subregions (denoted $\mathcal{R}_{\text{sp1}}, \mathcal{R}_{\text{sp2}}$) match the best previous results when auxiliaries are optimized accordingly.
• Numerical comparisons illustrate especially strong gains when the interference link is strong ($c_{21} > 1$), with the new region offering substantial rate improvement, particularly for Receiver 1, due to both collaborative decoding of sender 2’s message and effective pre-cancellation.

Figures in the paper plot rate regions under different channel parameters, demonstrating that the proposed coding scheme generally outperforms earlier methods under high interference.

5. Applications and Theoretical Implications

The implications of this work extend to multiple domains:

• Cognitive Radio: The IC-DMS models secondary user access where non-causal knowledge of the primary user allows not only interference avoidance but also proactive cooperation and spectrum efficiency enhancement.
• Wireless Sensor Networks: In networks with node cooperation or side information, the scheme provides a method to increase aggregate data rates and link reliability.
• Interference-Limited Wireless Systems: In dense wireless deployments where interference is managed through sophisticated coding, the combination of superposition, collaborative message splitting, and DPC is essential for spectral efficiency.

More broadly, the scheme bridges the interference and broadcast channel models, indicating a continuum between them and motivating new strategies where transmitter side information is non-uniformly distributed.

6. Methodological Advances and Future Work

This work provides a general coding framework for IC-DMS that can be specialized to recover several classic results, offering a unifying lens for understanding interference management with transmitter side information. The approach is directly extensible to:

• Other channels with partial message cognition
• Multiuser scenarios with hybrid broadcast/interference characteristics
• More general forms of transmitter and receiver cooperation via side information and feedback

The strong numerical evidence for rate region enlargement in high-interference regimes suggests practical utility for next-generation wireless systems operating in environments with layered access and opportunistic cooperation.

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Table: Coding Features in the IC-DMS Scheme

Technique	Role in Encoding	Impact on Rates
Cooperative Coding	Sender 2 helps Sender 1	Increases Receiver 1
Collaborative Coding	Split $w_2$ for both Rx	Mitigates interference
Dirty Paper Coding	DPC by Sender 2	Pre-cancels known state

This table summarizes the integrative role of the three key techniques and their impact on achievable rates in the IC-DMS setting.