Rate-Splitting & Robust Precoding

Updated 23 December 2025

Rate-splitting is a transmission strategy that partitions each user’s message into common and private streams to manage interference under imperfect CSIT.
Robust precoding methods optimize power allocation and interference constraints, ensuring improved sum-rate and fairness in challenging scenarios.
Advanced algorithmic approaches like CCCP and WMMSE enhance computational efficiency and resilience, adapting to diverse channel uncertainties.

Rate-splitting (RS) is a transmission strategy for multi-antenna downlink—especially under imperfect channel state information at the transmitter (CSIT)—in which each user’s message is partitioned into a common part (to be decoded by all users) and a private part (specific to each user). This joint encoding augments classical linear precoding with an additional layer of flexibility in interference management, substantially improving robustness and performance. Robust precoding methods for RS address design challenges arising from CSIT uncertainty, nonconvex rate constraints, and computational tractability.

1. System Model and Rate-Splitting Transmission

In the canonical RS setting, a transmitter with $M$ antennas serves $K$ single-antenna users. The transmitted vector $x\in\mathbb{C}^M$ is decomposed as: $x = p_c s_c + \sum_{k=1}^K p_k s_k$ where $s_c$ is a “common” symbol conveying parts of all users’ data, $s_k$ is the private symbol for user $k$ , and $p_c$ , $p_k\in\mathbb{C}^M$ are the corresponding linear precoders. The total BS power is limited: $\|p_c\|^2+\sum_k \|p_k\|^2 \leq P$ .

Each user $k$ receives: $Y_k = h_k^H x + Z_k$ where $h_k\in\mathbb{C}^M$ is the channel to user $k$ and $Z_k\sim\mathcal{CN}(0,1)$ is noise. Under imperfect CSIT, the transmitter knows only a noisy estimate $\widehat{h}_k$ ; $h_k = \widehat{h}_k + e_k$ , with various error models—stochastic (e.g., $e_k \sim \mathcal{CN}(0,\sigma_e^2 I)$ ) or bounded norm ( $\|e_k\| \leq \epsilon$ ) (Li et al., 2020).

The RS approach generalizes to settings with multiple receive antennas per user (MU-MIMO) (Zhou et al., 4 Dec 2025), cell-free architectures (Zheng et al., 2023, Flores et al., 26 Feb 2025), and integrated radar–communications systems (Loli et al., 2022), retaining the central paradigm of common/private split and superposed transmission.

2. Achievable Rates and Decoding Strategies

Each user uses successive interference cancellation (SIC): first decodes the common stream treating all private streams as noise, then removes $s_c$ and decodes its private stream. The per-user effective SINRs under linear precoding are: $\gamma_{c,k} = \frac{|h_k^H p_c|^2}{1 + \sum_{j=1}^K |h_k^H p_j|^2},\qquad \gamma_{k} = \frac{|h_k^H p_k|^2}{1 + \sum_{j\ne k} |h_k^H p_j|^2}$ The common rate is $\leq \min_k \log_2(1+\gamma_{c,k})$ ; the private rate is $\leq \log_2(1+\gamma_k)$ . The total rate to user $k$ is its private rate plus its assigned share of the common rate (Li et al., 2020).

In the most general RS, up to $2^K-1$ streams (for every nonempty user subset) may be generated, with SIC order affecting achievable rates. Complexity-optimized variants restrict the decoded subsets, use “stream elimination,” or enforce orderings based on group cardinality (Li et al., 2020).

Extensions exist for finite alphabet inputs—requiring non-Gaussian mutual information calculations and power allocation matching the discrete-input bottleneck (Salem et al., 2019)—and for both joint radar-communications and secure MIMO broadcasting with multiple eavesdroppers (Loli et al., 2022, Lee et al., 2024).

3. Formulations for Robust Precoder Optimization

Robust precoding is formulated as an optimization problem, typically maximizing a utility $U(R_1,\dots,R_K)$ —sum-rate, weighted sum-rate, or max-min rate—subject to power and (worst-case, stochastic, or expected-value) rate constraints derived from the RS strategy.

Maximizing utility under power constraint:

$\max U(R_1,\,\dots,\,R_K)$

subject to

$\|p_c\|^2+\sum_k\|p_k\|^2 \leq P;\qquad \text{RS rate constraints}$

Power minimization under rate constraints:

$\min\, \|p_c\|^2+\sum_k\|p_k\|^2$

subject to

$R_k \geq r_k~\forall k;~\text{RS constraints}$

Robustness to CSIT uncertainty: The constraints are imposed for all channel realizations in the (stochastic or deterministic) uncertainty set. E.g., for bounded error, the worst-case rate over all $h_k$ with $\|h_k-\widehat{h}_k\| \leq \epsilon$ is enforced (Joudeh et al., 2016, Joudeh et al., 2016).

Under stochastic error, outage/chance constraints or lower bounds on expected rates (e.g., via Jensen's inequality or Generalized Mutual Information) are adopted: $E_{\mathbf{H}| \widehat{\mathbf{H}}} [\log(1+ \text{SINR}(H))] \geq \log\left(1 + \frac{|\widehat{h}^H p|^2}{1 + \sum_j |\widehat{h}^H p_j|^2 + \sigma_e^2 \sum_j \|p_j\|^2}\right)$ (Li et al., 2020, Zhou et al., 4 Dec 2025).

4. Algorithmic Approaches: CCCP, WMMSE, SAA, and Stream Management

Concave-Convex Procedure (CCCP)

Key RS rate constraints are difference-of-convex (DC) functions; e.g., $R \leq \log(1+\text{signal+interference}) - \log(1+\text{interference})$ . CCCP linearizes the concave part and solves the resulting convex problem iteratively, converging to stationary points under mild regularity conditions. This approach applies both to sum-rate maximization and power minimization under RS (Li et al., 2020).

Weighted MMSE (WMMSE) and Alternating Optimization

WMMSE approaches reformulate rate maximization as an equivalent minimization of weighted MMSE, introducing receiver/weight updates and precoder updates in an alternating fashion. The method generalizes to:

imperfect CSIT (by lower-bounding the rates),
MIMO settings (multiple receive antennas),
sum-rate and secrecy objectives (log-sum-exp smoothing is often used for nonsmooth “min” and “max” in RS constraints) (Zhou et al., 4 Dec 2025, Mishra et al., 2021, Lee et al., 2024).

Sample Average Approximation (SAA) handles stochastic uncertainties by generating channel samples and solving deterministic problems over the ensemble (Mishra et al., 2021).

Stream Elimination and SIC Order Reduction

For general multi-layer RS, managing $2^K-1$ streams is prohibitive for large $K$ . Heuristics greedily eliminate streams that contribute negligibly to the sum-rate, retaining only a small subset (e.g., $N_s\ll 2^K$ ) (Li et al., 2020). SIC order optimization is also simplified by layer-based rules.

5. Performance and Robustness

Numerical and theoretical results demonstrate:

RS with robust precoding outperforms conventional schemes (linear MMSE, ZF, SDMA, and NOMA) in terms of sum-rate, fairness, and resilience to CSIT errors (Li et al., 2020, Zhou et al., 4 Dec 2025, Joudeh et al., 2016, Joudeh et al., 2016).
RS achieves unsaturated rates ("non-saturating DoF") in regimes where conventional methods suffer from DoF collapse due to non-scaling CSIT uncertainty (Joudeh et al., 2016, Joudeh et al., 2016).
With practical constraints (pilot contamination, asynchronous reception, channel aging, finite-constellation inputs), robust RS designs maintain substantial spectral efficiency gains (Mishra et al., 2022, Thomas et al., 2020, Zheng et al., 2022, Zheng et al., 2023, Salem et al., 2019).
Low-complexity robust algorithms (e.g., block-coordinate, AWAMSE minimization) achieve near-optimal rates with dramatically reduced compute requirements (Amor et al., 2024, Zhou et al., 4 Dec 2025).

Practical insights:

RS is especially advantageous at intermediate–high SNR, when the impact of interference and CSIT error is most pronounced.
The common stream efficiently absorbs interference that cannot be canceled by private streams due to CSIT limitations, and enables guaranteed user fairness via rate partitioning (the allocation of the common rate shares $C_k$ ) (Loli et al., 2022).
In cell-free and massive MIMO scenarios, RS enables robust common-stream transmission, leveraging bisection-based max–min fairness or robust MMSE frameworks (Zheng et al., 2022, Zheng et al., 2023, Flores et al., 26 Feb 2025).

6. Extensions: Joint Radar-Communications, Security, Finite Constellations

Integrated Radar–Communications: RSMA facilitates dual-functional systems by enabling the common stream to shape radar beampatterns while maintaining communication performance under partial CSIT; optimization jointly maximizes weighted sum-rate and minimizes radar beampattern error, subject to per-user QoS (Loli et al., 2022).
Physical Layer Security: RSMA’s inherent flexibility in message partitioning and rate allocation allows efficient secure transmission amid heterogeneous secrecy constraints, extended via log-sum-exp smoothing and generalized power iteration under limited or imperfect CSIT (Lee et al., 2024).
Finite Constellations and Constructive Interference: RS is effective for finite (e.g., PSK) alphabets; constructive interference precoding with RS further enhances rates under practical modulation constraints (Salem et al., 2019).

7. Computational Aspects and Scalability

The block-coordinate or alternating minimization algorithms used for robust precoding in RS typically require only $O(M^3)$ operations for $M$ antennas per iteration, with fast convergence (typically $\leq 10$ steps) (Zhou et al., 4 Dec 2025, Amor et al., 2024).
SAA–WMMSE or CCCP can be computationally heavy but stream selection, dimensionality reduction, and closed-form updates can provide order-of-magnitude runtime improvements without significant rate loss (Amor et al., 2024, Li et al., 2020).

The rate-splitting paradigm, underpinned by robust optimization and precoding design, systematically overcomes the limitations imposed by imperfect CSI in multi-antenna downlink. Modern robust RS methods, using CCCP, WMMSE, and related iterative algorithms, achieve near-capacity performance, enable efficient and secure transmission, and scale to massive deployment settings with manageable computational overhead (Li et al., 2020, Zhou et al., 4 Dec 2025, Loli et al., 2022, Lee et al., 2024).