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Beyond Dirty Paper Coding for Multi-Antenna Broadcast Channel with Partial CSIT: A Rate-Splitting Approach
Yijie Mao, Bruno Clerckx
TL;DR
Multi-antenna broadcast-channel capacity and its capacity-achieving strategy remain unknown under imperfect CSIT, although perfect-CSIT DPC is capacity-achieving. This paper studies Rate-Splitting and proposes DPCRS, which combines RS with DPC for private-message encoding. In partial-CSIT MISO BC, the reported rate regions exceed those of conventional DPC and linearly precoded RS for DPCRS.
Problem
Under imperfect CSIT, the capacity and capacity-achieving strategy of the multi-antenna broadcast channel remain unknown, while CSIT inaccuracy causes severe multi-user interference.
Source: §I · p. 1Method
The paper studies linearly precoded Rate-Splitting and proposes DPCRS, which splits messages into common and private parts and uses DPC to encode private parts.
Source: §I · p. 3Results
In partial-CSIT MISO BC, DPCRS enlarges the rate region beyond conventional DPC and linearly precoded RS, while linearly precoded RS outperforms DPC when CSIT is sufficiently inaccurate.
Source: §I · p. 3Takeaways & Limitations
DPCRS is presented as a new benchmark and a less CSIT-, network-load-, and user-deployment-sensitive achievable strategy for multi-antenna BC with partial CSIT.
Source: §I · p. 3
Abstract
from arXiv · showhide
Imperfect Channel State Information at the Transmitter (CSIT) is inevitable in modern wireless communication networks, and results in severe multi-user interference in multi-antenna Broadcast Channel (BC). While the capacity of the multi-antenna (Gaussian) BC with perfect CSIT is known and achieved by Dirty Paper Coding (DPC), the capacity and the capacity-achieving strategy of the multi-antenna BC with imperfect CSIT remain unknown. Conventional approaches therefore rely on applying communication strategies designed for perfect CSIT to the imperfect CSIT setting. In this work, we break this conventional routine and make two major contributions. First, we show that linearly precoded Rate-Splitting (RS), relying on the split of messages into common and private parts and linear precoding at the transmitter, and successive interference cancellation at the receivers, can achieve larger rate regions than DPC in multi-antenna BC with partial CSIT. Second, we propose a novel achievable scheme, denoted as Dirty Paper Coded Rate-Splitting (DPCRS), that relies on RS to split the user messages into common and private parts, and DPC to encode the private parts. We show that the rate region achieved by DPCRS in Multiple-Input Single-Output (MISO) BC with partial CSIT is enlarged beyond that of conventional DPC and that of linearly precoded RS. Gaining benefits from the capability of RS to partially decode the interference and partially treat interference as noise, DPCRS is less sensitive to CSIT inaccuracies, networks loads and user deployments compared with DPC and other existing transmission strategies. The proposed DPCRS acts as a new benchmark and the best-known achievable strategy for multi-antenna BC with partial CSIT.
I. INTRODUCTION
The introduction asks whether multi-antenna transmission should be designed for partial CSIT from the outset, since perfect-CSIT strategies can be fragile when channel knowledge is imperfect. It presents Rate-Splitting and Dirty Paper Coded Rate-Splitting as approaches that outperform conventional DPC in the stated partial-CSIT setting.
I. INTRODUCTION
- Imperfect CSIT is a major challenge because channel estimation, feedback overhead, delay, mobility, and hardware effects can impair channel knowledge and contribute to multi-user interference.The introduction reports that CSIT inaccuracy causes losses of at least 30% in cell-average throughput and 42% in cell-edge throughput for 4G LTE-Advanced MU-MIMO.Source: §I · p. 1
- The paper asks whether multi-user multi-antenna transmission strategies should be designed and optimized under partial CSIT rather than adapting perfect-CSIT strategies afterward.The conventional approach first identifies a channel's limits and capacity-achieving strategy, then simplifies it and incorporates partial CSIT; residual interference is treated as noise.Source: §I · pp. 1–2
- Rate-Splitting divides messages into common and private parts, with a common stream decodable by all receivers and private streams intended for their corresponding users.The common stream is superposed on private streams, bringing partial CSIT into the design early; RS is stated to be DoF-optimal in the K-user MISO BC with partial CSIT –.Source: §I · p. 2
- In MISO BC with partial CSIT, linearly precoded Rate-Splitting (RS) outperforms DPC, while DPCRS enlarges the rate region beyond both DPC and linearly precoded RS.DPCRS combines RS message splitting with DPC encoding of private parts and is described as less sensitive to CSIT inaccuracies, network loads, and user deployments.Source: §I · p. 3
A. Channel Model
- The channel model assumes perfect CSIR and partial CSIT, with the CSIT quality scaling factor α representing quality in the high-SNR regime.α = 0 represents finite-precision partial CSIT, while α = ∞ represents perfect CSIT; the joint distribution of actual and estimated CSI is assumed stationary and ergodic.Source: A. Channel Model · pp. 3–4
B. Conventional Dirty Paper Coding
Conventional DPC encodes user messages in a chosen order and precodes the resulting streams using channel estimates. With imperfect CSIT, interference cancellation is incomplete, and instantaneous rates may not be achievable.
B. Conventional Dirty Paper Coding
- DPC encodes messages according to a permutation-based user order, then precodes and superposes the resulting streams.The encoding order specifies which user's message is encoded before another user's message.Source: B. Conventional Dirty Paper Coding · p. 4
- With imperfect CSIT, DPC precoders use the estimated channel, so only part of the interference from previously encoded streams is removed at a receiver.Each user decodes its intended message while treating residual interference as noise.Source: B. Conventional Dirty Paper Coding · p. 4
- Because the transmitter lacks the exact channel, an instantaneous DPC rate based on a joint fading state may be overestimated and unachievable.The passage presents ergodic rate as a more robust precoder-design approach under delay-unlimited transmission, averaging over possible joint fading states.Source: B. Conventional Dirty Paper Coding · p. 4
1) Motivation:
The section motivates DPCRS through RS's gains under partial CSIT, then describes one- and multi-layer schemes combining common-stream rate-splitting with DPC-coded private streams. It also qualifies the rate-region claims by noting that private-stream ergodic-rate achievability is unresolved.
1) Motivation:
- In an underloaded K-user MISO BC, RS achieves sum DoF 1 + (K −1)α, matching the upper bound and exceeding the conventional approach's max{1, Kα}.The interpretation splits the DoF into Kα from a perfect-CSIT sub-network and 1−α from a no-CSIT sub-network.Source: 1) Motivation · p. 4
2) One-Layer Dirty Paper Coded Rate-Splitting (1-DPCRS):
- 1-DPCRS splits each user's message into common and private parts, sending the common stream for all users to decode and encoding private streams with DPC in a chosen order.The common message is decoded while treating private streams as noise; users then decode their intended private stream.Source: 2) One-Layer Dirty Paper Coded… · p. 5
3) Multi-Layer Dirty Paper Coded Rate-Splitting (M-DPCRS):
- M-DPCRS extends generalized RSMA with DPC, splitting each three-user message into four parts for global and pairwise common streams plus a private stream encoded with DPC.The scheme is described for three users and can be generalized to K users.Source: 3) Multi-Layer Dirty Paper Coded… · pp. 5–6
- Three-user M-DPCRS uses three SIC layers, decoding the all-user common stream first, intended pairwise common streams next, and private streams last.The order among pairwise common streams is jointly optimized with the precoders.Source: 3) Multi-Layer Dirty Paper Coded… · p. 6
- Achievability of the DPC-coded private-stream ergodic rates under partial CSIT is unknown, and a rigorous demonstration is beyond this work's scope.The paper's rate-region conclusions therefore consider both achievable-rate and outer-bound cases.Source: 3) Multi-Layer Dirty Paper Coded… · pp. 6–7
- Lower-bound analysis confirms RS and DPCRS outperform DPC; if their ergodic rates are achievable, DPCRS enlarges the region, while otherwise RS beats DPC's outer bound and nearly overlaps DPCRS's.The paper frames these as two cases depending on whether the relevant DPC and DPCRS ergodic rates are achievable.Source: 3) Multi-Layer Dirty Paper Coded… · p. 7
III. PROBLEM FORMULATION
The formulation optimizes long-term weighted sum-rate performance under partial CSIT. It uses average rates conditioned on the transmitter's channel estimate to decompose the design into per-estimate problems.
III. PROBLEM FORMULATION
- The design objective is weighted ergodic sum rate, rather than instantaneous weighted sum rate, which may yield undecodable rates under partial CSIT.The paper defines WESR for DPC, 1-DPCRS, and M-DPCRS.Source: §III · p. 7
- Average rate is the expected decoding rate over CSIT uncertainty for a given channel estimate, whereas ergodic rate captures long-term performance across joint channel states.The transmitter can predict average rates even though it cannot estimate instantaneous rates.Source: §III · p. 7
- The WESR problem is decomposed into a weighted average sum-rate problem for each channel estimate, using short-term rate and power constraints.The paper notes that replacing a long-term power constraint with per-channel-state short-term constraints avoids an intractable power-allocation problem across channel states.Source: §III · pp. 7–8
A. DPC
The DPC formulation maximizes weighted average sum rate for a fixed user-weight vector and encoding order, then considers encoding orders to maximize weighted ergodic sum rate.
A. DPC
- For a given weight vector and fixed DPC encoding order, the formulation poses a weighted average sum-rate problem for each channel estimate.The problem includes user quality-of-service rate constraints.Source: A. DPC · p. 8
- Maximizing weighted ergodic sum rate for a given weight vector requires optimizing over DPC encoding orders.The paper states that all possible orders are considered.Source: A. DPC · p. 8
- With perfect CSIT and zero QoS rate constraints, the DPC formulation achieves the MISO BC capacity region by optimizing precoders.For single-antenna users, the optimal DPC covariance matrices are rank one, so the capacity region can be achieved through precoder optimization.Source: A. DPC · p. 8
B. 1-DPCRS
The section formulates the weighted average sum-rate design for one-layer and multi-layer DPCRS. Both optimize precoders and rate allocation for a given channel estimate, with additional optimization over relevant stream orders.
B. 1-DPCRS
- For 1-DPCRS, moving minimization inside the expectation over rates gives a lower bound that removes joint dependence among channel instances.The resulting per-estimate problem jointly optimizes the precoder and allocation of the common-stream rate across users.Source: B. 1-DPCRS · p. 8
- The 1-DPCRS weighted ergodic sum-rate design also considers all possible DPC encoding orders for each user-weight vector.The common-stream rate allocation is jointly optimized with the precoder for each channel estimate.Source: B. 1-DPCRS · p. 8
C. M-DPCRS
- For three-user M-DPCRS, the design jointly optimizes the pairwise-common-stream decoding order and precoders, then selects the DPC private-stream encoding order with highest weighted average sum rate.The search evaluates possible common-stream decoding and private-stream encoding orders.Source: C. M-DPCRS · p. 8
IV. PROPOSED OPTIMIZATION FRAMEWORK
The framework addresses stochastic, non-convex rate optimization through a three-step procedure. Sample Average Approximation (SAA) replaces stochastic average rates with deterministic approximations based on sampled user channels.
IV. PROPOSED OPTIMIZATION FRAMEWORK
- The framework combines SAA, Weighted Minimum Mean Square Error (WMMSE), and Alternating Optimization (AO) to address stochasticity and non-convexity.SAA yields deterministic problems, WMMSE yields block-wise convex problems, and AO solves them.Source: §IV · pp. 8–9
A. Sample Average Approximation Approach
- SAA samples user channels conditioned on the channel estimate and approximates average rates with sampled rates.The precoder is held fixed over the M samples, and the sampled approximation approaches the average rate when M is sufficiently large.Source: A. Sample Average Approximation Approach · p. 9
- For M-DPCRS, the base station evaluates common-stream and private-stream precoders across decoding-order pairs and selects the pair with the highest WASR.The optimization designs the precoder P and common-stream allocation vector bc; as M grows, the sampled problem approximates the stochastic problem for a given common-stream decoding order.Source: A. Sample Average Approximation Approach · p. 9
B. Weighted Minimum Mean Square Error Approach
The WMMSE approach reformulates the sampled rate optimization using equalizers and MSE weights. This transformation produces a WMMSE problem equivalent to the sampled optimization problem.
B. Weighted Minimum Mean Square Error Approach
- The sampled rate optimization remains non-convex, so the authors extend a WMMSE algorithm to reformulate it.Users decode streams sequentially according to the decoding order, using equalizers at the receivers.Source: B. Weighted Minimum Mean Square Error… · p. 9
- The method defines stream-level mean-square errors and weighted mean-square errors for decoding at each user.The MSE depends on the remaining received power during stream decoding.Source: B. Weighted Minimum Mean Square Error… · p. 9
- The rate-WMMSE relationship extends from instantaneous channel realizations to averages over sampled user channels, yielding an equivalent WMMSE problem.The formulation includes a transformation of the common-rate vector and MSE weights and equalizers.Source: B. Weighted Minimum Mean Square Error… · pp. 9–10
C. Alternating Optimization Algorithm
The alternating optimization algorithm exploits block-wise convexity by updating equalizers, weights, and precoders with the other variable blocks fixed. Its objective decreases monotonically and is bounded below, ensuring convergence.
C. Alternating Optimization Algorithm
- The joint problem is non-convex but block-wise convex in the equalizers, weights, and precoders with common-rate variables when the other blocks are fixed.This structure motivates alternating optimization.Source: C. Alternating Optimization Algorithm · p. 10
- At each iteration, the algorithm updates equalizers, weights, and then precoders and common-rate variables.The equalizer and weight updates use the current precoder; the precoder and common-rate update solves a convex quadratically constrained quadratic program by interior-point methods.Source: C. Alternating Optimization Algorithm · p. 10
- The algorithm repeats these updates until the WASR converges.The objective decreases monotonically and is bounded below by the transmit-power constraint.Source: C. Alternating Optimization Algorithm · pp. 10–11
D. Convergence
The authors establish convergence of the proposed algorithm and discuss its computational cost. Numerical evaluation compares eight transmission strategies over varied user deployments and system conditions.
D. Convergence
- With a feasible initial point, Algorithm 1 converges; as M approaches infinity, its convergent solution is a stationary point of problem (20).The proof uses monotonic objective decrease and the equivalence between the sampled and stochastic problems as M grows.Source: D. Convergence · p. 11
E. Complexity
- M-DPCRS has much higher computational complexity than DPC and 1-DPCRS because its common-stream count grows exponentially with the number of users.The authors note that linearly precoded RS has much lower complexity and achieves performance close to M-DPCRS in their Section V results.Source: E. Complexity · p. 11
V. NUMERICAL RESULTS
- For the simulations, average rates are approximated using SAA with 1,000 channel samples, and WESR is averaged over 100 channel realizations.The channels and estimation errors are generated from specified complex Gaussian distributions.Source: §V · pp. 11–12
- The evaluation compares eight strategies across user deployments, CSIT qualities, QoS rate requirements, and channel-strength disparities.The study examines two-user and three-user MISO broadcast-channel cases with partial CSIT.Source: §V · p. 12
A. Two-user case
In the two-user case, DPCRS and RS are evaluated against DPC and other strategies under partial CSIT. DPCRS achieves the largest ergodic rate region, while RS and DPCRS are more robust to CSIT inaccuracy than DPC and MU–LP.
A. Two-user case
- For K = 2, M-DPCRS, generalized RS, and SC–SIC per group reduce respectively to 1-DPCRS, 1-layer RS, and SC–SIC.The two-user comparison considers ergodic sum rate versus SNR under diverse CSIT inaccuracies and channel-strength disparities, with zero individual QoS constraints and equal user weights.Source: A. Two-user case · p. 12
- At α = 0.3 and 30 dB, DPCRS and RS each achieve 16.12% rate improvement over DPC and MU–LP with equal channel variance.Their rate gain over other strategies grows with SNR; with 10 dB channel-variance disparity, RS’s gain decreases.Source: A. Two-user case · p. 13, Figure 4 · p. 13, Figure 5 · p. 13
- RS and DPCRS achieve the same DoF, with values close to the theoretically anticipated RS values of 1.3, 1.6, and 1.9 for α = 0.3, 0.6, and 0.9, respectively.DPC’s DoF matches MU–LP’s values of 1, 1.2, and 1.8 for those α values; its rate drops rapidly as α decreases.Source: A. Two-user case · p. 13
- SC–SIC has the worst performance in the two sum-rate comparisons and the worst DoF, limited to 1.The section attributes its performance to inefficient use of the spatial domain for interference management.Source: A. Two-user case · p. 13, Figure 4 · p. 13, Figure 5 · p. 13
- At SNR = 20 dB, DPCRS achieves the largest ergodic rate region among the compared strategies.As the number of transmit antennas decreases, the rate-region gap between DPCRS and DPC/MU–LP becomes larger.Source: A. Two-user case · p. 13, Figure 6 · p. 14
B. Three-user case
The paper concludes that DPCRS enlarges the MISO BC rate region under partial CSIT and is more robust to CSIT inaccuracies, network loads, and user deployments than DPC and other strategies. In three-user tests, RS-assisted strategies are more robust to CSIT inaccuracy, and multi-layer RS improves ergodic sum rate in overloaded networks.
B. Three-user case
- The three-user comparisons use SNR = 20 dB and vary QoS constraints, network loads, and user deployments.In one comparison, the per-user QoS rate constraint increases as α rises from 0.2 to 1, from 0.1 to 0.5 bit/s/Hz.Source: B. Three-user case · p. 14
- In overloaded three-user networks, M-DPCRS and generalized RS achieve higher ergodic sum rate than 1-DPCRS and 1-layer RS, respectively.Adding common-stream layers improves interference management and ergodic sum rate, even though it does not increase DoF.Source: B. Three-user case · p. 14
- As α decreases from 1 to 0.2, DPC, MU–LP, and SC–SIC-assisted strategies’ ergodic sum rates decrease dramatically, while all RS-assisted strategies are more robust to CSIT inaccuracy.Underloaded-network tests show generalized RS and 1-layer RS improve ergodic sum rate over DPC when α is less than 0.6.Source: B. Three-user case · p. 14
VI. CONCLUSION
- DPCRS enlarges the MISO BC rate region under partial CSIT and is more robust to CSIT inaccuracies, network loads, and user deployments than DPC and other strategies.It splits user messages into common and private parts, using DPC to encode the private parts; linearly precoded RS also outperforms DPC when CSIT is sufficiently inaccurate, with lower hardware and computational complexity.Source: §VI · p. 14
APPENDIX
The appendix compares ergodic rate regions obtained from lower bounds on achievable rates for DPC, DPCRS, and RS. The DPCRS lower-bound region is always larger than those of RS and DPC in the reported comparison.
APPENDIX
- The appendix derives achievable-rate lower bounds for DPC and DPCRS following Theorem 1 in, and derives an RS lower bound for comparison.The encoding order π in equation (43) determines which user’s rate is approximated by its lower bound.Source: APPENDIX · p. 15
- The DPCRS lower-bound ergodic rate region is always larger than the RS or DPC lower-bound region.The RS lower bound improves over DPC and is closer to the DPCRS lower bound; when Nt = 2, the RS lower-bound region exceeds DPC’s region with much lower complexity.Source: APPENDIX · p. 15
- The reported ergodic rate regions based on lower bound (41) are smaller than those obtained based on equations (5) and (8b).The comparison includes DPC, DPCRS, and RS regions and their corresponding lower-bound regions.Source: APPENDIX · p. 15