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arxiv: 1907.08457 · v1 · pith:XESCHRFMnew · submitted 2019-07-19 · 📡 eess.SP · cs.IT· math.IT

Rate Splitting with Finite Constellations: The Benefits of Interference Exploitation vs Suppression

Abdelhamid Salem , Christos Masouros , Bruno Clerckx This is my paper

Pith reviewed 2026-05-24 19:22 UTC · model grok-4.3

classification 📡 eess.SP cs.ITmath.IT
keywords rate splittingconstructive interferencefinite constellationsPSKMU-MIMOsum-rateprecodingergodic rate
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The pith

Rate splitting with constructive interference precoding for PSK inputs yields higher ergodic sum-rate than zero-forcing rate splitting or non-rate-splitting in MU-MIMO.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper shows that splitting user messages into common and private parts, then precoding the private parts to exploit rather than cancel interference, improves achievable rates when inputs are restricted to finite PSK alphabets. New closed-form expressions for the ergodic sum-rate are derived under both perfect and imperfect channel knowledge at the transmitter. A power allocation rule tailored to the discrete constellation is introduced and demonstrated to outperform conventional linear precoding that does not use rate splitting. The approach is evaluated for multi-antenna broadcast channels where practical modulation limits the usual Gaussian-input assumptions.

Core claim

Under phase-shift keying inputs, rate splitting combined with closed-form constructive-interference precoding of the private messages produces higher ergodic sum-rate than rate splitting that uses zero-forcing suppression of the private messages or than transmission without rate splitting; the advantage is shown analytically for perfect CSIT and extended to imperfect CSIT, with a new power-allocation procedure that accounts for the finite alphabet.

What carries the argument

Constructive-interference precoding of the private messages inside the rate-splitting structure, which rotates the interference vectors so they add constructively to the desired signal at each receiver.

If this is right

  • RS with CI precoding of private messages outperforms both RS with ZF and NoRS under perfect CSIT.
  • The same ordering holds when only imperfect CSIT is available.
  • Analytical ergodic sum-rate expressions are obtained in closed form for both the ZF and CI private-message precoders.
  • The finite-alphabet power allocation rule produces measurable sum-rate gains relative to conventional linear precoding.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same interference-exploitation logic may extend to other discrete constellations such as QAM once suitable phase-alignment rules are derived.
  • Hardware implementations could trade the extra phase-rotation logic for reduced transmit power or fewer antennas while keeping the same rate.
  • In systems already using finite alphabets, the benefit of rate splitting may be larger than Gaussian analyses predict because the interference-exploitation gain is explicitly captured.

Load-bearing premise

The novel power allocation derived for the finite-alphabet case is optimal and accounts for the observed performance advantage over zero-forcing rate splitting and non-rate-splitting baselines.

What would settle it

Monte-Carlo simulation of the ergodic sum-rate under the same PSK constellation, channel distribution, and power constraint that shows the proposed rate-splitting constructive-interference scheme with the new power allocation does not exceed the rates of rate-splitting zero-forcing or non-rate-splitting transmission.

Figures

Figures reproduced from arXiv: 1907.08457 by Abdelhamid Salem, Bruno Clerckx, Christos Masouros.

Figure 1
Figure 1. Figure 1: Sum-rate versus SNR for RS and NoRS with different t [PITH_FULL_IMAGE:figures/full_fig_p025_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Sum-rate versus SNR for RS and NoRS with different t [PITH_FULL_IMAGE:figures/full_fig_p025_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Sum-rate versus SNR for RS and NoRS with different t [PITH_FULL_IMAGE:figures/full_fig_p027_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Sum-rate versus SNR for RS and NoRS with different t [PITH_FULL_IMAGE:figures/full_fig_p027_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Sum-rate for RS with CI and QPSK modulation versus S [PITH_FULL_IMAGE:figures/full_fig_p028_5.png] view at source ↗
read the original abstract

Rate-Splitting (RS) has been proposed recently to enhance the performance of multi-user multiple-input multiple-output (MU-MIMO) systems. In RS, a user message is split into a common and a private part, where the common part is decoded by all users, while the private part is decoded only by the intended user. In this paper, we study RS under a phase-shift keying (PSK) input alphabet for multi-user multi-antenna system and propose a constructive interference (CI) exploitation approach to further enhance the sum-rate achieved by RS under PSK signaling. To that end, new analytical expressions for the ergodic sum-rate are derived for two precoding techniques of the private messages, namely, 1) a traditional interference suppression zero-forcing (ZF) precoding approach, 2) a closed-form CI precoding approach. Our analysis is presented for perfect channel state information at the transmitter (CSIT), and is extended to imperfect CSIT knowledge. A novel power allocation strategy, specifically suited for the finite alphabet setup, is derived and shown to lead to superior performance for RS over conventional linear precoding not relying on RS (NoRS). The results in this work validate the significant sum-rate gain of RS with CI over the conventional RS with ZF and NoRS.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 2 minor

Summary. The paper investigates rate-splitting (RS) in multi-user MIMO systems using phase-shift keying (PSK) finite constellations. It derives new analytical expressions for the ergodic sum-rate under zero-forcing (ZF) and constructive interference (CI) precoding for the private messages, considering both perfect and imperfect channel state information at the transmitter (CSIT). A novel power allocation strategy tailored to finite alphabets is proposed, and numerical results are presented to show that RS with CI achieves significant sum-rate gains compared to RS with ZF and conventional NoRS precoding.

Significance. If the analytical expressions are accurate and the power allocation strategy is effective, this work demonstrates the benefits of interference exploitation in finite-alphabet MU-MIMO systems, providing closed-form tools for performance evaluation that could aid in system design. The comparison between CI and ZF approaches under RS is a useful contribution to understanding interference management with discrete inputs. The derivation of closed-form CI precoder expressions under PSK is a clear strength.

major comments (1)
  1. [Power allocation strategy] Power allocation strategy (central to abstract and all numerical results): The novel power allocation between common and private streams is presented as specifically derived for the finite-alphabet case and responsible for the claimed superiority over NoRS. However, the manuscript provides no proof of optimality for this allocation in the non-convex finite-alphabet ergodic rate maximization; the split appears obtained via heuristic search or low-dimensional enumeration rather than a globally optimal solution. This assumption is load-bearing for the central claim that RS-CI yields significant gains.
minor comments (2)
  1. [Imperfect CSIT analysis] The extension of the ergodic rate expressions to imperfect CSIT would benefit from an explicit statement of the error model (e.g., variance of the estimation error) in the relevant section.
  2. [Numerical results] Figure captions for the sum-rate plots should specify the number of Monte Carlo channel realizations used to generate the curves.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the constructive feedback. We address the sole major comment below.

read point-by-point responses

  1. Referee: [Power allocation strategy] Power allocation strategy (central to abstract and all numerical results): The novel power allocation between common and private streams is presented as specifically derived for the finite-alphabet case and responsible for the claimed superiority over NoRS. However, the manuscript provides no proof of optimality for this allocation in the non-convex finite-alphabet ergodic rate maximization; the split appears obtained via heuristic search or low-dimensional enumeration rather than a globally optimal solution. This assumption is load-bearing for the central claim that RS-CI yields significant gains.

    Authors: We agree that the optimization is non-convex and that a closed-form global optimum is unavailable. The allocation is obtained by a one-dimensional numerical search over the common-stream power fraction using the derived ergodic-rate expressions (exact for PSK), which is computationally tractable. This procedure is tailored to finite alphabets because it employs the constellation-specific mutual-information expressions rather than Gaussian approximations. We do not claim global optimality but demonstrate consistent gains over NoRS. We will revise the manuscript to explicitly describe the numerical search method and its motivation in Section IV. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivations use standard rate expressions

full rationale

The paper derives closed-form ergodic sum-rate expressions for ZF and CI precoders under PSK inputs using conventional MIMO mutual information formulas and extends them to imperfect CSIT. The novel power allocation is presented as a derived strategy for the finite-alphabet case and validated numerically against baselines, but no step reduces a claimed prediction or result to a quantity defined by the same fitted parameters or self-citation chain. All load-bearing analytical steps rest on external standard techniques rather than internal redefinition or heuristic search renamed as optimality proof. This is the expected self-contained case.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claim rests on standard MIMO channel and noise models plus the validity of the derived closed-form rate expressions and the optimality of the new power allocation for finite alphabets.

free parameters (1)
  • power allocation factor between common and private streams
    Novel strategy specifically suited for finite alphabet setup; its values are optimized or chosen to achieve the reported gains.
axioms (1)
  • domain assumption Standard assumptions on i.i.d. Rayleigh fading channels and additive white Gaussian noise for ergodic rate calculation.
    Invoked to derive the analytical expressions for perfect and imperfect CSIT cases.

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Reference graph

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