Algorithm-Hardware Co-Design of a Lightweight PCG Equalizer with a Fixed Step Size for Massive MIMO

Junshuo Wang; Shuting Liu; Shihan Wang; Yan Tian; Jienan Chen

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Algorithm-Hardware Co-Design of a Lightweight PCG Equalizer with a Fixed Step Size for Massive MIMO

Junshuo Wang, Shuting Liu, Shihan Wang, Yan Tian, Jienan Chen

arXiv:2609.00890v1Sep 1, 2026cs.ITeess.SP

TL;DR

Coarse ADC quantization reduces power but creates signal-dependent clipping distortions, while BEM-based recovery and CG step-size evaluation impose hardware costs. The paper proposes a one-step fixed-step PCG correction with an O(U) feed-forward symbol-level path, supported by finite-dimensional descent and normalization analysis. In the evaluated setting, it approaches high-resolution references and indicates a projected net power reduction of 289.1 mW.

Problem
Coarse quantization lowers power but causes clipping distortions, while BEM matrix inversion and CG dynamic step-size evaluation create hardware bottlenecks.
Method
The paper uses the initial Jacobi-PCG direction with a fixed relaxation parameter to replace per-symbol matrix inversion and dynamic step-size evaluation.
Results
In the evaluated setting, the 6-bit design achieves 8.558 bps/Hz, closely approaches 12-bit references, and indicates a projected net power reduction of 289.1 mW.
Takeaways & Limitations
The co-design supports an O(U) feed-forward symbol-level update for energy-efficient massive-MIMO uplinks in the evaluated setting.

Abstract

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Coarse quantization in massive multiple-input multiple-output (MIMO) systems reduces power but causes clipping distortions. The Bayesian Expectation-Maximization (BEM) algorithm can recover clipped signals, but its matrix inversion and dynamic step-size evaluation are hardware bottlenecks. We propose a hardware-friendly one-step correction that uses the initial Jacobi-preconditioned Conjugate Gradient (PCG) direction with a fixed relaxation parameter. The resulting symbol-level update has an ultra-lightweight $\mathcal{O}(U)$ feed-forward datapath and approaches high-resolution reference detectors in the evaluated massive-MIMO setting. Our finite-dimensional analysis establishes the exact one-step descent law, proves that Jacobi normalization cancels the raw multiplicative near-far scaling while confining the loaded-system dependence to bounded attenuation factors, and gives verifiable sufficient conditions for fixed-step descent in terms of normalized channel coherence. System-level results indicate projected power savings for energy-efficient massive MIMO uplinks.

I. INTRODUCTION

Coarse ADC quantization lowers power but creates signal-dependent clipping distortions that challenge conventional linear detection. The paper targets BEM and iterative-solver hardware bottlenecks with a fixed-step, one-step PCG correction and an O(U) feed-forward datapath.

Coarse quantization introduces correlated clipping distortions that skew constellation geometry and challenge flat-Gaussian-noise detectors.
BEM leverages discrete constellation priors for clipped-amplitude reconstruction but requires dense O(U^3) matrix inversion per symbol.
GS has strict sequential dependencies, whereas CG’s dynamic step-size evaluation incurs dense O(U^2) operations and pipelined feedback dependencies.
The proposed co-design uses a fixed relaxation parameter with the initial PCG direction to remove dynamic step-size evaluation and enable an O(U) feed-forward symbol-level datapath.
The effective beam-domain formulation uses y = Fyq and H = FHphy, reducing Gram-matrix formation to O(U^2Nbeam) complex multiplications.

B. BEM-Inspired Quadratic Recovery Surrogate

The recovery stage replaces the exact quantized likelihood with a decision-directed quadratic surrogate conditioned on hard decisions and diagonal weights. Its implementation goal is to avoid per-symbol inversion and dynamic step-size computation while retaining a fixed-step one-update path.

The recovery stage uses a decision-directed quadratic surrogate motivated by BEM updates, conditioned on the hard-decision vector and diagonal weight matrix.
The surrogate minimization becomes the linear system Ax = b with A = H^HH + D and D = W + σ^2I.
All formal descent statements apply only to the conditioned quadratic surrogate, not the exact quantized-data likelihood, spectral efficiency, or multi-iteration convergence.
The block-level Gram computation and ZF factorization are amortized, while fast-varying diagonal loading makes repeated per-symbol inversion or refactorization the bottleneck.
The method executes one update using the initial PCG search direction and a fixed relaxation parameter, without invoking conjugacy across iterations.
The finite-dimensional analysis is conditioned on H, y, hard decisions, and W without requiring Gaussian or channel-independent quantization errors.

A. Well-Posedness, Residual Cancellation, and Exact Line Search

Under full-column-rank and positive-loading assumptions, the surrogate is well posed and the exact ZF initializer yields a simplified residual. The initial Jacobi-preconditioned direction admits an exact line-search characterization, with finite-precision and interpretation caveats.

Well-Posedness: Full column rank of H and δw > 0 imply G ≻ 0, A ≻ 0, and a unique surrogate minimizer A^-1b.
Residual Cancellation: With an exact ZF initializer, the residual cancels to r(0) = W x̄ − D x̂(0).
Residual Cancellation: Finite-precision ZF error changes the true residual, so the simplified residual’s descent behavior requires an additional real-part positivity condition not established by the synthesis data.
Exact Line Search: For p = M^-1r, the quadratic surrogate along x̂(0) + αp has a unique optimal real relaxation parameter.
Exact Line Search: The one-step mismatch loss measures descent lost on the quadratic surrogate only, not error rate or spectral-efficiency performance.

B. Why an Unnormalized Global Step Is Fragile

Without Jacobi normalization, a single fixed step is fragile under near-far power variation. Jacobi scaling removes the raw user-power factors and bounds the loaded system through normalized channel geometry and attenuation factors.

Unnormalized fixed-step fragility: For the diagonal family A = P, the unpreconditioned optimal step depends on the power range, while worst-case relative loss tends to one as κP →∞.Any α ≥2/pmax is non-descending for the strongest-user direction.
Unnormalized fixed-step fragility: The counterexample rules out a uniform unpreconditioned fixed-step guarantee but does not claim that every ill-conditioned realization diverges.
Jacobi normalization: Factoring H = ZP1/2 separates user powers from channel directions, whose normalized Gram matrix T has unit diagonal.
Jacobi normalization: Theorem 1 shows that Jacobi normalization cancels P exactly when D = 0 and, with loading, confines the remaining dependence to contraction factors 0 < γi < 1.
Jacobi normalization: The finite-dimensional identity remains valid for arbitrary near-far powers, finite Nbeam and U, channel correlation, and positive diagonal loading.

D. Spectral and Fixed-Step Guarantees

The paper converts normalized channel coherence into verifiable fixed-step descent guarantees and uses those results to remove dynamic step-size evaluation from the symbol-level hardware path.

Spectral guarantees: If ε = ∥T −I∥2 < 1, the unit step αfix = 1 minimizes the worst-case bound and gives strict descent for every nonzero residual.
Spectral guarantees: As ∥T −I∥2 →0, the normalized system deviation vanishes and the optimal step converges uniformly to one, independently of near-far powers.
Spectral guarantees: The favorable-propagation condition is stated spectrally as a verifiable condition rather than inferred from the invalid approximation HHH/Nbeam ≈I.
Fixed-step calibration: The interval-robust calibration is αmm ≈1.068 for the empirical 95% interval [0.971, 1.187], while αfix = 1.075 is described as empirical calibration.
Hardware architecture: The architecture targets the fast-varying symbol-level solver path by eliminating the dynamic step-size evaluator, yielding an O(U) feed-forward datapath.

A. Algorithmic Simplifications for Hardware

The proposed equalizer removes redundant symbol-level computations and organizes the remaining operations into a pipelined hardware architecture. Its evaluated setup combines amortized block-level processing with lightweight updates and calibrated fixed-step behavior.

Algebraic simplifications: Exact ZF initialization cancels the dense matrix-vector multiplication, reducing residual initialization from O(U^2) operations to O(U) element-wise processing.The residual simplifies because the relevant matrices are diagonal.
Algebraic simplifications: Implicit symmetric preconditioning uses M−1r(0) instead of M−1/2r(0), avoiding the hardware-prohibitive square-root matrix.
Pipelined macro-architecture: The architecture separates slow-varying channel computations from fast-varying symbol updates through a deeply pipelined macro-architecture controlled by the TLC.The PCU computes G and yMF, while the MIU amortizes O(U^3) LDLH-based ZF factorization across the coherence block.
Pipelined macro-architecture: The VSPU performs symbol-level one-step PCG updates using exclusively O(U) element-wise operations and removes the O(U^2) step-size evaluator in the evaluated regime.The design also eliminates the high-latency scalar divider required for dynamic step-size evaluation.
Evaluation setup: The evaluated uplink uses B = 2048, Nbeam = 256, U = 32, 6-bit quantized beam-domain signals, and mixed-QAM scheduling under a 20 dB near-far effect.The channel model is 3GPP CDL-B with spatial correlation ρ = 0.2 and practical LS channel estimation with PCHIP interpolation.
Evaluation setup: Jacobi normalization keeps the observed κ(eA) between 23.7 and 41.3 across the evaluated near-far range, while the unconditioned κ(A) grows strongly.The observed plateau height is empirical and is not numerically predicted by the theorem.

B. Macroscopic SE and Power Trade-off

The proposed fixed-step BEM-PCG equalizer nearly matches high-resolution spectral-efficiency references in the evaluated setting while reducing symbol-level hardware cost and projected system power.

Macroscopic SE: 8.558 bps/Hz at 32 dB lets the 6-bit Fixed-Step-Size BEM-PCG closely approach 12-bit references and the 8.6 bps/Hz scheduled-modulation ceiling.At 26 dB, maximum relative SE degradation versus 12-bit PIC-MMSE and MMSE references is 0.52% and 0.31%, respectively.
Macroscopic SE: At low-to-medium SNRs, the fixed-step implementation slightly outperforms the dynamic-step BEM-PCG in the reported simulation.The proposed scheme also gains over the BEM-GS and dynamic BEM-CG baselines in the evaluated macroscopic SE results.
Hardware amortization: The evaluation reuses block-level factorization across 14 OFDM symbols, while executing only the symbol-level VSPU update for each received symbol.This amortization separates slow-varying channel computations from fast-varying symbol-level processing.
Power trade-off: Replacing the O(U 2) dynamic step-size evaluator with an O(U) element-wise preconditioning array yields a 164.3 mW VSPU.The cited passage attributes this implementation change to the reported VSPU consumption.
Power trade-off: 452.4 mW of model-based 6-bit ADC front-end reduction corresponds to a projected net system-level power reduction of 289.1 mW under the stated assumptions.The estimate uses SMIC 55nm CMOS synthesis assumptions and block-fading, low-to-medium mobility conditions.
Conclusion: The evaluated detector closely approaches 12-bit reference baselines and 6-bit BEM-GS detection while indicating projected net power savings.The conclusion does not claim convergence of a multi-iteration PCG method or optimality for the exact quantized likelihood.

APPENDIX A PROOFS OF PROPOSITIONS 1 AND 2

The appendix derives exact one-step quadratic identities, including the optimal relaxation, fixed-step mismatch loss, and descent conditions for preconditioned and unpreconditioned directions.

Proposition proofs: For a quadratic objective, the exact one-step change is expressed through the optimal relaxation parameter and yields a necessary-and-sufficient descent condition.The proof establishes f(α) − f(0) = αt(α − 2α∗).
Proposition proofs: For the unpreconditioned direction p = r with A = P and r = e_i, the relative loss from fixed α is η_i = (αp_i − 1)^2.The minimax fixed step over p ∈ [pmin, pmax] equalizes endpoint magnitudes.
Proposition proofs: The minimax step is α = 2/(pmin + pmax), while the strongest-user direction descends when 0 < α < 2/pmax.These results follow from endpoint equalization and the one-step descent condition.

APPENDIX C PROOF OF THEOREM 1

Theorem 1 shows that Jacobi normalization removes raw diagonal power scaling from normalized Gram entries, leaving loaded-system effects as bounded attenuation factors.

Jacobi normalization: Because the effective matrix has positive diagonal entries, its Jacobi-normalized diagonal entries are one.The proof begins from nonzero columns implying Cii > 0.
Jacobi normalization: For off-diagonal entries, the normalized identities are equivalent to the theorem’s stated representation, and spectral-norm submultiplicativity supplies the bound.The proof uses the fact that 0 < γi ≤ 1.
Jacobi normalization: When D = 0, the attenuation matrix satisfies Γ = I, giving exact cancellation of the raw multiplicative near-far factors.With loading, the remaining dependence is confined by attenuation factors bounded in (0, 1].

APPENDIX D PROOFS OF COROLLARIES 1 AND 2

These corollaries bound fixed-step loss and establish conditions under which normalized systems support uniform one-step descent.

Corollary proofs: When ε < 1, every eigenvalue of the normalized matrix lies in [1 − ε, 1 + ε], enabling interval-based one-step loss bounds.The Rayleigh quotient establishes the relevant quadratic bound.
Corollary proofs: The minimax unit calibration αfix = 1 equalizes endpoint errors and has worst-case loss ε2 over the eigenvalue interval.The loss is derived from η = (αfixλq − 1)^2.
Corollary proofs: Since 1 + ε < 2, the unit step satisfies strict descent, with the same argument providing a uniform descent condition.The proof explicitly connects the eigenvalue interval to the sufficient condition.
Corollary proofs: As ∥T − I∥2 → 0, the normalized matrix approaches the identity uniformly over all nonzero residuals.The eigenvalue interval contracts to one in this limit.
Corollary proofs: For eigenvalues in [ℓ, u], the minimax positive step is α = 2ℓu/(ℓ + u), with endpoint error (u − ℓ)/(u + ℓ).Squaring the endpoint error gives the corresponding worst-case relative loss.