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

• Coarse quantization introduces correlated clipping distortions that skew constellation geometry and challenge flat-Gaussian-noise detectors.Source: §I · p. 1
• BEM leverages discrete constellation priors for clipped-amplitude reconstruction but requires dense O(U^3) matrix inversion per symbol.Source: §I · p. 1
• GS has strict sequential dependencies, whereas CG’s dynamic step-size evaluation incurs dense O(U^2) operations and pipelined feedback dependencies.Source: §I · p. 1
• 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.Source: §I · p. 1
• The effective beam-domain formulation uses y = Fyq and H = FHphy, reducing Gram-matrix formation to O(U^2Nbeam) complex multiplications.Source: A. Quantized Massive MIMO Uplink · p. 2

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

Well-Posedness

• Full column rank of H and δw > 0 imply G ≻ 0, A ≻ 0, and a unique surrogate minimizer A^-1b.Source: A. Well-Posedness, Residual… · p. 2

Residual Cancellation

• With an exact ZF initializer, the residual cancels to r(0) = W x̄ − D x̂(0).Source: A. Well-Posedness, Residual… · p. 2
• 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.Source: A. Well-Posedness, Residual… · p. 2

Exact Line Search

• For p = M^-1r, the quadratic surrogate along x̂(0) + αp has a unique optimal real relaxation parameter.Source: A. Well-Posedness, Residual… · p. 2
• The one-step mismatch loss measures descent lost on the quadratic surrogate only, not error rate or spectral-efficiency performance.Source: A. Well-Posedness, Residual… · p. 2

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.Source: B. Why an Unnormalized Global Step Is… · p. 2
• The counterexample rules out a uniform unpreconditioned fixed-step guarantee but does not claim that every ill-conditioned realization diverges.Source: B. Why an Unnormalized Global Step Is… · p. 3

Jacobi normalization

• Factoring H = ZP1/2 separates user powers from channel directions, whose normalized Gram matrix T has unit diagonal.Source: C. Exact Removal of Near-Far Power… · p. 3
• 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.Source: C. Exact Removal of Near-Far Power… · p. 3
• The finite-dimensional identity remains valid for arbitrary near-far powers, finite Nbeam and U, channel correlation, and positive diagonal loading.Source: C. Exact Removal of Near-Far Power… · p. 3

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.Source: D. Spectral and Fixed-Step Guarantees · p. 3
• As ∥T −I∥2 →0, the normalized system deviation vanishes and the optimal step converges uniformly to one, independently of near-far powers.Source: D. Spectral and Fixed-Step Guarantees · p. 3
• The favorable-propagation condition is stated spectrally as a verifiable condition rather than inferred from the invalid approximation HHH/Nbeam ≈I.Source: D. Spectral and Fixed-Step Guarantees · p. 3

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.Source: D. Spectral and Fixed-Step Guarantees · p. 3

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.Source: §IV · p. 3

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.Source: A. Algorithmic Simplifications for… · p. 3
• Implicit symmetric preconditioning uses M−1r(0) instead of M−1/2r(0), avoiding the hardware-prohibitive square-root matrix.Source: A. Algorithmic Simplifications for… · p. 3

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.Source: B. Pipelined Macro-Architecture · p. 3
• 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.Source: B. Pipelined Macro-Architecture · p. 4

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.Source: §V · p. 4
• 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.Source: A. Condition Number Dynamics and… · p. 4

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.Source: B. Macroscopic SE and Power Trade-off · p. 5
• 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.Source: B. Macroscopic SE and Power Trade-off · p. 5

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.Source: C. Hardware Synthesis and Power… · p. 5

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.Source: C. Hardware Synthesis and Power… · p. 5
• 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.Source: C. Hardware Synthesis and Power… · p. 5

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.Source: §VI · p. 5

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∗).Source: Appendix A · pp. 5–6
• 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.Source: Appendix B · p. 6
• 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.Source: Appendix B · p. 6

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.Source: Appendix C · p. 6
• 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.Source: Appendix C · p. 6
• 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].Source: Appendix C · p. 6

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.Source: Appendix D · p. 6
• 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.Source: Appendix D · p. 6
• 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.Source: Appendix D · p. 6
• 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.Source: Appendix D · p. 6
• 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.Source: Appendix E · p. 6