Linear-Quantized Precoding Model

• Linear-quantized precoding is a transmitter design technique for MIMO systems that combines linear transformation with coarse quantization to overcome hardware limitations.
• It leverages Bussgang decomposition to reformulate the nonlinear quantization effect into an equivalent linear channel model, simplifying performance analysis.
• Advanced methods like quantization-aware MMSE and sparse precoding optimize system performance and reduce distortion, achieving near-infinite resolution gains.

A linear-quantized precoding model refers to a transmitter architecture in multi-antenna (MIMO or massive MIMO) systems where information symbols are precoded by a linear transformation, then subjected to coarse quantization (typically by low-power, low-resolution DACs) before transmission. This design paradigm is motivated by hardware limitations in large-scale antenna arrays and fundamentally alters system analysis and optimization. The model illuminates the performance tradeoffs, algorithmic developments, and asymptotic behaviors in modern wireless downlink systems utilizing low-resolution front ends.

1. Signal and System Model

A canonical linear-quantized precoding system employs an $N$-antenna base station serving $K$ single-antenna users ($N > K$). The transmitter maps the user data vector $\mathbf{s} \in \mathcal{S}_M^K$ (from constellation $\mathcal{S}_M$) through a linear precoding matrix $\mathbf{P} \in \mathbb{C}^{N \times K}$ and applies an element-wise scalar quantizer $q(\cdot)$ to yield the physical transmit vector $\mathbf{x}$: $\mathbf{x} = \eta\,q(\mathbf{P} \mathbf{s}),$ where $\eta$ normalizes average power and $q$ represents quantization to a finite output alphabet $\mathcal{X}_L$ (e.g., $L$-level QAM, PSK, or constant-envelope sets) (Jacobsson et al., 2016, Wu et al., 2023, Zhang et al., 5 Dec 2025). The received signal at the $K$ users is

$\mathbf{y} = \mathbf{H}\mathbf{x} + \mathbf{n},$

with channel $\mathbf{H} \in \mathbb{C}^{K \times N}$ (entries typically i.i.d. $\mathcal{CN}(0,1/N)$) and additive noise $\mathbf{n}\sim\mathcal{CN}(0,\sigma^2\mathbf{I}_K)$.

When $q$ is a low-resolution quantizer (down to 1-bit per real/imag. part), as in massive MIMO with DAC constraints, the effective transmit alphabet is highly discrete and nonlinear distortion dominates system design (Jacobsson et al., 2016, Nadeem et al., 2023).

2. Fundamental Bussgang Decomposition and Equivalent Channel

The analysis and design of the linear-quantized precoding model rests on the Bussgang decomposition. When $\mathbf{P}\mathbf{s}$ is approximately jointly Gaussian (large $K$, i.i.d. symbols), the quantizer $q(\cdot)$ admits the affine decomposition: $$q(\mathbf{P}\mathbf{s}) = \mathbf{G}(\mathbf{P}) \mathbf{P}\mathbf{s} + \mathbf{d},$$ where $\mathbf{G}(\mathbf{P})$ is a diagonal gain matrix (function of input variance) and $\mathbf{d}$ is a distortion vector uncorrelated with $\mathbf{P}\mathbf{s}$ (Jacobsson et al., 2016, Wu et al., 2023). The key consequence is that the system can be equivalently modeled as: $$\mathbf{y} = \eta\,\mathbf{H}\mathbf{G}(\mathbf{P})\mathbf{P}\mathbf{s} + \eta\,\mathbf{H}\mathbf{d} + \mathbf{n},$$ facilitating tractable performance analysis as a linear MIMO channel subject to colored, non-Gaussian (but uncorrelated) additive noise from quantization (Saxena et al., 2016, Zhang et al., 5 Dec 2025).

In the large ($N,K \rightarrow \infty$) regime with $N/K \rightarrow \gamma > 1$, random matrix arguments further justify that $\mathbf{y}$ for each user can be approximated by a scalar AWGN channel with deterministic effective gain and distortion (Wu et al., 2023, Zhang et al., 5 Dec 2025).

3. Precoder Design Methodologies

3.1 Classical Linear-Quantized Precoding

Common linear precoding choices under quantization include:

• Zero-forcing (ZF): $$\mathbf{P}_{\text{ZF}} = \mathbf{H}^H(\mathbf{H}\mathbf{H}^H)^{-1}$$
• Regularized ZF (RZF): $$\mathbf{P}_{\text{RZF}} = \mathbf{H}^H(\mathbf{H}\mathbf{H}^H + \rho \mathbf{I})^{-1}$$
• Maximal Ratio Transmission (MRT): $\mathbf{P}_{\text{MRT}} = \mathbf{H}^H$

These classical structures persist, but the quantization nonlinearity fundamentally degrades both multi-user interference suppression and SNR even for optimal $\mathbf{P}$, especially for low DAC resolutions and high SNR (Jacobsson et al., 2016, Nadeem et al., 2023).

3.2 Bussgang-Adapted Linear Precoding

To partially compensate for quantization-induced distortion, the Bussgang-adapted linear precoder scales the columns of $\mathbf{P}$ (or introduces a diagonal gain) to make effective channel gains more nearly orthogonal after quantization. Optimizing $\mathbf{P}$ under constraints

$$\operatorname{diag}(\mathbf{P}\mathbf{P}^H)^{-1/2}\mathbf{P} = \mathbf{H}^\dagger \mathbf{D}$$

yields strictly better rates and symbol error rates, as the cross-correlation matrix $\mathbf{G}$ may be forced to be diagonal (Saxena et al., 2016).

3.3 Quantization-Aware MMSE, SLNR, and Power-Allocated Linear Precoding

Recent advances optimize the linear precoder using mean-square error (MMSE) (Usman et al., 2017), signal-to-leakage-plus-noise ratio (SLNR) (Yapıcı et al., 2019), and sum-rate expressions derived via Bussgang equivalents and mutual information bounds (Pinto et al., 2021). Closed-form design rules are given for diagonal scaling factors, regularization, and per-stream power allocation, exploiting both channel diagonality and quantizer statistics.

In multiuser/multistream contexts, block-diagonalization and MAAS (most advantageous allocation strategy) approaches extend this design to multi-user, multi-antenna receivers (Pinto et al., 2021). Water-filling-like power allocation over effective singular values (after precoder and quantization gain factors) recovers 20–30% sum-rate gain versus uniform allocation with negligible additional complexity.

3.4 Sparse Linear Precoding

By promoting row-sparsity and low PAPR in $\mathbf{P}$, it is possible to further mitigate quantizer nonlinearity. Mixed $\ell_{1,2}$ (group Lasso) and elastic-net regularizations are incorporated in the precoder design to reduce peaks before quantization and enable hardware-efficient, quantization-tolerant precoding (Mezghani et al., 2021).

4. Fundamental Performance Limits and Asymptotics

Rigorous analysis reveals several key behaviors for linear-quantized precoding:

• Performance with 3–4 DAC bits: Linear-quantized precoding approaches infinite-resolution MIMO performance; the excess SNR loss is negligible for sum-rate (Jacobsson et al., 2016).
• For 1-bit quantizers, classical linear precoding (including ZF and MRT) suffers a fixed SNR penalty (e.g., a ≈8 dB loss for QPSK at $10^{-3}$ BER in $N=128, K=16$) relative to perfect DACs, as predicted by arcsin-law limits and confirmed via large-system analysis (Jacobsson et al., 2016, Nadeem et al., 2023).
• Under asymptotic random-matrix models, the effective per-user decision SNR is diminished by a factor $2/\pi$ (MRT/mixed ADCs), $4/\pi^2$ (MRT, 1-bit DAC+ADC), but can be restored by antenna array scaling ($M_{1\text{-bit}} \approx 2.47 M_{\text{conv}}$ for MRT) (Nadeem et al., 2023).
• Asymptotic and finite-$N,K$ performance converge rapidly: SINR, SEP, and achievable rate error between the finite and asymptotic models decays as $O((\ln K/K)^{1/3})$, and empirical results demonstrate <1 dB performance loss for $K \gtrsim 100$ (Zhang et al., 5 Dec 2025).
• Error bounds for performance estimators (SINR, SEP) have been explicitly quantified using the Ky-Fan metric and probabilistic concentration (Zhang et al., 5 Dec 2025).

5. Algorithmic and Computational Considerations

Linear-quantized precoding, particularly in high-dimensional settings, is computationally tractable.

• Bussgang-based MMSE/SINR expressions and gradient-projection optimization converge in $O(N^3)$ per iteration with low constants (e.g., 14–20 iterations for $N=128$) (Jedda et al., 2017, Usman et al., 2017).
• Power allocation via CQA-MAAS and similar water-filling algorithms adds only $O(N_u)$ complexity and closes the rate loss to less than 1–2 dB (Pinto et al., 2021).
• The cost of computing the Bussgang gain factors and distortion covariance is negligible compared to SVD or matrix inversion for MIMO precoding (Pinto et al., 2021).
• Fully iterative symbol-wise quantized precoding (MAGIQ, ADMM, SQUID) achieves marginal additional performance at much higher computational loads and is generally reserved for high-order modulation or extremely coarse quantization regimes (Nedelcu et al., 2017).

6. Key Theoretical and Practical Implications

Resolution (bits)	SNR Loss vs. Full Res	Array Gain (M/K) Needed for Small BER	Hardware Implication
3–4 (uniform DAC)	~0 dB	Small (e.g. $\gamma>4$)	Low-penalty, practical
1	$5$–$8$ dB	Large ($M$ ≳ 2–3$K$ for SER $<10^{-3}$)	2.5–3$\times$ more DACs
1 + Power Alloc.	$3$–$5$ dB	Reduced penalty, near-nonlinear	Hardware feasible

• The main performance penalty of linear-quantized precoding appears with very low-resolution DACs, particularly 1–2 bits per real dimension, but can be partially offset by intelligent precoder design and resource allocation (Jacobsson et al., 2016, Nadeem et al., 2023, Pinto et al., 2021).
• With statistically or asymptotically optimal RZF-type precoders, the closed-form design (e.g. $f^*(d) = d/(d^2+\phi^*/\gamma)$) yields provable minimizers of SER/SINR in both finite and infinite system limits (Wu et al., 2023, Zhang et al., 5 Dec 2025).
• In MIMO relay channels and with quantized channel feedback, the model extends to two-hop systems, with feedback scaling linearly with SNR to maintain bounded rate loss (Xu et al., 2012, 0704.0217).
• Real-world design is enabled by explicit error bounds and practical recipes for parameter tuning, such as selecting bit-depth versus antenna scaling to meet energy efficiency or throughput targets.

7. Extensions and Research Directions

Recent research has further explored:

• Asymptotic optimality and convergence proofs for precoded, quantized systems in both statistical equivalence and practical design (Zhang et al., 5 Dec 2025).
• Novel sparse and low-PAPR linear precoding formulations that reduce nonlinear distortion without non-linear (e.g., symbol-wise or ADMM-based) optimization (Mezghani et al., 2021).
• Multi-user and multi-cell impacts, including pilot contamination and joint quantization at both transmit and receive arrays (Nadeem et al., 2023).
• Linear precoding under constrained feedback (quantized precoding matrix selection via RVQ, Grassmannian codebooks) and precise “$\log$(SNR)” scaling laws to preserve multiplexing gains (0704.0217, Xu et al., 2012).

The linear-quantized precoding model is established as a core paradigm in the design and analysis of massive MIMO and energy-efficient downlink systems with low-resolution radio hardware. Through rigorous analysis, constructive design frameworks, and practical error bounding, it informs both theoretical research and real-world wireless platform development (Jacobsson et al., 2016, Wu et al., 2023, Zhang et al., 5 Dec 2025).