One-Bit Quantized Observations in MIMO

• One-bit quantized observations are discrete measurements derived by applying a sign function to the real and imaginary parts of analog signals, enabling energy-efficient MIMO estimation.
• The methodology contrasts nonlinear MMSE estimators, which require high-dimensional probability integrals, with tractable linear Bussgang decomposition useful under specific channel conditions.
• Practical designs leverage these techniques to balance estimation performance and computational complexity in massive MIMO setups with low-resolution ADCs.

One-bit quantized observations are discrete-valued measurements obtained by applying a one-bit quantizer to the real and imaginary components of a continuous-valued analog signal or vector. In the context of multi-antenna (MIMO) wireless systems and estimation theory, one-bit quantization emerges as an energy- and cost-efficient approach for analog-to-digital conversion, especially in large-scale antenna arrays where high-resolution ADCs are impractical. This severe quantization introduces pronounced nonlinearity, fundamentally altering the statistical and algorithmic properties of inference and signal processing tasks, most notably minimum mean-square error (MMSE) channel estimation. The theoretical and practical underpinnings of the field, associated estimator designs, and optimality conditions for linear and nonlinear methods constitute an active research area.

1. Definition and System Model

A one-bit quantizer maps each real or imaginary part of the analog input $\mathbf{y}$ to its sign: $$\mathbf{r} = Q(\mathbf{y}) = \operatorname{sgn}(\Re\{\mathbf{y}\}) + j \operatorname{sgn}(\Im\{\mathbf{y}\})$$ Hence $\mathbf{r}$ takes values in the set $\{\pm 1 \pm j\}^n$ for $n$-dimensional complex input. In one-bit quantized MIMO channel estimation, the observed data $\mathbf{r}$ results from the quantization of the noisy linear superposition of training signals and unknown channel coefficients. For a basic MIMO model: $$\mathbf{y} = H\mathbf{s} + \mathbf{n}, \quad \mathbf{r} = Q(\mathbf{y})$$ where $H$ is the channel, $\mathbf{s}$ is the transmitted symbol vector, and $\mathbf{n}$ is complex Gaussian noise. Both the channel and noise might exhibit arbitrary correlation.

2. Statistical Properties and Nonlinear Estimation

Unlike in unquantized or high-resolution quantized systems, the conditional distribution $p(\mathbf{r}|\mathbf{h})$ under one-bit quantization is no longer Gaussian but is determined by the orthant probability of a multivariate normal distribution. The optimal MMSE estimate of the channel is then: $$\hat{\mathbf h}_{\mathrm{MMSE}} = \mathbb{E}[\mathbf h|\mathbf r]$$ For vectorized models with Gaussian priors and noise, this estimator can be computed as a ratio of high-dimensional orthant probability integrals involving the precision matrix of the sufficient statistics (see extensive derivations in (Ding et al., 2024, Ding et al., 2024)). Specifically: $$\hat{\mathbf h}_{\mathrm{MMSE}} = \frac{ \int \mathbf h\, P(\mathbf{r}|\mathbf h) p(\mathbf h) \, d\mathbf h }{ \int P(\mathbf{r}|\mathbf h) p(\mathbf h) \, d\mathbf h }$$ where $P(\mathbf{r}|\mathbf h)$ factors into products of cumulative Gaussian probabilities over half-spaces governed by the quantization pattern $\mathbf{r}$ (Ding et al., 2024).

This estimator is generally nonlinear in $\mathbf{r}$, except in specific structurally favorable scenarios. The complexity of computing the MMSE estimator generally grows exponentially with the dimension, except in carefully structured cases (see Sec. 4).

3. Linear Estimation: Bussgang Decomposition and BLMMSE

Given the intractability of the full posterior mean for high dimensional settings, the Bussgang decomposition provides a tractable linear surrogate. By expressing the quantized output $\mathbf{r}$ as a linear function of the pre-quantized signal plus uncorrelated distortion, one obtains the Bussgang linear MMSE (BLMMSE) estimator: $\hat{\mathbf h}_{\mathrm{BLMMSE}} = G \mathbf r$ where the gain matrix $G$ is obtained via closed-form expressions involving the covariance of the pre-quantized input, arcsine transforms, and the Bussgang gain, typically: $$G = \frac{\sqrt{\pi}}{2} \Sigma A^H D_{\Omega}^{-1/2} \left( \arcsin(D_{\Omega}^{-1/2} \Re \Omega_b D_{\Omega}^{-1/2}) + j \arcsin(D_{\Omega}^{-1/2} \Im \Omega_b D_{\Omega}^{-1/2}) \right)^{-1}$$ where $D_\Omega=\mathrm{diag}(\Omega_b)$ and $\Omega_b$ is the covariance of the unquantized observation vector (Ding et al., 2024, Ding et al., 2024).

While the BLMMSE is always optimal among linear estimators, it generally does not coincide with the true nonlinear MMSE estimator unless strong structural conditions are met.

4. MMSE–BLMMSE Optimality Condition and Special Cases

A central theoretical advance is the characterization of when the BLMMSE estimator is MMSE-optimal (i.e., the optimal estimator is linear in $\mathbf{r}$ even after one-bit quantization). The necessary and sufficient condition is given in terms of the precision matrix $C$ appearing in the orthant-probability representation. The MMSE channel estimator is linear in $\mathbf{r}$, and BLMMSE is MMSE-optimal, if and only if each row (or column) of $C$ contains at most one nonzero off-diagonal entry, so that $C$ is block diagonal with $1\times1$ or $2\times2$ blocks (Ding et al., 2024, Ding et al., 2024). This includes:

• Uncorrelated channels with orthogonal pilots: $C$ is diagonal.
• Channels with transmit-only correlation and pilots aligned to the eigenbasis: $C$ remains diagonal.
• SIMO systems with $N_R=2$: $C$ splits into two $2\times2$ blocks.
• SISO and scalar channels: always holds.
• Some special forms of spatial correlation and/or block-diagonal pilot allocation.

Whenever this condition is not satisfied—e.g., with fully correlated MIMO channels, correlated noise, or non-orthogonal pilots—the estimator is necessarily nonlinear, and the MMSE–BLMMSE gap may become substantial, especially in moderate SNR regimes and high-dimensional arrays (Ding et al., 2024).

5. Impact of Channel and Noise Correlation

Recent work investigates the influence of channel and noise correlation on the structure and efficacy of the MMSE estimator from one-bit quantized measurements. In the presence of spatially correlated additive noise, the MMSE estimator remains nonlinear in general but can again become linear for certain combinations of channel and noise correlation parameters (Ding et al., 18 Jan 2026). Specifically, in models with constant channel correlation $\phi$ and noise correlation $\xi$, linearity is induced when the effective correlation $(\gamma \phi + \xi)$ vanishes for appropriately chosen SNR. Noise correlation may either enhance or degrade estimator performance depending on the degree of channel correlation and SNR; for uncorrelated channels at low-to-medium SNR, increased noise correlation can improve MMSE, while the opposite is true for highly correlated channels (Ding et al., 18 Jan 2026).

In all these cases, closed-form expressions or low-dimensional single-integral forms exist for SISO and certain SIMO cases, but for general MIMO with arbitrary correlation, one must evaluate high-dimensional orthant probabilities, incurring exponential complexity.

6. Computational Considerations and Specialized Algorithms

A summary of computational features for practical channel estimation under one-bit quantization:

• BLMMSE: Requires evaluating covariance-related quantities and inverting an appropriately dimensioned matrix; complexity typically $O((\tau N_R)^3)$. Widely used for large systems due to tractability (Ding et al., 2024).
• MMSE (nonlinear): Requires computing the ratio of $2\tau N_R$-dimensional orthant probability integrals (optionally, and often necessarily, via Monte Carlo or numerical integration); infeasible for $N_R>4$ except in special cases with symmetry/exchangeability (Ding et al., 2024, Ding et al., 2024).
• Special-case (SIMO/SISO): When correlation is symmetric (e.g., equal correlation), the necessary integrals can be reduced to one-dimensional forms (e.g., via Tong’s formula), enabling feasible computation even for large $N_R$ (Ding et al., 2024).

For massive arrays and systems lacking special symmetry, scalable nonlinear MMSE estimation relies on approximate algorithms such as expectation-propagation, approximate message passing, or utilizing generative models, but these approaches depart from provable optimality.

7. Applications and Design Implications

The theoretical findings on one-bit quantized channel estimation have direct implications for design and operation of large-scale MIMO systems with low-resolution ADCs:

• BLMMSE estimators are theoretically justified and often optimal in low or high SNR, and under orthogonality/no-correlation conditions. They provide reasonable performance–complexity tradeoffs for practical systems (Ding et al., 2024, Ding et al., 2024).
• True nonlinear MMSE estimators should be preferred in regimes with strong spatial correlation or non-orthogonal pilot allocation, particularly in moderate SNR regimes where estimator nonlinearity leads to marked MSE reduction (Ding et al., 2024).
• System designers may tune pilot sequences or artificially induce noise correlation to enforce conditions for linearity when possible, benefitting from the simplicity and optimality of BLMMSE (Ding et al., 18 Jan 2026).
• In high-dimensional and unsymmetrical MIMO settings, the MMSE estimator is intractable; approximate inference algorithms or data-driven nonlinear estimators (e.g., via deep learning) may mitigate the complexity (Nguyen, 2020).

These results provide mathematically rigorous foundations and clear design criteria for developing estimation strategies in large-scale one-bit quantized systems.

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References:

(Ding et al., 2024) On Optimal MMSE Channel Estimation for One-Bit Quantized MIMO Systems (Ding et al., 2024) Optimality of the Bussgang Linear MMSE Channel Estimator for MIMO Systems with 1-Bit ADCs (Ding et al., 18 Jan 2026) The Effect of Noise Correlation on MMSE Channel Estimation in One-Bit Quantized Systems (Nguyen, 2020) Neural Network-Optimized Channel Estimator and Training Signal Design for MIMO Systems with Few-Bit ADCs