Low-Complexity MIMO Channel Estimation

• Low-complexity MIMO channel estimation is a collection of algorithmic strategies designed to acquire channel state information with reduced computational and memory burdens in large-scale systems.
• Techniques such as polynomial expansion, FFT-based filtering, and sparsity exploitation achieve near-optimal estimation performance while scaling from cubic to nearly linear complexity.
• Innovations in deep learning and distributed architectures further improve efficiency by balancing pilot overhead, robustness, and hardware limitations in diverse MIMO scenarios.

Low-complexity MIMO channel estimation refers to a class of algorithmic and architectural strategies that enable accurate channel state information (CSI) acquisition in multiple-input multiple-output (MIMO) wireless systems under severe complexity constraints. These techniques address the prohibitive computational scaling, memory consumption, pilot overhead, and hardware limitations that arise in large-scale, high-dimensional, or resource-constrained communication scenarios. The central challenge is to achieve estimation accuracy (measured by normalized mean-square error or bit-error rate with downstream detectors) comparable to classical methods such as MMSE or maximum-likelihood (ML), but with polynomial or even nearly linear computational complexity—even as system size or pilot dimension grows by orders of magnitude.

1. Canonical Models and Complexity Bottlenecks

In MIMO systems, pilot-based CSI acquisition is usually described by a linear model

$\mathbf{Y} = \mathbf{H}\mathbf{P} + \mathbf{N}$

where $\mathbf{H}$ is the unknown channel, $\mathbf{P}$ the pilot matrix, and $\mathbf{N}$ additive noise. The statistical MMSE estimator or related schemes (e.g., LMMSE in correlated models) involve inversion of large covariance or Gram matrices, causing complexities scaling as $\mathcal{O}(N_t^3)$, $\mathcal{O}(N_t^2N_r)$, or higher in the number of transmit ($N_t$) and receive ($N_r$) antennas. In massive MIMO, RIS-aided, or extra-large (XL-MIMO) configurations, direct implementation of these methods is infeasible due to both computational and memory burden.

Low-complexity MIMO channel estimation thus focuses on:

• Polynomial or sub-cubic scaling in $N_t$, $N_r$
• Pilot/sample overhead reduction
• Reduction of arithmetic operations per update (e.g., by FFTs, message passing, or one-shot learning)
• Hardware and implementation efficiency (e.g., decentralized architectures)

2. Classical Low-Complexity Approaches

2.1 Polynomial Expansion and Fast Filtering

One established paradigm replaces expensive matrix inversions by low-order matrix polynomials. The PEACH estimator (Shariati et al., 2013) approximates the inversion in MMSE estimation by

$$(\mathbf{A})^{-1} \approx \alpha\,\sum_{\ell=0}^L (I - \alpha\mathbf{A})^\ell$$

where the polynomial order $L$ (typically 5–20) is chosen to balance complexity and bias. This reduces complexity to $\mathcal{O}(L\,N^2)$, significantly lower than the $\mathcal{O}(N^3)$ of standard MMSE; weighting strategies can further speed convergence. Under pilot contamination, the necessary $L$ drops, as the MSE floor dominates performance.

2.2 Structure Exploitation: Circulant Covariances and FFTs

When the array geometry and propagation ensure a circulant covariance (e.g., in extra-large MIMO with uniform circular arrays and line-of-sight), the LMMSE solution can be diagonalized by DFTs (Pu et al., 2023). All $\mathcal{O}(N^3)$ matrix operations reduce to scalar divisions in the FFT domain, yielding complexity $\mathcal{O}(N\log N)$ for matrix-vector products and $\mathcal{O}(N^2\log N)$ for the full estimator. When channel statistics are not perfectly known, sample-DFT methods provide nearly optimal eigenvalue estimates with small additional cost.

2.3 Sparsity and Matrix Completion

In mmWave and switching-based hybrid architectures, MIMO channels are often approximately low-rank or sparse in the angular or delay domain. Matrix completion (Hu et al., 2016) exploits low-rankness directly, reconstructing the channel from random entry samples via a singular-value projection (SVP) algorithm that avoids explicit basis dictionaries. Each iteration costs only $\mathcal{O}(N_t^2N_r + N_t^3 + N_t^2L)$ flops, and in practical settings, SVP achieves NMSE and spectral efficiency close to compressive-sensing baselines, but with 6–26$\times$ lower complexity and mitigated basis mismatch issues.

3. Algorithmic Innovations: Message Passing and Hierarchical Search

3.1 Message Passing and Structured Bayesian Models

Low-complexity message passing algorithms, such as UAMP-SBL (Li et al., 2023) and GAMP-EM (Hu et al., 2021), leverage sparsity or block structure within MIMO/RIS channels. These methods use variational inference and structured priors (e.g., hierarchical or partial-common support) to decompose large inference problems into scalar updates with rigorous complexity bounds: $O(JTN P_{BR}I)$ for RIS-aided systems ($J$ users, $T$ training, $P_{BR}$ support), several orders lower than conventional SBL ($O(N^3)$ per iteration). Row and column support prunings, as in partial-common identification, further accelerate convergence while preserving performance.

3.2 Hierarchical and Two-Stage Search

In hybrid mmWave systems, training overhead and search complexity can be reduced via hierarchical codebook-based multi-beam search (Xiao et al., 2016). The method progressively refines AoA/AoD estimates over logarithmic lattice search grids, reducing time slots required for $N_S$ streams from $O(M_A N_A)$ (sequential) or $O(N_S^3M^2)$ (sparse hybrid search) to $O(N_S M)$ (Xiao et al., 2016). Analog and digital precoding are decoupled, enabling most heavy computation to be performed in a low-$N_S$-dimensional baseband, rather than across the full array.

Another family uses two-stage or 2D compressive sensing, splitting support search in AoA and AoD (e.g., two-stage SOMP + OMP or 2D-OMP), reducing dictionary and matching complexity from $O(N^4)$ (1D-OMP) to $O(N^3)$ (Yang et al., 2022) with negligible accuracy loss.

4. Deep Learning and Generative Low-Complexity Channel Estimation

Recent advances employ deep generative models, particularly diffusion models (DMs), to learn the complex non-Gaussian structure of massive MIMO channels and provide Bayesian MMSE estimates directly from pilot observations (Fesl et al., 2024, Chen et al., 2 Feb 2026, Fan et al., 24 Oct 2025). Techniques include:

• Diffusion-based estimation with CNNs (Fesl et al., 2024): DMs learn the angular-domain distribution via a lightweight convolutional network (∼$5\times 10^4$ parameters), with deterministic, truncated reverse diffusion guided by pilot SNR positional embeddings. Complexity is reduced to one or a few forward passes, and performance tracks or exceeds classical score-based DMs, LMMSE, and GMM methods.
• Sampling-Free Diffusion Transformers (Chen et al., 2 Feb 2026): A sampling-free diffusion transformer maps noisy LS (least-squares) angular-domain estimates directly to denoised channels in a single inference pass, leveraging angular sparsity and transformer scalability. With $0.67\times 10^6$ parameters, inference latency is under $2$ ms and NMSE outperforms state-of-the-art (up to $5.65$ dB gain), while being robust to channel statistics shift.
• Latent Diffusion Models (Fan et al., 24 Oct 2025): Channels are compressed to a smooth low-dimensional manifold via a VAE, and diffusion is performed in latent space. Posterior sampling with approximated gradients and self-consistency leads to computation reduction by $10$–$100\times$ vs. image-scale DMs, and improved NMSE compared to LDAMP, SGM, and other deep-learning baselines.
• CNN-Structured MMSE Estimators (Fesl et al., 2021): By casting the unconditional Bayesian MMSE estimator as a two-layer circular convolutional neural network under FFT and pilot-structure assumptions, complexity is reduced to $\mathcal{O}(SU\log(SU))$ for $S \times U$ arrays.

5. Domain-Specific Methods: RIS, XL-MIMO, and OFDM

Several low-complexity estimators address domain-specific challenges:

• RIS-Assisted MIMO: UAMPSBL-PCI (Li et al., 2023) exploits RIS-induced sparsity and multiuser partial support. The unitary AMP backbone avoids matrix inversions, and partial support identification ties hyperparameters across users and clusters, providing $2$–$3$ dB NMSE improvement over DS-OMP at constant complexity.
• Semi-Passive RIS with 1-bit Quantization: A two-stage ADMM+nuclear-norm low-rank recovery and EM-GAMP sparse Bayesian inference achieves high accuracy in joint spatio-frequency domain with just $8\%$ of hardware active, compared to 1-bit CS, QIHT, and OMP (Hu et al., 2021).
• Near-Field XL-MIMO: Sequential angle-distance estimation (SADCE (Huang et al., 2023)) decomposes 3D parameter search into 2D FFT-based angle estimation and closed-form distance computation, reducing complexity from $O(I_uI_vI_rM^2)$ as in 3D-MUSIC to $O(G_YG_ZM+M^3)$, with comparable or better RMSE.
• Hybrid Precoding: Hierarchical codebooks and beam subtraction (Xiao et al., 2016) reduce both time-slots and computation for analog/digital stage design, mapping most baseband computation to a reduced equivalent channel.

6. Distributed and Decentralized Estimation Architectures

Low-complexity channel estimation is further enabled by architectural decentralization (Xu et al., 2022). In massive MIMO with distributed baseband units (DBUs), diagonal MMSE estimators are computed either after local summary statistics aggregation (ATE) or per-cluster with weighted fusion (ETA). These schemes attain near-centralized NMSE with per-round communication as low as $O(MK+M)$ compared to $O(MK\tau)$ in centralized designs, and reduce per-node complexity from cubic to linear or nearly linear in antenna count.

7. Implementation Trade-offs, Robustness, and Practical Guidelines

The key general trade-offs involve:

• Complexity vs. MSE: Increased model-based or learned structure (sparsity, low rank, angular, or statistical prior knowledge) yields rapid complexity reductions without MSE sacrifice for most practical SNR/pilot regimes.
• Pilot training vs. throughput: Iterative schemes (e.g., (0809.2446)) reduce pilot overhead by leveraging larger data blocks, requiring accurate detection to avoid error propagation, with optimal pilot fractions $N_t/T=1/(1+N_d)$.
• Robustness: Basis-free (matrix completion) and generative methods avoid significant performance loss under array response mismatch and changing propagation environments.
• Distributed schemes enable a spectrum of deployment options balancing computation, communication, and latency.

• (0809.2446): M-LAS iterative detection and estimation for large space-time block-coded MIMO
• (Shariati et al., 2013): PEACH polynomial expansion estimators
• (Pu et al., 2023): SWM-based LMMSE with FFT acceleration for extra-large MIMO
• (Hu et al., 2016): Basis-free matrix completion for mmWave/switch-based MIMO
• (Li et al., 2023): UAMPSBL-PCI for RIS-assisted systems
• (Fesl et al., 2024): Diffusion-based DM estimators with SNR embedding
• (Chen et al., 2 Feb 2026): Sampling-free transformer-based diffusion
• (Fan et al., 24 Oct 2025): Posterior sampling with latent diffusion
• (Fesl et al., 2021): FFT-based CNN MMSE estimator
• (Xu et al., 2022): Decentralized low-complexity MMSE for massive MIMO
• (Huang et al., 2023): SADCE for near-field XL-MIMO
• (Hu et al., 2021): 1-bit quantized RIS and ADMM+GAMP
• (Xiao et al., 2016): Hierarchical multi-beam search in mmWave hybrid precoding
• (Yang et al., 2022): Two-stage and 2D-OMP compressed sensing
• (Khan et al., 2024): Peak-power-aided data-assisted channel estimation for MIMO-OFDM
• (Kim et al., 14 Feb 2025): On-grid, subarray-based estimation for XL-MIMO
• (Yuan et al., 2017): DP-message passing with common support clustering in sparse massive MIMO

These works collectively represent the current state-of-the-art in low-complexity MIMO channel estimation, spanning Bayesian, deterministic, learning-based, and distributed paradigms.