Semi-Blind Channel Estimation Methods

• Semi-blind channel estimation is a technique that jointly exploits known pilot symbols and unknown data for channel inference, reducing training overhead.
• It employs advanced methodologies such as tensor decompositions, expectation-maximization, and subspace-assisted MMSE to accurately estimate channels.
• This approach is pivotal in systems like MIMO-OFDM, massive MIMO, and RIS-assisted networks where maintaining identifiability while controlling ambiguity is essential.

Semi-blind channel estimation denotes a class of channel inference methods that use both explicit side information and unknown payload data, rather than relying exclusively on pilots or exclusively on blind statistical structure. In the cited literature, the side information includes pilot symbols, orthogonal pilot decorrelation, constellation knowledge, DMA training matrices, IRS or BD-RIS configuration matrices, and state-space dynamics, while the unknowns can include channel coefficients, hardware-induced factors, and transmitted symbols estimated jointly with the channel (Karami et al., 2018, Abdallah et al., 2013, Araújo et al., 2022, Magalhães et al., 14 Jun 2025). The common objective is to reduce pilot overhead, improve spectral efficiency, or track channels whose dynamics make pilot-only estimation inaccurate, while preserving identifiability and controlling ambiguity.

1. Conceptual scope and terminology

Semi-blind channel estimation occupies the space between training-based and blind estimation. In the MIMO-OFDM formulation of Shahbazpanahi and coauthors, training-based methods use known pilot symbols, blind methods use no explicit pilots, and semi-blind methods combine a small amount of training with blind processing on data symbols (Karami et al., 2018). In the massive MIMO literature, the same term is used for pilot-based channel estimation enhanced by a subspace estimated from all received symbols, including payload data (Weißer et al., 2023, Weißer et al., 24 Apr 2025). In DMA, IRS, and BD-RIS formulations, semi-blind also appears as a data-aided or data-driven joint channel and symbol estimation problem in which no dedicated pilot-only phase is required, but structured matrices such as $\mathbf{F}$, $\mathbf{S}$, $\mathbf{P}$, or $\mathbf{W}$ remain known (Magalhães et al., 14 Jun 2025, Araújo et al., 2022, Araujo et al., 2024).

Two usages recur. The first is the classical pilot-plus-data setting, where a short pilot phase initializes or anchors the estimate and the data phase refines it. The second is a no-dedicated-pilot formulation in which identifiability is provided by multilinear structure, known coding matrices, or symbol alphabets rather than by explicit pilot symbols. This distinction is important because “semi-blind” does not mean “pilot-free” in every system. In AF two-way relay networks, for example, semi-blind estimation uses pilots, known self-interference symbols, and the modulation structure of the partner’s data (Abdallah et al., 2013, Abdallah et al., 2012). In contrast, the DMA and BD-RIS formulations explicitly state that no dedicated pilot-only stage is required (Magalhães et al., 14 Jun 2025, Araujo et al., 2024).

A recurring misconception is that semi-blind estimation is merely decision-directed tracking after a conventional pilot estimate. The literature is broader. It includes tensor decomposition, expectation-maximization, expectation propagation, subspace-constrained MMSE, projected MMSE, adaptive Bussgang methods, and direct equalizer estimation through mutually referenced equalizers (Naraghi-Pour et al., 2020, Weißer et al., 24 Apr 2025, Son et al., 2023). Another misconception is that semi-blind methods are blind methods with a few pilots added. In several works, the pilots are only one part of a larger structural model, and the dominant source of identifiability can be multilinear diversity, latent-variable regularization, or graph structure (Magalhães et al., 14 Jun 2025, Araujo et al., 2024, Forsch et al., 20 May 2026).

2. Signal models and estimation objectives

The underlying estimation problem is usually bilinear or multilinear. In uplink massive MIMO, the received block is written as

$$\mathbf{Y} = [\mathbf{Y}_p,\mathbf{Y}_d] = \mathbf{H}[\mathbf{X}_p,\mathbf{X}_d] + [\mathbf{N}_p,\mathbf{N}_d],$$

with $\mathbf{X}_p$ known and $\mathbf{X}_d$ unknown (Weißer et al., 2023). In cell-free massive MIMO identifiability analysis, the noiseless semi-blind JCD model is reduced to

$\mathbf{Y} = \mathbf{H}_I \mathbf{X},$

where the non-negligible links define a sparse support pattern induced by geometry (Forsch et al., 20 May 2026). In time-varying multi-cell massive MIMO, the observation is

$y_t = H_t s_t + w_t,\qquad h_t = A h_{t-1} + v_t,$

so semi-blind estimation becomes a joint smoothing and detection problem under an AR(1) channel model with spatial correlation (Naraghi-Pour et al., 2020).

Several system classes add hardware structure to the channel model. In AF two-way relay networks, estimation is carried out for composite parameters

$a \triangleq h^2,\qquad b \triangleq hg,$

which are sufficient for self-interference cancellation and coherent detection at the terminal (Abdallah et al., 2013). In DMA-assisted MISO-OFDM, the effective channel on subcarrier $\mathbf{S}$0 is

$\mathbf{S}$1

where $\mathbf{S}$2 is the wireless channel and $\mathbf{S}$3 is the inner waveguide propagation vector (Magalhães et al., 14 Jun 2025). The corresponding received tensor follows a PARAFAC model,

$\mathbf{S}$4

with $\mathbf{S}$5 (Magalhães et al., 14 Jun 2025). In BD-RIS and IRS-assisted systems, the received slices are modeled through PARATUCK or generalized PARATUCK structures that separate BS–RIS, UT–RIS, and symbol factors (Araujo et al., 2024, Araújo et al., 2022).

The estimation objectives differ accordingly. Some formulations estimate the physical channel itself, some estimate composite channels, and some estimate an effective inverse or equalizer. SB-MRE, for instance, estimates an effective channel inverse $\mathbf{S}$6 rather than $\mathbf{S}$7 directly, through a semi-blind equalizer design (Son et al., 2023). In DMA, the target is the decoupled recovery of $\mathbf{S}$8, $\mathbf{S}$9, and $\mathbf{P}$0 (Magalhães et al., 14 Jun 2025). In IRS-assisted MU-MIMO, the target is joint recovery of $\mathbf{P}$1, $\mathbf{P}$2, and $\mathbf{P}$3 through Khatri–Rao and Kronecker factorizations (Araújo et al., 2022).

3. Principal estimation mechanisms

A major family uses multilinear algebra. In DMA-assisted downlink MISO-OFDM, Alternating Least Squares is applied to the PARAFAC unfoldings

$\mathbf{P}$4

followed by a best rank-1 approximation of $\mathbf{P}$5 to separate $\mathbf{P}$6 and $\mathbf{P}$7 (Magalhães et al., 14 Jun 2025). In BD-RIS, the PARE receiver uses Trilinear Alternating Least Squares on three unfoldings to update $\mathbf{P}$8, $\mathbf{P}$9, and $\mathbf{W}$0 from a PARATUCK model (Araujo et al., 2024). In IRS-assisted MU-MIMO, KAKF performs a Khatri–Rao factorization stage and then a Kronecker factorization stage, each solved by truncated SVD, yielding a non-iterative semi-blind receiver (Araújo et al., 2022).

A second family is decision-directed and adaptive. In time-domain semi-blind MIMO-OFDM, an LMS update is extended by replacing unknown data symbols with a nonlinear function $\mathbf{W}$1, leading to DD-LMS and ABA-LMS variants. The adaptive Bussgang Algorithm uses

$\mathbf{W}$2

with $\mathbf{W}$3 and $\mathbf{W}$4 updated adaptively, and blind-mode LMS is run multiple times with a decreased step size to form some kind of annealing (Karami et al., 2018). In massive MIMO LEO satellite communications, DD-SB first estimates the channel from pilots, then detects data, and finally re-estimates the channel using both pilot and detected data symbols; MDD-SB re-estimates from the most recent detected data block only, specifically to mitigate the channel-aging effect (Darya et al., 2024).

A third family uses latent-variable probabilistic inference. In AF two-way relay networks, an EM algorithm treats the partner’s data symbols as hidden variables and alternates between posterior symbol probabilities and closed-form updates of the composite channel parameters (Abdallah et al., 2013). In multiband OFDM UWB, the EM algorithm is embedded in a wavelet-domain Bayesian framework with a Bernoulli–Gaussian prior on wavelet coefficients; the M-step becomes a hard-thresholding plus shrinkage rule that discards unsignificant wavelet coefficients from the estimation process (0708.1414). In multi-cell massive MIMO on time-varying channels, expectation propagation approximates the joint a posteriori distribution of the unknown channel matrix and transmitted data symbols by a distribution from an exponential family, and a modified Kalman filtering algorithm referred to as KF-M emerges from the derivation (Naraghi-Pour et al., 2020).

A fourth family is subspace-assisted MMSE with learned priors. In massive MIMO, a channel subspace is estimated from all received symbols and then used either to estimate within the subspace or as a preprocessing projection before LMMSE. The projected LMMSE estimator reduces the effective noise variance to $\mathbf{W}$5 under uncorrelated Rayleigh fading and is shown to have superior estimation performance in terms of mean square error (Weißer et al., 24 Apr 2025). The same paper then parameterizes the estimator with conditional Gaussian latent models based on a Gaussian mixture model and a variational autoencoder learned from channel data (Weißer et al., 24 Apr 2025). A closely related earlier formulation uses GMM-based pilot-only estimation and then introduces subspace GMM and projected GMM as semi-blind extensions (Weißer et al., 2023).

4. Identifiability, ambiguities, and bounds

Identifiability is central because semi-blind estimation infers both channel and data from partially unknown inputs. In PARAFAC-based DMA estimation, the relevant uniqueness statement is Kruskal’s condition,

$\mathbf{W}$6

under which $\mathbf{W}$7 and $\mathbf{W}$8 are essentially unique up to permutation and scaling (Magalhães et al., 14 Jun 2025). In BD-RIS, the LS subproblems imply the joint uniqueness conditions

$\mathbf{W}$9

which summarize the degrees of freedom needed to resolve $$\mathbf{Y} = [\mathbf{Y}_p,\mathbf{Y}_d] = \mathbf{H}[\mathbf{X}_p,\mathbf{X}_d] + [\mathbf{N}_p,\mathbf{N}_d],$$0, $$\mathbf{Y} = [\mathbf{Y}_p,\mathbf{Y}_d] = \mathbf{H}[\mathbf{X}_p,\mathbf{X}_d] + [\mathbf{N}_p,\mathbf{N}_d],$$1, and $$\mathbf{Y} = [\mathbf{Y}_p,\mathbf{Y}_d] = \mathbf{H}[\mathbf{X}_p,\mathbf{X}_d] + [\mathbf{N}_p,\mathbf{N}_d],$$2 (Araujo et al., 2024).

In cell-free massive MIMO, identifiability is analyzed from a large-scale system design perspective. Access points and user equipments are modeled by Poisson point processes, the resulting topology is represented by a bipartite random geometric graph, and semi-blind recovery succeeds when a Karp–Sipser-like leaf-removal procedure empties the residual graph with high probability (Forsch et al., 20 May 2026). The analysis yields an identifiability region as a function of AP density, UE density, and connectivity radius, which provides system-level conditions under which semi-blind JCD is possible (Forsch et al., 20 May 2026). This suggests that identifiability in large networks is not only an algebraic property of matrix ranks but also a geometric property of sparse connectivity.

Ambiguities remain a persistent issue. Blind MRE for MIMO systems exhibits scalar ambiguity, phase ambiguity, permutation ambiguity, and possibly delay ambiguity; SB-MRE resolves these by adding a pilot-based least-squares term to the blind MRE criterion (Son et al., 2023). Tensor methods also have scaling ambiguities that are usually fixed by a known row, power normalization, or constellation normalization (Araújo et al., 2022, Magalhães et al., 14 Jun 2025). A common misconception is that the presence of some pilots automatically eliminates all ambiguities. The literature shows instead that ambiguity removal depends on the particular factorization, the chosen normalization, and whether the known structure is sufficient to anchor every latent factor.

Lower bounds reinforce these points. In AF two-way relay networks, exact CRBs and modified CRBs are derived for semi-blind estimation with PSK and square QAM. The exact semi-blind CRB is smaller than the training-based CRB because the data symbols provide additional Fisher information, while the modified CRB is accurate at high SNR for low modulation orders but does not capture modulation-order effects as completely as the exact CRB (Abdallah et al., 2012). The bound analysis therefore places semi-blind gains on a statistical footing rather than treating them as a purely algorithmic artifact.

5. Representative system classes and reported performance

In DMA-assisted downlink MISO-OFDM, the reported setup uses $$\mathbf{Y} = [\mathbf{Y}_p,\mathbf{Y}_d] = \mathbf{H}[\mathbf{X}_p,\mathbf{X}_d] + [\mathbf{N}_p,\mathbf{N}_d],$$3, $$\mathbf{Y} = [\mathbf{Y}_p,\mathbf{Y}_d] = \mathbf{H}[\mathbf{X}_p,\mathbf{X}_d] + [\mathbf{N}_p,\mathbf{N}_d],$$4, $$\mathbf{Y} = [\mathbf{Y}_p,\mathbf{Y}_d] = \mathbf{H}[\mathbf{X}_p,\mathbf{X}_d] + [\mathbf{N}_p,\mathbf{N}_d],$$5, $$\mathbf{Y} = [\mathbf{Y}_p,\mathbf{Y}_d] = \mathbf{H}[\mathbf{X}_p,\mathbf{X}_d] + [\mathbf{N}_p,\mathbf{N}_d],$$6, 64-QAM, and $$\mathbf{Y} = [\mathbf{Y}_p,\mathbf{Y}_d] = \mathbf{H}[\mathbf{X}_p,\mathbf{X}_d] + [\mathbf{N}_p,\mathbf{N}_d],$$7 Monte Carlo runs. The notable findings are that $$\mathbf{Y} = [\mathbf{Y}_p,\mathbf{Y}_d] = \mathbf{H}[\mathbf{X}_p,\mathbf{X}_d] + [\mathbf{N}_p,\mathbf{N}_d],$$8 is estimated more accurately than $$\mathbf{Y} = [\mathbf{Y}_p,\mathbf{Y}_d] = \mathbf{H}[\mathbf{X}_p,\mathbf{X}_d] + [\mathbf{N}_p,\mathbf{N}_d],$$9, the realistic setting shows a loss of about $\mathbf{X}_p$0 dB in NMSE compared to the benchmark with ideal orthogonal $\mathbf{X}_p$1, and the number of ALS iterations and runtime are comparable between the proposed scheme and the benchmark in the medium-to-high SNR regime (Magalhães et al., 14 Jun 2025). The significance lies in the decoupled recovery of the wireless channel and the inner DMA channel, which enables separate calibration and beamformer compensation.

In time-domain semi-blind MIMO-OFDM, the uncoded $\mathbf{X}_p$2 and $\mathbf{X}_p$3 simulations show that ABA-LMS gives very close MSE to full training after about 3 iterations and that usually 3–5 iterations is enough to get the best result (Karami et al., 2018). In coded experiments, LMS remains approximately constant or slightly decreasing over the data symbols while LS cannot track symbol-by-symbol channel variations, and the LMS-based estimator avoids the high error floor observed with LS (Karami et al., 2018). These results position semi-blind adaptive tracking as particularly relevant for fast fading and partial-band operation.

In data-aided massive MIMO with GMM priors, the projected GMM estimator is best across most SNRs, with about $\mathbf{X}_p$4 dB NMSE gain in mid-SNR over the pilot-only GMM and all other baselines, while the subspace GMM closely follows it (Weißer et al., 2023). In the later generative-prior formulation, projected and subspace variants based on GMM and VAE outperform state-of-the-art semi-blind estimators with respect to the MSE on both real-world measurement data and spatial channel models (Weißer et al., 24 Apr 2025). A plausible implication is that subspace information and non-Gaussian environment-specific priors are complementary rather than competing design choices.

In RIS and BD-RIS systems, performance outcomes are more differentiated. For IRS-assisted MU-MIMO, KAKF yields significantly lower NMSE than BALS for $\mathbf{X}_p$5; at SNR $\mathbf{X}_p$6 dB, the gain is described as an order-of-magnitude NMSE improvement, and runtime is much lower than BALS while remaining essentially flat versus SNR (Araújo et al., 2022). In BD-RIS, PARE outperforms LS in NMSE of the cascaded channel, shows a performance gap versus BTKF, and typically converges in fewer than 20 iterations in the moderate SNR range for the considered parameters; its SER approaches that of a ZF receiver with perfect cascaded CSI at moderate-to-high SNR (Araujo et al., 2024). The contrast between IRS and BD-RIS results reflects different trade-offs between training-based tensor baselines and pilot-free multilinear identifiability.

In massive MIMO LEO satellite communications, DD-SB outperforms P-LS in NMSE for SNR $\mathbf{X}_p$7 dB, while MDD-SB is specially designed to mitigate the channel-aging effect and consequently outperforms an optimal pilot-based estimator in terms of normalized mean square error (Darya et al., 2024). The SER of MDD-SB is reported to be comparable to that of a Genie-aided detector, whereas pilot-based detection degrades strongly across blocks because the channel estimate becomes outdated (Darya et al., 2024). This makes channel aging a concrete example where the semi-blind advantage is driven less by spectral efficiency alone than by estimation freshness.

6. Practical issues, recurring misconceptions, and research directions

The practical limitations are consistent across otherwise dissimilar models. Alternating least squares and trilinear least squares can converge to local minima or stationary points, which makes initialization sensitivity a real issue in DMA and BD-RIS estimation (Magalhães et al., 14 Jun 2025, Araujo et al., 2024). Decision-directed methods can suffer from error propagation at low SNR, so they typically use smaller step sizes in blind mode, repeated passes, annealing schedules, or more local updates that emphasize recent data blocks (Karami et al., 2018, Darya et al., 2024). Generative-prior methods require representative training datasets and can lose performance under model mismatch, while exact knowledge of IRS, BD-RIS, or DMA configuration matrices is itself an assumption that some papers explicitly propose to relax (Weißer et al., 24 Apr 2025, Magalhães et al., 14 Jun 2025).

Several misconceptions recur. Semi-blind estimation does not uniformly dominate pilot-only estimation at every SNR; the LEO and MIMO-OFDM results show that incorrect symbol decisions can make pilot-only schemes preferable in sufficiently noisy regimes (Darya et al., 2024, Karami et al., 2018). Semi-blind estimation also does not remove the need for structure: constellation knowledge, orthogonal pilots, spatial covariance, array geometry, or designed switching matrices are repeatedly used as prior information (Weißer et al., 2023, Magalhães et al., 14 Jun 2025). Finally, “more data” is not automatically better. In the GMM literature, very small $\mathbf{X}_p$8 yields poor subspace estimates, while in time-varying settings outdated data can worsen tracking if the update window is too long (Weißer et al., 2023, Darya et al., 2024).

The research directions stated in the cited works are likewise structured. DMA estimation points to scenarios where $\mathbf{X}_p$9 is unknown and must be estimated or refined (Magalhães et al., 14 Jun 2025). BD-RIS work points to multi-user and wideband extensions, as well as alternative scattering-network designs (Araujo et al., 2024). Adaptive MIMO-OFDM work suggests combining DD and ABA more explicitly (Karami et al., 2018). The cell-free massive MIMO identifiability analysis leaves exact bipartite random geometric graph analysis, finite-size effects, and the gap between identifiability and practical algorithmic performance as open problems (Forsch et al., 20 May 2026). The generative-prior line suggests that environment learning, subspace estimation, and MMSE channel recovery can be unified into a single semi-blind framework with offline learning and online inference (Weißer et al., 24 Apr 2025).

Semi-blind channel estimation is therefore best understood not as a single algorithmic recipe but as a family of inference problems in which partially known signaling structure, latent data symbols, and system-specific channel models are exploited jointly. Across tensor models, EM and EP formulations, adaptive decision-directed tracking, and subspace-assisted MMSE with learned priors, the shared principle is the same: data symbols are not discarded as mere payload, but reused as statistical evidence for channel inference under carefully specified structural assumptions (Abdallah et al., 2013, Naraghi-Pour et al., 2020, Weißer et al., 24 Apr 2025).