Iterative Channel Estimation Using LSE and Sparse Message Passing for MmWave MIMO Systems

I. INTRODUCTION

• MmWave path loss is severe—40 dB worse at 28 GHz than at 2.8 GHz—and relatively few channel paths make the channel approximately sparse.Measurements cited in the introduction report typically 3–4 scattering clusters in dense-urban non-line-of-sight environments; hybrid analog/digital architectures also distinguish mmWave systems from conventional MIMO models.Source: §I · p. 1
• Sparse estimators differ in required channel priors: AMP needs distribution and structure knowledge, while ITD-SE, ACS, and OMP need partial information such as sparsity –.LASSO and EM-BG-AMP may not produce the sparsest solutions, and EM-BG-AMP requires an i.i.d. zero-mean Gaussian training matrix [32],; modified MF message passing is also described as promising.Source: §I · pp. 1–2
• The proposed iterative estimator combines LSE, EM, and Sparse Message Passing, modeling the beamspace channel as Bernoulli-Gaussian to exploit its sparsity.The paper states that combining LSE and SMP leverages both algorithms’ virtues and can improve on existing sparse channel estimation methods.Source: §I · p. 2
• The analysis derives an upper bound and CRLB and establishes the proposed algorithm as an MVUE when partial prior channel knowledge is available.The paper also analyzes convergence using EXIT charts, Gaussian-approximated message densities, density evolution, and message LLRs.Source: §I · p. 2
• Simulations show the proposed algorithm performs far better than classical LSE and existing sparse estimators, and approximately achieves the CRLB with fast convergence.The reported comparisons include LASSO, ITD-SE, and EM-BG-AMP.Source: §I · p. 2

II. SYSTEM MODEL

• The system model uses hybrid analog-digital transmission with transmit and receive antennas, RF chains, and data streams, and assumes a geometric channel with L scatterers, each contributing one path.The model assumes frequency-flat fading and Gaussian receiver noise; large arrays and few reflected paths imply L ≪ min{Nr, Nt}.Source: §II · pp. 2–3
• The beamspace representation maps spatial-domain signals to beams using a discrete lens array, and channel estimation is formulated as recovering the beamspace vector hv from the equivalent training matrix and observations.The beamspace entries represent gains between transmit and receive beams.Source: Figure 1 · p. 3, §II · p. 3

III. SPARSE CHANNEL ESTIMATION

• LSE-SMP has four phases: coarse LSE estimation, SMP detection, LSE and sparsity-ratio updates, and decision and output.The processing repeats the detection and update phases until the MSE requirement is met or the iteration limit is reached.Source: §III · p. 3, Figure 2 · p. 4
• With no prior knowledge of the channel vector hv, the algorithm first obtains a coarse estimate using LSE.It then treats finding hv's nonzero positions as a detection problem and applies SMP using the estimated sparsity ratio.Source: §III · p. 3
• After SMP detects nonzero positions, LSE uses them for a fine estimate of hv, while Expectation-Maximization estimates the sparsity ratio.The decision phase evaluates performance requirements such as MSE and iteration count before outputting the final estimate or repeating detection and updates.Source: §III · p. 3

A. LSE Coarse Estimation

• The coarse LS estimator is an MVUE for a deterministic signal when no prior knowledge of the channel’s sparsity structure or degree is available.It targets the coarse estimate of hv from the observed vector.Source: A. LSE Coarse Estimation · p. 4

B. Sparse Message Passing Algorithm

• After coarse estimation, sparse message passing uses channel sparsity to iteratively find the positions of nonzero entries.The proposed SMP algorithm is described as similar to belief-propagation decoding: each edge’s extrinsic message is computed from messages on the other edges connected to the same node.Source: B. Sparse Message Passing Algorithm · p. 4, 1) Factor Graph Representation of the… · p. 4

1) Factor Graph Representation of the mmWave Channel:

• The factor-graph representation decomposes hv into a diagonal coefficient matrix and a position vector whose Bernoulli entries indicate nonzero positions.The position vector is b, with bij in {0, 1}; the resulting observation model is expressed as y = S̄Uhv b + n̄.Source: 1) Factor Graph Representation of the… · p. 4
• The factor graph has sum nodes for the observation terms and variable nodes for channel coefficients.The graph represents the equations for the channel model and is used by the proposed SMP detection algorithm.Source: 1) Factor Graph Representation of the… · p. 4, Figure 3 · p. 5

2) Message Update at Sum Nodes :

• The sparsity ratio is η = L/(NrNt); with i.i.d. position indicators, the Bernoulli probability is estimated as η̂, initialized at 0.5 and updated by EM.The Bernoulli probabilities are p0(bij = 1) = η̂ and p0(bij = 0) = 1 − η̂.Source: 2) Message Update at Sum Nodes · p. 4
• For large Nr or Nt, the sum of other variable-node contributions is approximated as Gaussian noise, enabling sum-node message updates.The approximation is justified by the law of large numbers; the method computes the equivalent noise’s mean and variance before updating messages.Source: 2) Message Update at Sum Nodes · p. 4
• Sum-node messages use the equivalent Gaussian noise statistics to compute probabilities for each position indicator being 1 or 0.The incoming messages from other variable nodes are used to form the equivalent noise, and the resulting probabilities are passed to the variable node.Source: 2) Message Update at Sum Nodes · p. 5

3) Message Update at Variable Nodes :

• Variable nodes are modeled as a broadcast process, for updating messages.The messages passed on each edge are Bernoulli probabilities; Fig. 4 describes the extrinsic message as calculated from messages on the node’s other edges.Source: 3) Message Update at Variable Nodes · p. 5, Figure 4 · p. 5
• At iteration τ + 1, the probability message for bij = 1 is passed from sum node i to variable node ij, and bij is estimated.The supplied passage introduces both the probability message and the Bernoulli-variable estimate at that iteration.Source: 3) Message Update at Variable Nodes · p. 5
• The extrinsic probability message updates sum-node messages in the next iteration, while the full-information estimate ˆbij(τ + 1) is used in Step 3 to estimate hv.Fig. 4 notes that the sum nodes update Gaussian means and variances, which are used to compute Bernoulli probabilities.Source: 3) Message Update at Variable Nodes · p. 5, Figure 4 · p. 5
• Approximating the sum term as Gaussian enables the proposed SMP to obtain near-optimal performance without heavy computational complexity.The passage attributes the approximation’s accuracy to large transmit-antenna counts in mmWave systems and says this specializes the algorithm for mmWave.Source: 3) Message Update at Variable Nodes · p. 5

1) LSE Fine Estimation:

• The LSE fine-estimation strategy swaps the positions of h_ij and b_ij in (14), leveraging the sparsity of b to estimate the coefficient matrix U_hv.This step follows estimation of the non-zero-entry positions.Source: 1) LSE Fine Estimation · p. 5
• With the SMP-estimated vector b̂, the observation is rewritten as ȳ = S̄Û_bh_v + n̄ to estimate h_v.Here h_v is in C^(N_tN_r×1), and b̂ contains the estimated entries.Source: 1) LSE Fine Estimation · p. 5
• The h_v estimate is obtained by solving a least-squares problem, and its estimated value and variance are carried into the next iteration’s equivalent Gaussian-noise calculation.The estimates replace the preceding iteration’s ĥ_v and v̂_h values.Source: 1) LSE Fine Estimation · p. 6

2) EM update for the sparsity ratio :

• A classical EM algorithm learns the sparsity ratio η under the assumption that channel vector h_v is i.i.d. Bernoulli-Gaussian.The marginal distribution includes a Dirac-delta component; the non-zero entries have mean u_hv and variance v_hv.Source: 2) EM update for the sparsity ratio · p. 6
• The EM update for η uses the estimated parameter vector r̂(τ) = [η̂(τ), û_hv(τ), v̂_hv(τ)], with r̂ excluding η denoted r̂\η.These parameters support the update derived for the Bernoulli-Gaussian model.Source: 2) EM update for the sparsity ratio · p. 6
• The η update is obtained by maximizing the stated objective and setting the derivative of its sum to zero, yielding the solution to (32).The supplied passages do not state the solution’s explicit formula.Source: 2) EM update for the sparsity ratio · p. 6

D. Decision and Output of LSE-SMP

• When the MSE requirement is met or the iteration limit is reached, LSE-SMP outputs its final channel-vector estimate based on SMP and LSE fine estimation.Source: D. Decision and Output of LSE-SMP · p. 6
• SMP and LSE refine each other iteratively, improving estimation until the MSE reaches a minimum or system requirement; the benefit grows as the channel becomes sparser.LSE uses sparse information estimated by SMP, while its estimates improve SMP accuracy in the next iteration.Source: D. Decision and Output of LSE-SMP · p. 6

E. LLRs of LSE-SMP

• LLRs replace non-zero-probability message calculations to prevent simulation overflow and reduce computation by converting probability multiplications into additions.The passage also notes that the LLR form facilitates later analysis and performance prediction.Source: E. LLRs of LSE-SMP · pp. 6–7
• The LLR scheme rewrites message updates at sum and variable nodes; each variable-node update uses information from all sum nodes to calculate the next estimate of bij.Source: E. LLRs of LSE-SMP · p. 6

F. LSE-SMP in the Matrix Form

• To reduce matrix inversion and multiplication complexity, the formulation splits high-dimensional diagonal and block-diagonal matrices into lower-dimensional matrices, relying on transmit- and receive-antenna independence.Source: F. LSE-SMP in the Matrix Form · p. 7
• The section defines matrix quantities and presents Algorithm 1 as the detailed LSE-SMP procedure in LLR matrix form.Source: F. LSE-SMP in the Matrix Form · p. 7

A. Cram´er-Rao Low Bound Of LSE-SMP

• For deterministic non-sparse channels, the conventional LSE is an MVUE with covariance bound n(¯SH¯S)−1,.The passage identifies this as the conventional LSE CRLB.Source: A. Cram´er-Rao Low Bound Of LSE-SMP · p. 7
• The CRLBs differ because LSE-SMP estimates nonzero positions with SMP and EM, while the conventional LSE does not use that sparsity information.The paper denotes the bounds CRLBLSE and CRLBLSE−SMP, respectively.Source: A. Cram´er-Rao Low Bound Of LSE-SMP · p. 7
• The analysis separates the channel into Uhv and b and analyzes each part while treating the other as given.This alternating-minimization-style approach is used because the full LSE-SMP loop is difficult to model analytically; Assumption 1 specifies known nonzero positions b.Source: A. Cram´er-Rao Low Bound Of LSE-SMP · p. 7
• Prior work shows message passing can converge to the LMMSE under Gaussian distributions, and supplemental simulations report accurate nonzero-position estimation, especially with few nonzero entries,.The passage reports the position-estimation simulations under Assumption 1.Source: A. Cram´er-Rao Low Bound Of LSE-SMP · p. 7
• The sparse-case Fisher information matrix is singular, so the analysis checks a constraint to ensure the proposed estimator has finite variance.The constraint is shown to hold for Ub, after which the CRLB is computed for the singular Fisher information matrix [27],.Source: A. Cram´er-Rao Low Bound Of LSE-SMP · p. 8
• Under Assumption 1, LSE-SMP is an unbiased MVUE that achieves its CRLB.Assumption 1 treats the channel as deterministic and assumes prior knowledge of the nonzero positions.Source: A. Cram´er-Rao Low Bound Of LSE-SMP · pp. 7–8
• For deterministic sparse channels, the LSE-SMP MSE is no greater than the LSE MSE.The paper states this as MSELSE−SMP ≤ MSELSE.Source: A. Cram´er-Rao Low Bound Of LSE-SMP · p. 8

B. Analysis of Iterative Evolution of LSE-SMP

• The analysis focuses on SMP because its convergence behavior determines that of LSE-SMP, using alternating minimization to rule out LSE and EM effects.Direct analysis with existing density-evolution methods is described as difficult.Source: B. Analysis of Iterative Evolution of… · p. 8
• Under Assumption 2, Gaussian message densities and the symmetry condition allow variable-node updates to retain only message means and yield a closed-form update rule.Assumption 2 includes prior knowledge of nonzero channel entries and their mean and variance, and zero-mean Gaussian training signals; the symmetry condition is stated as σ²v = 2uv.Source: B. Analysis of Iterative Evolution of… · pp. 8–9
• The EXIT-chart analysis uses variable-node variance as extrinsic information to predict estimator performance, unlike prior charts using mutual information or MSE.The exact message mean and variance are difficult to solve; the cited analysis states that variance messages can be transformed into mutual information and MSE.Source: B. Analysis of Iterative Evolution of… · p. 9
• If LSE-SMP has converged, larger variable-node convergence variance σ²v indicates better estimation performance, consistent with BER decreasing as σ²v increases.The BER relationship is based on EXIT-chart analysis and the cited equation (26).Source: B. Analysis of Iterative Evolution of… · p. 9
• The proposed LSE-SMP estimator has the best average NMSE among the tested algorithms across the SNR comparison.The figure uses Nt = 32, Nr = 64, η = 0.007, β = 10, σ²h = 10, and training sequence length T = 64.Source: Figure 5 · p. 9

• Simulations evaluate LSE-SMP on 32 × 64 mmWave MIMO channels with Gaussian nonzero entries across SNRs from −10 to 40 dB, using NMSE.Source: §V · p. 9
• Results are averaged over 500 different channel realizations.Source: §V · p. 9

A. Performance Comparison

• LSE-SMP has the best NMSE among the tested algorithms, reducing NMSE by 3.5 dB versus EM-BG-AMP and 1.5 dB versus BP-MF SBL.The proposed estimator’s CRLB is the lowest and is consistent with classical LSE when non-zero positions are perfectly known.Source: A. Performance Comparison · p. 9
• LSE-SMP reaches convergence after five turbo iterations under the reported setting.The comparison uses Nt = 32, Nr = 64, η = 0.031, β = 10, σ²_h = 10, and T = 64.Source: A. Performance Comparison · p. 10, Figure 6 · p. 10
• Across sparsity ratios η ∈ {0.007, 0.125, 0.50, 0.80}, the LSE-SMP CRLB is the lower bound for LSE-SMP, while the LSE CRLB is the upper bound.The comparison uses six iterations with Nt = 32, Nr = 64, β = 10, and T = 64.Source: A. Performance Comparison · p. 10, Figure 7 · p. 10
• The gap between LSE-SMP and its CRLB is about 1.8 dB, partly due to non-zero-position detection errors and reliance on a coarse initial channel estimate.The passage also attributes the gap partly to the coarse initialization used by all detection strategies.Source: A. Performance Comparison · pp. 9–10

B. Effect of Iterations

• After five turbo iterations, LSE-SMP shows no significant NMSE improvement and approaches its analyzed CRLB.NMSE decreases with iterations, with smaller gains between adjacent iterations as estimates of sparsity ratio η and nonzero-position vector b become more accurate.Source: B. Effect of Iterations · p. 10

C. Effect of Sparsity Ratios

• As the channel becomes sparser, LSE-SMP NMSE decreases toward its CRLB, while LSE NMSE remains unchanged across sparsity ratios.The simulations vary sparsity ratio η from 0.007 to 0.80; the authors interpret the results as showing LSE-SMP exploits channel sparsity.Source: C. Effect of Sparsity Ratios · p. 10

D. Effect of Training Sequence Length

• Longer training sequences improve BER and reduce the iterations needed, while increasing computational time; T = 16 is the shortest sequence shown to converge.EXIT-chart analysis can guide training-length selection to balance performance and runtime and predict the length needed for a given BER; multiple intersections of input and output variance traces indicate failure to converge.Source: D. Effect of Training Sequence Length · p. 10, Figure 8 · p. 11

E. Effect of the Coefficient of Variation of the Channel

• Greater channel coefficient variation raises the convergent variance point and lowers system performance, but reduces the time needed to converge.The higher convergent variance point requires more iterations; the narrower bottleneck region between input and output variance traces remains unchanged.Source: E. Effect of the Coefficient of… · p. 11, Figure 9 · p. 11

F. Complexity of the Algorithms

• The best NMSE-runtime trade-off shifts from ITD-SE above −66dB to LASSO, EM-BG-AMP, and BP-MF SBL across successively lower NMSE ranges before LSE-SMP below −80dB.The reported ranges are LASSO from −66dB to −73dB, EM-BG-AMP from −73dB to −76dB, and BP-MF SBL from −76dB to −80dB; LSE-SMP may require more time.Source: F. Complexity of the Algorithms · p. 11, Figure 10 · p. 11

VI. CONCLUSION

• LSE-SMP reduces NMSE by about 2dB relative to the best existing algorithms and has the best performance-complexity trade-off below −80dB NMSE.The runtime comparison evaluates NMSE-runtime frontiers for several algorithms; the conclusion reports the approximate 2dB reduction.Source: Figure 10 · p. 11, F. Complexity of the Algorithms · p. 11, §VI · p. 11
• The proposed LSE-SMP combines SMP and LSE, is an MVUE under partial prior channel knowledge, and uses EXIT-chart analysis for convergence and design-parameter selection.The paper identifies accounting for correlation among adjacent beam-domain channel entries as future work.Source: §VI · pp. 11–12

APPENDIX

• The update is obtained by substituting (22) into (60) after simplifying (40).Source: APPENDIX · p. 12
• Equation (62) is defined to make the subsequent analysis more convenient.Source: APPENDIX · p. 12
• Under transmit- and receive-antenna independence and symmetry, the derivation suppresses link subscripts and denotes the message quantity by lv(τ).Source: APPENDIX · p. 12
• Substituting (20), (21), and (39) yields an expression that is reorganized using snr and common terms, with β, a1, and a2 defined for simplification.Source: APPENDIX · p. 12
• With lv Gaussian of mean uv and variance 2uv, evaluating the expectation gives an update for uv(τ+1) and leads to Theorem 2.Source: APPENDIX · p. 12