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Massive MIMO with Spatially Correlated Rician Fading Channels
Özgecan Özdogan, Emil Björnson, Erik G. Larsson
TL;DR
Prior work largely used spatially uncorrelated channels, Rayleigh inter-cell fading, or approximate spectral-efficiency expressions. This paper analyzes multi-cell Massive MIMO with spatially correlated Rician fading, derives rigorous closed-form uplink and downlink expressions for several estimators, and finds MMSE generally performs best.
Problem
Prior studies did not fully capture spatial correlation, inter-cell LoS paths, and exact closed-form spectral-efficiency behavior in practical multi-cell Rician channels.
Method
The paper derives channel-estimate statistics and rigorous closed-form uplink and downlink spectral-efficiency expressions for MMSE, EW-MMSE, and LS estimation with MR processing.
Results
MMSE estimation achieves higher spectral efficiency than EW-MMSE and LS for spatially correlated Rayleigh and Rician fading, while LS gives the lowest spectral efficiency.
Takeaways & Limitations
The derived expressions provide insights into Rician-channel operation and interference and can support user scheduling, pilot allocation, power control, and other resource allocation.
Abstract
from arXiv · showhide
This paper considers multi-cell Massive MIMO (multiple-input multiple-output) systems where the channels are spatially correlated Rician fading. The channel model is composed of a deterministic line-of-sight (LoS) path and a stochastic non-line-of-sight (NLoS) component describing a practical spatially correlated multipath environment. We derive the statistical properties of the minimum mean squared error (MMSE), element-wise MMSE (EW-MMSE), and least-square (LS) channel estimates for this model. Using these estimates for maximum ratio (MR) combining and precoding, rigorous closed-form uplink (UL) and downlink (DL) spectral efficiency (SE) expressions are derived and analyzed. The asymptotic SE behavior when using the different channel estimators are also analyzed. Numerical results show that the SE is higher when using the MMSE estimator than the other estimators, and the performance gap increases with the number of antennas.
I. INTRODUCTION
The paper addresses gaps in multi-cell Massive MIMO analysis by modeling spatially correlated Rician fading and deriving rigorous spectral-efficiency results for multiple channel estimators.
- Motivation: Practical channels combine a deterministic LoS path with small-scale multipath fading, motivating the spatially correlated Rician model.
- Research gap: Prior multi-cell Rician studies largely assumed spatially uncorrelated channels, Rayleigh inter-cell fading, and approximate closed-form SE expressions.
- Main contributions: The paper considers multi-cell spatially correlated Rician channels, with channel pairs statistically determined as Rician or Rayleigh.
- Main contributions: It derives statistics for MMSE, EW-MMSE, and LS channel estimates and uses them with MR combining and precoding.
- Main contributions: The work derives rigorous closed-form UL and DL SE expressions and analyzes their asymptotic behavior for the different estimators.
- Main contributions: Numerical comparisons cover MMSE, EW-MMSE, and LS estimation under correlated and uncorrelated Rician and Rayleigh fading.
B. Element-wise MMSE Channel Estimator
EW-MMSE estimation uses only covariance diagonals, reducing complexity by ignoring antenna-element correlations, while its resulting statistics support closed-form MR-based SE analysis.
- Estimator definition: EW-MMSE estimation requires only covariance-matrix diagonals and ignores correlations between channel elements.
- Estimator definition: Because EW-MMSE avoids matrix inversions, its computational complexity is greatly reduced relative to full MMSE estimation.
- Estimator statistics: The EW-MMSE estimate and estimation error are characterized through their distributions and correlation statistics.
- Estimator statistics: Unlike MMSE estimation, EW-MMSE does not generally yield uncorrelated estimates and errors except when all covariance matrices are diagonal.
- Spectral-efficiency analysis: The paper derives closed-form effective SINR and achievable SE expressions for MR combining with the channel estimators.
A. Uplink Spectral Efficiency with MMSE estimator
The MMSE-based uplink analysis derives a closed-form spectral-efficiency expression whose terms expose estimation quality, LoS structure, covariance relationships, and pilot contamination. Comparisons with EW-MMSE and LS show why exploiting mean and covariance information is beneficial under Rician fading.
- MMSE estimator: The MMSE estimator yields a rigorous closed-form uplink SINR and SE expression for MR combining in spatially correlated Rician fading.The expression separates LoS-related, non-coherent, and coherent interference contributions.
- MMSE estimator: The desired-signal terms depend on channel-estimation quality and the desired UE’s LoS component, while pilot contamination reduces estimation quality.The relevant signal and interference terms are determined by the channel estimate covariance, covariance matrices, and LoS inner products.
- Interference behavior: Non-coherent interference generally does not increase with antenna number, whereas coherent interference from pilot-contaminating UEs grows linearly with the number of antennas.LoS-related interference depends on the norms and inner products of LoS components, while NLoS interference depends on covariance relationships.
- EW-MMSE estimator: EW-MMSE provides a closed-form uplink SE expression with lower computational complexity because it uses only covariance diagonals and avoids matrix inversions.Its estimator ignores correlations between channel elements.
- LS estimator: Under Rician fading, LS can incur a large SE loss because it does not use mean values as prior information, increasing interference relative to MMSE.The loss depends on the dominance of LoS paths; in Rayleigh fading, the MMSE and LS/EW-MMSE gap can be rather small.
- Mean Only estimator: The MO estimator can reduce computation by avoiding pilot-based estimation across coherence blocks, but it is useful only for UEs with LoS paths.Without an LoS component, the corresponding MO-estimator SINR becomes zero.
V. DOWNLINK SPECTRAL EFFICIENCY WITH MR PRECODING
The downlink analysis models MR precoding and derives closed-form SE expressions for MMSE and EW-MMSE channel estimates. The resulting SINR interpretation parallels the uplink through UL-DL duality, with interference divided into intra-cell and inter-cell components.
- System model: MR precoding transmits each user’s DL data through a spatially directable vector selected from channel-state information.The received signal contains desired, intra-cell interference, inter-cell interference, and receiver-noise terms.
- SE formulation: The downlink ergodic capacity is lower bounded by an SE expression based on an effective SINR, whose randomness includes all sources in the system.The effective SINR is computed for MR precoding with each channel estimator.
- MMSE estimator: The MMSE estimator gives a closed-form downlink SE expression whose SINR resembles the uplink MMSE counterpart.The transmit power, noise variance, and interference indices differ as expected from UL-DL duality.
- MMSE estimator: The downlink MMSE SINR separates non-coherent and coherent interference terms, enabling an interpretation qualitatively analogous to the uplink.The interference structure reflects the corresponding uplink terms after the UL-DL transformation.
- EW-MMSE estimator: EW-MMSE also produces a closed-form downlink SE expression for MR precoding.Its derivation follows the effective-SINR formulation used for the MMSE case.
C. Downlink Spectral Efficiency with LS Estimator
The asymptotic analysis studies MR-based SE under technical assumptions on covariance matrices and LoS components. It shows that unbounded SE growth occurs only under asymptotic spatial orthogonality, while practical angular separation may be unlikely.
- Asymptotic analysis: The asymptotic analysis examines Rician-fading SE behavior with MR processing under three technical assumptions.The assumptions constrain covariance scaling and the relationships among LoS components.
- Spatial orthogonality: Asymptotic spatial orthogonality is defined through covariance-matrix relationships as the numbers of BS antennas grow without bound.The definition concerns covariance matrices associated with different users and cells.
- Practical boundary: The orthogonality condition corresponds to non-overlapping multipath angular supports for a ULA, but such angular separation is unlikely in practical sub-6 GHz cellular networks.The condition implies strongly rank-deficient covariance matrices.
- Non-orthogonal covariance matrices: When covariance matrices are not asymptotically spatially orthogonal, the asymptotic SINR follows a bounded simplified expression under the stated assumptions.The theorems characterize the non-orthogonal case separately from the unbounded-growth case.
- MMSE estimator: Under the assumptions, the MMSE downlink SINR grows without bound when the relevant covariance matrices are asymptotically spatially orthogonal.The condition excludes the desired UE from the set of pilot-contaminating UEs.
- Asymptotic conclusion: Only asymptotically spatially orthogonal matrices yield SE growth without limit at rate log2(M), consistent with correlated-Rayleigh results.The asymptotic expressions depend on covariance matrices and mean values.
B. Asymptotic Analysis of Spectral Efficiency with EW-MMSE Estimator
The EW-MMSE asymptotic analysis identifies conditions under which spectral efficiency becomes unbounded and shows that these conditions are more restrictive than for MMSE. For Rician channels, the relevant asymptotic orthogonality concerns covariance-matrix diagonals, while LoS components are generally orthogonal except at identical AoAs.
- EW-MMSE asymptotic behavior: EW-MMSE spectral efficiency grows without bound when the relevant diagonal covariance terms are asymptotically spatially orthogonal.The theorem statements characterize this behavior as the number of base-station antennas grows under Assumptions 1–3.
- Comparison with MMSE: Unlike MMSE, EW-MMSE requires the diagonals of covariance matrices to be asymptotically spatially orthogonal for unbounded spectral efficiency.The paper explicitly identifies this diagonal-orthogonality requirement as more restrictive than the corresponding MMSE condition.
- Uplink and downlink: The EW-MMSE uplink and downlink asymptotic results are established under the stated assumptions, with corresponding interference terms differing by swapped user indices.The downlink theorem follows the same general asymptotic pattern as the uplink, while the interference expressions exchange the involved indices.
- Comparison with LS: The LS estimator does not achieve unbounded spectral efficiency in contrast to the MMSE and EW-MMSE estimators.This distinction is stated for the asymptotic analysis under the paper’s assumptions.
- LoS assumptions: For a uniform linear array, the LoS-component norm remains finite for any antenna count, supporting the relevant asymptotic assumption.The array model uses antenna spacing in fractions of the wavelength and the angle of arrival to characterize the LoS component.
- LoS assumptions: The inner product of distinct LoS components supports asymptotic spatial orthogonality except when their angles of arrival are exactly equal.Thus, the exceptional case is coincident AoAs rather than ordinary distinct user directions.
VII. NUMERICAL RESULTS
The simulations validate the closed-form SE expressions and compare MMSE, EW-MMSE, and LS estimation under correlated Rician and Rayleigh fading. MMSE generally provides the highest SE, while pilot reuse creates a trade-off between pilot contamination and pre-log loss.
- Simulation setup: A 16-cell wrap-around network with K = 10 UEs per cell evaluates the analytical SE expressions under realistic UE locations and shadow fading.Each BS uses a half-wavelength-spaced ULA; the simulations include spatially correlated channels modeled with six scattering clusters and 5° angular standard deviation.
- Uplink results: The MMSE estimator achieves the highest UL SE, because it exploits the LoS component and spatial correlation; Monte Carlo markers overlap the analytical curves.The comparison uses MR combining with MMSE, LS, and EW-MMSE estimates, alongside a Rayleigh reference with the same covariance matrices.
- Uplink results: For weak-channel UEs, MMSE outperforms EW-MMSE noticeably, whereas their SEs are similar for UEs with good channels.The CDF analysis uses M = 100 BS antennas and random UE locations and shadow-fading realizations.
- Downlink results: The downlink shows the same estimator behavior as the uplink, with average sum DL SE evaluated using MR precoding.The comparison is made across different numbers of BS antennas and estimator choices.
- Fading comparisons: In uncorrelated fading, EW-MMSE and MMSE coincide; both outperform LS for Rician fading, while all estimators have the same Rayleigh performance.For Rayleigh fading, the estimates differ only by a deterministic scaling factor that cancels in the SINR expressions.
- LoS scenarios: When every UE-BS pair has an LoS path, average UL sum SE is higher than when only some pairs have LoS.This scenario is identified as potentially impractical but isolates the SE benefit of universal LoS connectivity.
- Pilot reuse: A larger pilot reuse factor reduces pilot contamination but incurs a larger pre-log penalty, producing estimator-dependent optimal reuse factors.LS is more sensitive to pilot contamination, whereas MMSE estimators can suppress pilot interference through spatial processing.
- Conclusion: The study concludes that spatial correlation matters, LoS improves achievable SE, and MMSE outperforms EW-MMSE and LS, with LS giving the lowest SE.The expressions are intended to support analysis of scheduling, pilot allocation, power control, and other resource-allocation decisions.
APPENDIX A USEFUL RESULTS
The appendix develops Gaussian-vector identities for independent and correlated vectors, including quadratic-form expectations and spectral-norm bounds.
- Gaussian-vector identities: Lemma 4 treats independent complex Gaussian vectors with specified means and covariance matrices under a deterministic matrix transformation.
- Gaussian-vector identities: Lemma 5 treats correlated Gaussian vectors constructed from a shared Gaussian component and derives the corresponding transformed-moment expression.
- Matrix bounds: Lemma 6 states a bound for positive semi-definite matrices using the spectral norm, defined as the largest eigenvalue.
- Proof strategy: The proofs eliminate remaining terms using circular symmetry, independence, and zero-mean properties of the Gaussian vectors.
APPENDIX C PROOF OF LEMMA 3
The proof characterizes least-square estimate statistics and evaluates estimation-error terms by exploiting Gaussian representations, independence, and the MMSE estimator's error properties.
- LS estimate: The LS estimate is Gaussian distributed, and its mean and covariance are computed explicitly.
- Estimation error: The estimation-error mean and covariance are derived and substituted into the result stated in Lemma 3.
- MMSE estimator: For the MMSE estimator, estimate and estimation error are independent, making the relevant cross term zero.
- Channel-estimate calculations: The channel-estimate calculations distinguish index pairs inside and outside P_jk and use Lemmas 4 and 5 accordingly.
APPENDIX E PROOF OF THEOREM 2
The proof of Theorem 2 decomposes expectations in the spectral-efficiency expressions and evaluates them separately for independent and non-independent channel-related vectors.
- Expectation decomposition: The proof identifies the expectation terms required in the SINR expression and evaluates them component by component.
- Independence cases: Terms involving indices outside P_jk use independence and Lemma 4, while terms inside P_jk require handling non-independent vectors.
- Noise term: The noise term is modeled with a Gaussian distribution during the decomposition of the SINR terms.
- Final combination: The proof combines the results for both index cases to obtain the final expectation terms used in Theorem 2.
APPENDIX G PROOF OF THEOREM 7 AND THEOREM 8
The asymptotic proofs analyze uplink and downlink SINR terms as the antenna count grows, showing that interference vanishes under asymptotic spatial orthogonality while specified residual terms may remain otherwise.
- Interference terms: For distinct user indices, non-coherent interference asymptotically vanishes under Assumptions 1–3.
- Interference terms: The coherent-interference analysis shows bounded behavior because the denominator scales with M_j while the relevant traces cannot grow faster than M_j.
- MMSE asymptotics: Under asymptotic spatial orthogonality, the uplink MMSE SINR grows without bound because the relevant normalized trace terms converge to zero.
- MMSE asymptotics: Without asymptotic spatial orthogonality, the corresponding interference terms remain in the asymptotic denominator.
- Downlink asymptotics: The downlink proof follows the same asymptotic procedure, with uplink and downlink interference indices exchanged.
- EW-MMSE asymptotics: The EW-MMSE analysis similarly finds an unbounded SINR under asymptotic spatial orthogonality and residual denominator terms otherwise.
APPENDIX I PROOF OF THEOREM 11
The proof establishes the uplink result by bounding terms under Assumptions 1–3 and then begins the analogous downlink analysis by normalizing by M_j.
- Uplink proof: The uplink analysis proceeds by dividing numerator and denominator terms by M_j and applying inequalities involving correlation matrices.
- Uplink proof: Interference-related terms are shown to approach zero using trace bounds and Assumptions 1–3.
- Uplink proof: The uplink proof concludes with a finite positive asymptotic limit, completing the proof of the UL result.
- Downlink proof: For the downlink, the proof likewise divides the numerator and denominator by M_j, with the numerator described as positive and finite.
- Downlink proof: The downlink interference analysis follows the approach used for the uplink and yields equation (62).