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Achievable Rates of FDD Massive MIMO Systems with Spatial Channel Correlation
Zhiyuan Jiang, Andreas F. Molisch, Giuseppe Caire, Zhisheng Niu
TL;DR
FDD massive MIMO with i.i.d. channels faces prohibitively large CSIT-acquisition overhead, motivating methods that exploit spatial correlation. The paper designs eigenspace-based training and feedback schemes and evaluates their achievable rates against TDD, i.i.d. FDD, and JSDM. The proposed schemes narrow the FDD–TDD rate gap, with higher throughput than JSDM in stated moderate-antenna and coherence-time conditions.
Problem
FDD massive MIMO suffers from prohibitively large CSIT-acquisition overhead, motivating achievable-rate analysis and reduced-dimensionality acquisition for spatially correlated channels.
Method
The paper designs eigenspace channel-training sequences and KL-transform scalar-quantization feedback codebooks, deriving achievable-rate deterministic equivalents with RZF precoding and imperfect CSIT.
Results
The proposed eigenspace schemes significantly improve rates, narrow the FDD–TDD gap, can approach or outperform TDD, and outperform JSDM under some channel conditions.
Takeaways & Limitations
Spatial correlation enables dimensionality-reduced FDD channel estimation without channel pre-projection, while the feedback design remains computationally efficient and near-optimal.
Abstract
from arXiv · showhide
It is well known that the performance of frequency-division-duplex (FDD) massive MIMO systems with i.i.d. channels is disappointing compared with that of time-division-duplex (TDD) systems, due to the prohibitively large overhead for acquiring channel state information at the transmitter (CSIT). In this paper, we investigate the achievable rates of FDD massive MIMO systems with spatially correlated channels, considering the CSIT acquisition dimensionality loss, the imperfection of CSIT and the regularized-zero-forcing linear precoder. The achievable rates are optimized by judiciously designing the downlink channel training sequences and user CSIT feedback codebooks, exploiting the multiuser spatial channel correlation. We compare our achievable rates with TDD massive MIMO systems, i.i.d. FDD systems, and the joint spatial division and multiplexing (JSDM) scheme, by deriving the deterministic equivalents of the achievable rates, based on popular channel models. It is shown that, based on the proposed eigenspace channel estimation schemes, the rate-gap between FDD systems and TDD systems is significantly narrowed, even approached under moderate number of base station antennas. Compared to the JSDM scheme, our proposal achieves dimensionality-reduction channel estimation without channel pre-projection, and higher throughput for moderate number of antennas and moderate to large channel coherence time, though at higher computational complexity.
I. INTRODUCTION
The paper addresses the severe CSIT-acquisition overhead of FDD massive MIMO by exploiting spatial channel correlation to reduce estimation dimensionality and improve achievable rates. It designs correlation-aware training and feedback schemes and compares them with TDD, i.i.d. FDD, and JSDM systems.
- Motivation: Without CSIT, the total DoF reduces to one for identically distributed i.i.d. user channels, even when CSIT error does not decrease with SNR.The paper therefore treats CSIT acquisition as central to FDD massive-MIMO performance.
- Motivation: FDD CSIT-acquisition overhead scales with the number of BS antennas and becomes prohibitively large as massive MIMO scales up.This motivates continued interest in realizing massive-MIMO gains in currently prevalent FDD systems.
- Spatial correlation: Channel correlation matrices vary more slowly than instantaneous CSIT and can facilitate FDD transmission at substantially lower estimation cost.The paper uses these second-order statistics to exploit multiuser spatial correlation.
- Contributions: The proposed eigenspace schemes exploit low-rank covariance matrices for dimensionality-reduced training and feedback, potentially requiring fewer pilots than BS antennas.They are presented as an alternative to JSDM’s pre-projection and effective-channel approach.
- Contributions: An iterative algorithm optimizes training sequences for distinct user CCMs, and the resulting sequences substantially improve achievable rates over i.i.d.-optimized sequences.The optimization maximizes mutual information between channel coefficients and received training signals.
- Contributions: KLSQ uses a KL transform, entropy-coded scalar quantization, and reverse-water-filling bit loading to approach optimal VQ performance for correlated Gaussian channel vectors.The design is emphasized as simple and computationally efficient.
II. SYSTEM MODEL
The system model is a downlink broadcast channel in which an M-antenna base station serves N single-antenna users over mutually independent spatially correlated channels. Each user’s channel is modeled as a zero-mean complex Gaussian vector with its own channel correlation matrix, assumed known to both sides.
- System model: An M-antenna base station serves N single-antenna users in a downlink broadcast channel.The transmitted data vector is precoded by a matrix W before users receive their signals.
- Channel model: User n’s channel vector follows a zero-mean complex Gaussian distribution with covariance R_n.The compound channel matrix collects the users’ channel vectors.
- Channel model: The users’ channel vectors are assumed mutually independent because users are usually well separated.This is a modeling assumption for the spatially correlated channel matrix.
- Channel model: The BS and users are assumed to have perfect knowledge of the second-order channel statistics, namely the CCMs.These statistics support the paper’s correlation-aware channel estimation and transmission design.
B. Dominant Eigenspace Representation of CCM
The paper represents correlated channels through dominant covariance eigenspaces and uses this structure within a pilot-assisted FDD transmission strategy. It optimizes multiuser training sequences through an iterative mutual-information procedure while accounting for estimation errors and dimensionality loss.
- Dominant eigenspace representation: The order-r_n dominant eigenspace representation retains the dominant r_n eigenvalues and eigenvectors of user n’s CCM.The resulting approximation is used to improve CSIT feedback efficiency.
- Transmission strategy: The transmission strategy retains the conventional pilot-assisted FDD structure of downlink training followed by uplink CSIT feedback.The paper optimizes these stages with minimal modifications to the current transmission strategy.
- Training design: Because users with distinct CCMs share downlink training sequences, the paper develops an iterative algorithm to optimize sequences by maximizing conditional mutual information.The optimization is posed for a fixed training length and total transmit power.
- Training design: The training optimization problem is generally nonconvex, so the proposed heuristic algorithm uses a KKT-based iteration rather than a guaranteed convex optimization procedure.Simulations report that the algorithm performs fairly well and converges fast.
- Training design: The paper uses CMI rather than direct MSE minimization because the MSE-based algorithm does not converge under ill-conditioned matrices, while the CMI problem is well-conditioned.The obtained sequences nevertheless show very good MSE performance.
B. Uplink CSIT Feedback
For uplink CSIT feedback, the paper exploits spatial correlation by quantizing transformed channel estimates efficiently and comparing the resulting method with vector-quantization alternatives.
- Feedback design: The proposed feedback method applies entropy-coded scalar quantization after a Karhunen-Loeve transform to exploit spatially correlated channel estimates.The method is introduced as a simple approach to efficient CSIT feedback.
- Feedback design: Its performance is compared with two vector-quantization approaches designed for spatially correlated channels.The vector-quantization methods also serve as implementation options.
1) Entropy Coded Scalar Quantization:
The section develops scalar and vector quantization schemes that concentrate finite feedback on dominant channel eigenspaces, while characterizing their distortion and rank tradeoffs.
- Scalar quantization: The scalar quantizer operates component by component on the KL-transformed channel vector and can be implemented efficiently without vector quantization.When the transform matrix is a DFT slice, the KL transform can be approximated by an FFT, enabling highly efficient parallel implementation.
- Error decomposition: The quantized channel estimate combines channel-training error, CSI quantization error, and error from neglecting subdominant eigenspaces.The reconstructed channel is obtained by applying the inverse KL transform after feedback recovery.
- Feedback dimensionality: Dominant-eigenspace feedback compresses overhead by retaining only the order-r_n dominant channel eigenmodes.The approach exploits rank-deficient channel correlation matrices and is reported to outperform feeding back the full channel space when B is finite.
- Skewed codebooks: Skewed codebooks match the dominant channel eigenspace, addressing eigenvalue imbalance that makes isotropical RVQ suboptimal.Their correlation matrix is designed to match the dominant eigenspace, and Theorem 2 upper-bounds the resulting quantization error.
- Distortion behavior: The skewed-codebook error contains both quantization of the retained channel and distortion from neglecting subdominant eigenmodes.With fixed r_n, the bound does not vanish as B grows; choosing r_n optimally makes the error scale to zero as B approaches infinity.
- Rank selection: The dominant rank r_n trades channel-approximation accuracy against quantization error because larger ranks increase quantization dimension.The paper selects the optimal rank using a one-dimensional search over 1:M.
C. Data Transmission
This section formulates data transmission with RZF precoding and optimizes training and feedback lengths under imperfect channel estimates and dimensionality loss.
- Precoding: The RZF precoder treats the channel estimates as the actual channel coefficients when computing downlink transmission.Equal power allocation is assumed, with a normalization scalar enforcing the power constraint.
- Rate model: The SINR of user n is used to compute achievable rates under the RZF transmission model.The analysis assumes a common feedback-bit allocation B_n = B across users and incorporates the MIMO-MAC feedback constraint.
- Rate optimization: The achievable sum rate is optimized over training and feedback lengths under imperfect channel training and feedback.The paper uses exhaustive search to determine the optimal lengths for correlated-channel downlink rates.
- Overhead tradeoff: Longer training and feedback improve channel-estimation accuracy but increase dimensionality loss.The training dimensionality loss is determined by the training-sequence length τ.
IV. PERFORMANCE ANALYSIS
The performance analysis derives deterministic equivalents for downlink achievable sum rates under pre-user CCMs and RZF precoding as M grows without bound.
- Deterministic-equivalent analysis: The analysis derives downlink achievable sum-rate expressions using deterministic-equivalent techniques with necessary modifications for pre-user CCMs.The exposition assumes identical dominant feedback ranks, r_n = r, for all users.
- Asymptotic SINR: As M approaches infinity, each user’s RZF SINR converges to a deterministic quantity computable from parameters specified in equations (49)–(61).The derivation generalizes prior results to uncorrelated channel-estimation error matrices.
V. NUMERICAL RESULTS
Numerical results evaluate correlated-channel FDD massive MIMO under multiple channel models and compare it with TDD, i.i.d. FDD, and JSDM baselines. Spatial correlation and eigenspace-based estimation substantially improve FDD rates and narrow the TDD gap, particularly with moderate or large antenna arrays.
- Channel models and effective rank: The simulations use one-ring and Laplacian channel-correlation models to evaluate singular-value distributions and effective rank.Effective rank decreases with smaller antenna spacing or angular spread, and is generally larger under the Laplacian model than the one-ring model.
- Eigenspace estimation: The iterative training sequences and KLSQ feedback codebook are used to obtain correlated-channel FDD achievable rates.Simulation settings include N = 8 users, channel block length T = 200, and RZF regularization α = 0.01 in the stated experiments.
- FDD rate comparisons: The proposed correlated-channel FDD scheme achieves higher sum rates than i.i.d. FDD, especially as the number of BS antennas grows.The improvement is attributed to dominant-channel estimation based on the per-user correlation matrices.
- FDD rate comparisons: The rate gap between correlated-channel FDD and TDD narrows significantly when the number of BS antennas is moderate.By contrast, FDD dimensionality loss becomes non-negligible as the antenna count increases, while TDD rates can continue increasing with sufficiently favorable uplink conditions.
- FDD rate comparisons: When downlink and uplink SNR are equal, TDD performs better, whereas the reported FDD-over-TDD cases use uplink SNR 10 dB below downlink SNR.The lower uplink SNR makes TDD channel training more imperfect in those simulations.
C. Comparison with JSDM
The proposed eigenspace channel-estimation scheme is compared with JSDM under different coherence times, antenna counts, and angular spreads. It reduces estimation dimensionality without pre-projection and can outperform JSDM in several moderate-to-large-system regimes, while requiring more computation.
- Eigenspace channel estimation: Grouping users by their channel-correlation matrices and choosing virtual-sector dimensions balances power gain against channel-estimation overhead.The equivalent antenna count bg can be reduced toward the number of users in a sector, reducing estimation dimensionality.
- Comparison with JSDM: The proposed dominant-channel estimation incorporates all user CCMs and significantly mitigates residual inter-group interference.JSDM’s pre-beamforming cannot completely suppress inter-group interference when users within a group have different CCMs.
- Comparison with JSDM: JSDM achieves better sum rate when channel coherence time is small and the number of BS antennas is large.These conditions place greater weight on suppressing channel-estimation overhead.
- Comparison with JSDM: The proposed eigenspace estimator achieves better rates when coherence time is larger or users have larger angular spreads.Larger angular spreads increase residual inter-group interference in JSDM.
- Complexity and scope: The eigenspace scheme has lower complexity than JSDM, although JSDM uses a lower-dimensional channel matrix and can be better after parameter tuning.The paper leaves optimization of JSDM parameters outside its scope.
D. Performance Gain Leveraging Eigenspace Channel Estimation
The paper evaluates gains from designing training and feedback in the dominant channel eigenspaces. These designs reduce estimation error and produce substantial FDD rate improvements across channel block lengths.
- Training design: The iterative training algorithm achieves lower total MSE than orthogonal training by concentrating power on the eigenspace requiring more accurate estimation.Unitary training becomes asymptotically optimal only when SNR and the number of training symbols are sufficiently large.
- Feedback design: KLSQ achieves better feedback MSE than VQ, including when shaping loss is present and the number of feedback symbols is large.After the KL transform, the channel vector has independent Gaussian components with unequal variances, motivating rate-weighted bit allocation.
- Feedback design: An optimal dominant rank minimizes total feedback error by balancing effective-channel quantization accuracy against neglected-eigenspace error.The optimal rank increases with the number of feedback symbols.
- Feedback design: Feedback-codebook refinements provide little MSE gain at the optimal dominant rank because error mainly comes from the neglected non-dominant eigenspace.Skewed codebooks help more when the dominant rank is large and quantization error dominates.
- Rate improvement: Training error dominates feedback error because training MSE decreases inversely with training symbols, whereas feedback MSE decreases exponentially with feedback symbols.The simulations show that eigenspace channel estimation can improve FDD sum rate by up to two-fold across block lengths.
VI. CONCLUSIONS
The paper shows that spatial correlation can make FDD massive MIMO more practical by reducing channel-estimation dimensionality and narrowing its rate gap with TDD. Its eigenspace training and feedback schemes outperform or complement JSDM under specified channel conditions, while incurring higher computational complexity.
- An iterative algorithm jointly designs optimal channel-training sequences under multiuser spatial correlation, while KL-transform/SQ feedback with reverse-water-filling bit loading is computationally efficient and near-optimal.The approach is designed for practical implementation while preserving near-optimal feedback performance.
- The proposed schemes exploit spatial channel correlation to reduce FDD channel-estimation dimensionality without channel pre-projection.They are presented as an alternative to JSDM’s projection and effective-channel approach.
- Significant rate improvements are obtained over i.i.d. FDD systems when the proposed eigenspace channel-estimation approaches leverage spatially correlated channels.The reported improvements are especially relevant to the paper’s FDD setting and its comparison with i.i.d. channels.
- When channel correlation is strong and the BS antenna count is moderate, FDD achievable sum rate can exceed that of TDD systems.The benefit is attributed to strongly correlated, effectively rank-deficient channel correlation matrices.
- Compared with JSDM, the proposed schemes perform better for large coherence time or large user angular spread, but require higher computational complexity.The complexity increase results from operating on a higher-dimensional matrix.
- Increasing spatial correlation, such as through smaller antenna spacing or more line-of-sight transmission, can benefit FDD massive MIMO despite reducing downlink degrees of freedom.The paper identifies a tradeoff between downlink broadcast-channel degrees of freedom and spatial correlation.
APPENDIX A
Appendix A derives the MSE of the skewed codebook using integral transformations and asymptotic approximations supported by prior results.
- The appendix expresses the skewed-codebook MSE in equation (59).
- The derivation uses Q_n as a unitary matrix and z_n distributed as CN(0, I_rn).
- Equality (60) follows by integrating equation (59) by parts, while the approximation in (63) uses a dominant-integral-term result.
- Equations (64) and (65) follow from the approximation in (63) and Corollary 1 of reference [22].