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Large System Analysis of Linear Precoding in Correlated MISO Broadcast Channels under Limited Feedback

Sebastian Wagner, Romain Couillet, Merouane Debbah, Dirk. T. M. Slock

arXiv:0906.3682v5cs.IT

TL;DR

The paper addresses how to characterize and optimize ZF and RZF sum rates when large MISO broadcast systems have imperfect CSIT and per-user channel correlation. It develops deterministic SINR equivalents using large-dimensional random matrix theory, then applies them to precoding, power, user-loading, training, and feedback optimization. Simulations indicate that the approximations are accurate even for small system dimensions.

  • Problem

    The paper studies the lack of tractable large-system analysis for MISO broadcast precoding with imperfect CSIT and per-user channel correlation.

  • Method

    The authors derive deterministic SINR equivalents using large-dimensional random matrix theory and generalized-variance-profile Stieltjes-transform results.

  • Results

    The approximations are reported as almost surely exact as M,K →∞ and accurate even for small system dimensions, while enabling optimization of regularization, users, power, training, and feedback.

  • Takeaways & Limitations

    The framework provides a consistent way to analyze correlated, imperfect-CSIT systems and apply the resulting SINR approximations to several limited-feedback optimization problems.

  • Takeaways & Limitations

    For small dimensions, the feedback-scaling results cannot guarantee that the per-user rate gap remains below the target bound.

Abstract

from arXiv · show

In this paper, we study the sum rate performance of zero-forcing (ZF) and regularized ZF (RZF) precoding in large MISO broadcast systems under the assumptions of imperfect channel state information at the transmitter and per-user channel transmit correlation. Our analysis assumes that the number of transmit antennas $M$ and the number of single-antenna users $K$ are large while their ratio remains bounded. We derive deterministic approximations of the empirical signal-to-interference plus noise ratio (SINR) at the receivers, which are tight as $M,K\to\infty$. In the course of this derivation, the per-user channel correlation model requires the development of a novel deterministic equivalent of the empirical Stieltjes transform of large dimensional random matrices with generalized variance profile. The deterministic SINR approximations enable us to solve various practical optimization problems. Under sum rate maximization, we derive (i) for RZF the optimal regularization parameter, (ii) for ZF the optimal number of users, (iii) for ZF and RZF the optimal power allocation scheme and (iv) the optimal amount of feedback in large FDD/TDD multi-user systems. Numerical simulations suggest that the deterministic approximations are accurate even for small $M,K$.

I. INTRODUCTION

The paper analyzes ZF and RZF precoding in large MISO broadcast channels with imperfect CSIT and per-user channel correlation, using deterministic SINR approximations to study practical design problems.

  • Motivation: Multi-user transmission can mitigate the antenna-correlation limitations that constrain single-user MIMO by exploiting diversity across non-cooperative receivers.
  • Problem Setting: Imperfect CSIT is a practical constraint because transmitters acquire downlink CSI through feedback over finite channel coherence intervals.
  • Problem Setting: The study considers M-antenna transmitters serving K single-antenna users with M ≥ K, under imperfect CSIT and per-user channel correlation.
  • Large-System Analysis: The proposed deterministic SINR equivalent is a tight large-system approximation rather than a bound and is suggested by simulations to remain accurate for small dimensions such as M = K = 16.The analysis lets the system dimensions grow jointly with bounded ratio.
  • Practical Optimization: The SINR approximations support optimization of RZF regularization, ZF user loading, ZF/RZF power allocation, and feedback or training in large FDD/TDD systems.The framework extends prior large-system results by accounting for per-user correlation and imperfect CSIT.
  • Mathematical Framework: Large-dimensional random matrix theory is used to derive a deterministic equivalent for matrices with generalized variance profile, extending analysis to the per-user correlation model.

III. A DETERMINISTIC EQUIVALENT OF THE SINR

This section develops deterministic SINR approximations for RZF and ZF precoding in correlated, imperfect-CSIT systems using generalized random-matrix results and fixed-point equations.

  • Overview: Deterministic SINR approximations are introduced for RZF and ZF and then used to solve practical optimization problems.
  • Random-Matrix Analysis: The generalized-variance-profile theorem provides the mathematical basis for the large-system analysis and yields unique Stieltjes-transform solutions through coupled equations.The fixed-point iteration converges to a unique nonnegative solution under the stated conditions.
  • RZF Precoding: The RZF model uses a normalized channel estimate, a regularization parameter α > 0, and a scaling by M so α converges to a constant.
  • RZF Precoding: For RZF, the user SINR converges almost surely to a deterministic equivalent characterized by unique positive fixed-point solutions.
  • RZF Precoding: With common correlation, the RZF equivalent reduces to quantities m◦ and eij; with uncorrelated channels, the approximation becomes explicit.
  • RZF Precoding: RZF-CDU fixes α = 1/(βρ), whereas RZF-CDA accounts for imperfect CSIT and uses an optimal regularization parameter derived later.Under imperfect CSIT, the two precoders are not generally identical, and the optimal parameter depends on transmit correlation.

B. Zero-forcing Precoding

The ZF analysis derives deterministic SINR equivalents under assumptions ensuring well-conditioned channel matrices, with explicit simplifications for common and uncorrelated correlation.

  • ZF Precoding: For α = 0, the RZF precoding matrix reduces to the ZF precoding matrix.
  • ZF Precoding: For common correlation, the equivalent is expressed through the corresponding fixed-point quantities and their derivatives, with power normalization enforcing the transmit constraint.
  • Assumptions: The ZF deterministic equivalent requires the minimum eigenvalue of the normalized estimated-channel Gram matrix to remain bounded away from zero.
  • Assumptions: With common correlation, invertible Θ, and β > 1, the eigenvalue condition holds under the stated lower-bound assumptions.
  • ZF Precoding: Under the required assumptions, each ZF user SINR converges almost surely to a deterministic equivalent characterized by a unique positive solution.
  • ZF Precoding: For uncorrelated channels and β > 1, the ZF SINR equivalent takes an explicit form.

C. Rate Approximations

The paper develops deterministic SINR and rate approximations for ZF and RZF under imperfect CSIT and channel correlation, then validates their accuracy and applies them to regularization optimization.

  • Deterministic approximations: Replacing instantaneous SINR γ_k with γ°_k yields an approximation R̂_sum of the ergodic sum rate.The same convergence argument applies to per-user rates.
  • Numerical validation: 10,000 independent channel realizations show that the sum-rate approximation becomes more accurate as M increases, including under per-user correlation.The correlated model uses d_ij/λ = 0.5 in the reported simulation.
  • Numerical validation: The approximation lies roughly within one standard deviation of Monte Carlo simulations for RZF and ZF sum-rate comparisons.Figures 2 and 3 evaluate the approximations against ergodic sum rates.
  • Numerical validation: Under imperfect CSIT, RZF sum rate decreases at high SNR when α = 1/ρ because the estimated-channel matrix becomes ill-conditioned.For M > K, ZF does not show this high-SNR decrease because the CSIT estimate is better conditioned.
  • Regularization optimization: At asymptotically high SNR, imperfect CSIT keeps α⋆° non-zero, so RZF-CDA is not equivalent to ZF.The non-zero regularization is attributed to residual interference.
  • Regularization optimization: Adapting α produces a significant performance increase, while the proposed α⋆° performs close to optimal even for small system dimensions.Under perfect CSIT, α⋆° = 1/(βρ) and is independent of Θ; under imperfect CSIT, it depends on transmit correlation.
  • Regularization optimization: For imperfect CSIT and highly correlated channels, correlation-aware RZF significantly outperforms correlation-unaware RZF at medium to high SNR.At low SNR, the two precoders perform equally well in the reported comparison with v = 0.9.

V. OPTIMAL NUMBER OF USERS AND POWER ALLOCATION

This section optimizes the served-user count and power distribution using deterministic rate approximations. For unequal CSIT qualities under common correlation, the optimal power allocation is obtained by water-filling.

  • Scope: The section addresses the sum-rate-maximizing number of users for fixed M and the optimal power allocation for users with unequal CSIT qualities.Both problems are formulated using the paper’s approximated SINR framework.
  • Optimal number of users: Serving more users can eventually reduce sum rate because additional interference outweighs the aggregate rate gain.The closed-form ZF result is derived for a fair setting with equal approximated SINRs.
  • Power allocation: For common correlation and different CSIT qualities, the approximated optimal power distribution is the solution of a classical water-filling algorithm.The optimization is over the power allocation matrix for a given K.

A. Sum Rate Maximizing Number of Users

The paper derives and evaluates the sum-rate-maximizing user loading for ZF under imperfect CSIT. The approximation remains accurate in small systems, while the preferred user count depends on SNR and CSIT quality.

  • Analytical user-number optimization: A closed-form approximate solution determines the number of users K maximizing sum rate per transmit antenna for fixed M under ZF.For uncorrelated antennas and common CSIT distortion, the derivation uses a one-dimensional optimization that admits a Lambert W-based solution.
  • SNR-dependent loading: As SNR increases, both the simulated and approximated optimal user counts increase, but with imperfect CSIT they saturate below M at high SNR.Thus, serving all M users is not asymptotically optimal under nonzero CSIT distortion.
  • Numerical validation: The approximation fits simulation results even for small dimensions and achieves most of the ergodic sum rate.The study compares K⋆◦ = M/β⋆◦ with exhaustive selection of K ∈ {1, 2, ..., M}.
  • Numerical validation: Adapting the number of users to SNR is beneficial compared with using a fixed K.For M = 16, K = 8 is optimal at medium SNR but becomes suboptimal at low and high SNR, while K = 4 is highly suboptimal in medium and high SNR.
  • Related optimization results: When CSIT qualities differ substantially, optimal power allocation produces significant gains across the SNR range, especially at high SNR.The water-filling allocation can effectively turn off users with the lowest CSIT accuracy as SNR increases, under its stated large-system assumption.

A. Channel Distortion Aware Regularized Zero-forcing Precoding

The paper derives CSIT-distortion scaling laws for channel-distortion-aware and channel-distortion-unaware RZF, relating feedback quality to a target per-user rate gap. The results are asymptotic and have explicit scope limitations.

  • RZF-CDA: For RZF-CDA, the paper derives CSIT-distortion scaling that maintains a target instantaneous per-user rate gap log2 b as M,K → ∞.The scaling is obtained by setting the deterministic rate-gap expression equal to log2 b.
  • High-SNR behavior: At high SNR, the relevant rate-gap scaling limits converge to expressions involving b − 1, with distinct forms depending on β.For β = 1, the RZF-CDU limit is stated as 2(b − 1), whereas for β > 1 it is b − 1.
  • RZF-CDU: For RZF-CDU with α = 1/(βρ), the paper likewise derives CSIT-distortion scaling for maintaining a target per-user rate gap.The result is based on the deterministic equivalent for the RZF-CDU rate gap.
  • Scope and approximation: The analysis requires β > 1 for several ZF and feedback-scaling results, although high-SNR RZF-CDU results can approximate ZF even when β = 1.At finite SNR, the approximation in Proposition 5 gives τ^2 > 0.

D. Discussion and Numerical Results

The paper uses deterministic large-system approximations to analyze feedback and training design for ZF and RZF under imperfect CSIT. Numerical results show advantages for distortion-aware RZF and validate efficient training approximations, while exposing scope limits for some scaling laws.

  • Feedback scaling: RZF-CDA requires fewer feedback bits than RZF-CDU and ZF at high SNR when β = 1, while the schemes have the same CSIT-distortion scaling for β > 1.For β > 1, the channel is well conditioned, so RZF and ZF perform similarly and are equally sensitive to imperfect CSIT.
  • Feedback scaling: At high SNR and β = 1, the RZF-CDA feedback scaling requires (M−1) log2(b + 1) bits less than RZF-CDU and ZF.The proposed scaling also improves on the scaling in [45, Theorem 3] by (M−1) log2(b + 1) bits.
  • Model scope: The feedback model combines generic additive CSIT distortion with i.i.d. block fading, while temporal, frequency, and spatial correlation could reduce feedback overhead.The distortion may represent channel estimation, feedback delay, or feedback errors when modeled as additive noise.
  • Numerical feedback results: RZF-CDA achieves significantly higher sum rate than RZF-CDU for equal feedback, reaching about 2.5 bits/s/Hz at 20 dB.The compared feedback scaling in [45, Theorem 3] is described as pessimistic, with rate offsets of about 6 and 7 bits/s/Hz at 20 dB under the two power constraints.
  • Training optimization: For TDD training optimization, the net sum rate accounts for the pre-log factor 1−Tt/T after spending Tt channel uses on uplink pilots.The training length is optimized using deterministic SINR approximations for ZF and RZF-CDA.
  • Training optimization: The ZF and RZF-CDA training objectives are strictly concave over [K, T] in the stated loading regimes, enabling standard convex optimization.Equal power allocation is used because it is optimal for large M and τ^2 under the considered setting.

1) Case 1: finite ratio ρdl/ρul:

For finite downlink-to-uplink SNR ratio, the paper derives and evaluates optimal TDD training for ZF and RZF-CDA. Training decreases relative to coherence time as SNR rises, approaches half the interval at low SNR, and reaches the pilot minimum at high SNR.

  • High-SNR training: Approximate explicit solutions are derived for the sum-rate-maximizing training intervals of ZF and RZF-CDA when downlink and uplink SNRs grow with finite ratio.The analysis treats the high-SNR regime and also derives limiting behavior at asymptotically low SNR.
  • High-SNR training: For fixed downlink SNR and coherence interval, the optimal training interval scales with the coherence interval and with log(ρdl) under the corresponding fixed-parameter regimes.These scaling laws are stated for both RZF-CDA and ZF precoding.
  • High-SNR training: As ρdl →∞, the optimal training interval tends to K, the minimum required by orthogonal pilot sequences.RZF-CDA requires less training than ZF except that both converge to the same training interval at asymptotically high SNR.
  • Low-SNR training: At low SNR with constant ρdl/ρul and T ≥2K, Proposition 10 gives the sum-rate-maximizing training intervals for ZF and RZF-CDA.The result follows by applying the low-SNR expansion log(1+x) = x+O(x^2) and optimizing the resulting expressions.
  • Scope boundary: No accurate closed-form solution has been found for RZF-CDA in the finite-uplink-SNR, high-downlink-SNR scenario.In that regime, the system is interference-limited because the uplink transmit power remains finite.
  • Numerical results: The training approximations become very accurate for K = 16, while ZF and RZF-CDA require approximately the same training for K = 2.These observations agree with the theoretical expressions for the two schemes.
  • Numerical results: Numerically, the relative training amount converges to 1/2 at low SNR and to K/T at high SNR for both ZF and RZF-CDA.The optimal amount is lower for RZF-CDA than for ZF over the relevant finite-SNR range.
  • Numerical results: With fixed uplink SNR, the high-SNR training approximation performs well, whereas using Tt,zf = K at every SNR causes significant performance loss.The approximation has only a small performance loss at low and medium SNR and can compute training efficiently.

APPENDIX I PROOF OF THEOREM 1

The proof establishes existence, uniqueness, and convergence of the fixed-point sequence defining the deterministic equivalent, then extends convergence across the complex plane.

  • Fixed-point convergence: The fixed-point sequence converges to a limit eN,i under suitable initialization and remains a Stieltjes transform.
  • Convergence extension: Vitali’s convergence theorem extends convergence from a restricted subset of C+ to all z ∈ C\R+.
  • Uniqueness and initialization: For z < 0, standard interference-function theory gives convergence to a unique nonnegative solution from any real initial point.
  • Uniqueness and characterization: The limit eN is the unique Stieltjes-transform solution to the fixed-point equation, completing the deterministic-equivalent characterization.

C. Proof of Convergence of the Deterministic Equivalent

The proof decomposes the RZF SINR into signal, interference, and normalization terms, derives deterministic equivalents for each, and verifies their almost-sure convergence.

  • SINR decomposition: The RZF SINR is decomposed into scaled signal power, scaled interference power, and a power-normalization term.
  • Deterministic equivalents: Each SINR component is replaced by a deterministic equivalent, yielding almost-sure convergence of the normalization term and related quadratic forms.
  • Technical derivation: The derivation uses resolvent identities, trace functionals, matrix lemmas, and assumptions ensuring bounded correlation and power matrices.
  • Regularity conditions: Assumptions preventing concentration of all transmit power on one user ensure the RZF SINR convergence result remains valid.

APPENDIX III PROOF OF THEOREM 3

The proof shows that RZF approaches ZF as the regularization parameter vanishes and that the resulting deterministic SINR approximation is valid under the stated assumptions.

  • SINR convergence: For sufficiently small α, the RZF and ZF SINRs differ by less than any ε in the large-system limit.
  • RZF-to-ZF limit: As α approaches zero, the RZF precoder converges to the ZF precoder through a matrix-inversion representation.
  • Fixed-point solution: The fixed-point iteration converges to a positive solution independently of M and K under Assumption 5.
  • Boundedness: The limiting normalization terms exist and remain uniformly bounded, completing the deterministic-equivalent proof for ZF.

APPENDIX IV PROOF OF PROPOSITION 2

The proof derives the optimal RZF regularization parameter by differentiating the deterministic SINR with respect to regularization and solving the resulting condition.

  • SINR reformulation: The deterministic SINR is rewritten using auxiliary quantities χ, ψ, and φ to expose its dependence on α.
  • Optimization condition: Differentiating the SINR along α produces the stationarity condition used to identify the optimal regularization parameter.
  • Optimal regularization: Because Ω ≠ 0 for ρ > 0 and τ^2 < 1, the optimal regularization parameter α⋆◦ is given by (53).
  • Feedback optimization: Solving the feedback-training rate-gap equations for finite c yields the optimal training lengths Tt,zf and Tt,rzf.

APPENDIX VI IMPORTANT LEMMAS

The appendix collects matrix identities, probabilistic trace tools, rank-1 perturbation results, and random-matrix lemmas used in the analysis. These results establish conditions for invertibility, quadratic-form control, and deterministic-equivalent derivations.

  • Matrix identities: The Matrix Inversion Lemma and Resolvent Identity provide algebraic tools for manipulating inverses of perturbed or distinct invertible matrices.The inversion lemma treats rank-1 updates, while the resolvent identity relates the inverses of two matrices.
  • Random-vector tools: The trace and quadratic-form lemmas control random-vector expressions under zero-mean, variance-1/N entries and bounded moment assumptions.The assumptions include independent vectors, bounded spectral norms, and eighth-order moments of order O(1/N^4).
  • Probabilistic foundations: Several lemmas establish almost-sure convergence and boundedness using the Markov inequality, Borel-Cantelli Lemma, Tonelli Theorem, and trace-lemma arguments.The probabilistic proofs transfer almost-sure statements across realizations and product probability spaces.
  • Invertibility and perturbations: The appendix gives conditions ensuring random Hermitian matrices and rank-1 perturbations remain invertible, including uniform lower bounds on the smallest eigenvalue.For appropriate realizations, the smallest eigenvalue is greater than a positive ε, so both the original and perturbed matrices are invertible.
  • Correlated quadratic forms: A generalized quadratic-form lemma handles correlated random vectors x = Θ^1/2z and y = Θ^1/2q with independent i.i.d. entries and bounded spectral norms.Its proof combines inverse identities with limits for terms such as x^HUA^-1x, y^HA^-1y, and cross terms.
  • Correlated quadratic forms: The quadratic-form derivation remains valid for nonnegative coefficients satisfying c0c1 ≥ c2^2 because the associated polynomial is bounded away from zero.The proof applies the resolvent identity and auxiliary lemmas to expressions involving c0xx^H, c1yy^H, and cross terms.
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