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Secrecy Sum-Rates for Multi-User MIMO Regularized Channel Inversion Precoding

Giovanni Geraci, Malcolm Egan, Jinhong Yuan, Adeel Razi, Iain B. Collings

arXiv:1207.5063v1cs.IT

TL;DR

Secrecy rates for multi-user networks with potentially eavesdropping intended users remain an open problem. The paper proposes an RCI precoder with optimized regularization and power allocation, showing that it outperforms several linear precoders while retaining the scaling factor of optimum RCI without secrecy requirements.

  • Problem

    Secrecy rates for multi-user networks in which intended users may eavesdrop remain an open problem.

  • Method

    The paper proposes a linear precoder based on regularized channel inversion with a regularization parameter and power allocation vector that maximize achievable secrecy sum-rate.

  • Results

    RCI with equal power allocation and the optimal regularization parameter outperforms several other linear precoding schemes and achieves the same sum-rate scaling factor as optimum RCI without secrecy requirements.

  • Takeaways & Limitations

    Optimizing RCI regularization and power allocation provides a secrecy-oriented precoding approach that preserves the scaling factor of optimum non-secrecy RCI.

  • Takeaways & Limitations

    The secrecy sum-rate is limited by the eavesdropper receiving the largest information leakage.

Abstract

from arXiv · show

In this paper, we propose a linear precoder for the downlink of a multi-user MIMO system with multiple users that potentially act as eavesdroppers. The proposed precoder is based on regularized channel inversion (RCI) with a regularization parameter $α$ and power allocation vector chosen in such a way that the achievable secrecy sum-rate is maximized. We consider the worst-case scenario for the multi-user MIMO system, where the transmitter assumes users cooperate to eavesdrop on other users. We derive the achievable secrecy sum-rate and obtain the closed-form expression for the optimal regularization parameter $α_{\mathrm{LS}}$ of the precoder using large-system analysis. We show that the RCI precoder with $α_{\mathrm{LS}}$ outperforms several other linear precoding schemes, and it achieves a secrecy sum-rate that has same scaling factor as the sum-rate achieved by the optimum RCI precoder without secrecy requirements. We propose a power allocation algorithm to maximize the secrecy sum-rate for fixed $α$. We then extend our algorithm to maximize the secrecy sum-rate by jointly optimizing $α$ and the power allocation vector. The jointly optimized precoder outperforms RCI with $α_{\mathrm{LS}}$ and equal power allocation by up to 20 percent at practical values of the signal-to-noise ratio and for 4 users and 4 transmit antennas.

I. INTRODUCTION

The paper addresses secrecy in multi-user MIMO downlinks where intended users may cooperate as eavesdroppers. It proposes RCI-based precoding with optimized regularization and power allocation, and reports improved secrecy performance.

  • The secrecy-capacity region for multi-user networks where any number of intended users may eavesdrop remains an open problem.
  • RCI precoding trades off interuser interference and desired signal through a regularization parameter.
  • The proposed linear precoder targets multiple single-antenna users that cooperate to jointly eavesdrop on other users.
  • Large-system analysis derives the optimal regularization parameter αLS and its corresponding achievable secrecy sum-rate.
  • The RCI precoder with αLS outperforms several linear precoding schemes and has the same secrecy-sum-rate scaling factor as optimum RCI without secrecy requirements.
  • Jointly optimizing α and power allocation outperforms RCI-EP by up to 20 percent at practical SNR for 4 users and 4 transmit antennas.

II. SYSTEM MODEL

The system is a narrowband multi-user MIMO downlink in which a multi-antenna base station sends confidential messages to spatially dispersed single-antenna users. Security is evaluated under a worst-case model where the other users cooperate as eavesdroppers, and linear RCI precoding controls interference.

  • The base station has M antennas and simultaneously transmits K independent confidential messages to K users.
  • The block-fading model assumes channel coherence time is much longer than one symbol interval.
  • The received signal uses fading gains hk,j distributed as CN(0, 1), with receiver noise nk distributed as CN(0, σ2).
  • The model imposes E[∥x∥2] = 1 and defines the signal-to-noise ratio as ρ = 1/σ2.
  • For each intended receiver, the remaining K − 1 users cooperate as a single eavesdropper with K − 1 receive antennas.
  • Linear precoding maps confidential messages through a deterministic transformation, while RCI controls crosstalk and performs better than plain channel inversion especially at low SNR.

B. Achievable Secrecy Sum-Rates with Linear Precoding

The paper derives achievable secrecy sum-rates for multi-user MIMO linear precoding by treating each user’s transmission as a MISOME wiretap channel. Independent codebooks and linear beamforming yield a secrecy sum-rate built from the users’ individual achievable secrecy rates.

  • Equivalent wiretap channels: The multi-user MIMO system is modeled through equivalent MISOME wiretap channels with one intended receiver and K−1 cooperating eavesdroppers.The eavesdropper is strengthened to observe other confidential messages and cancel interference, producing a lower bound on secrecy rates.
  • Code construction: Independent scalar wiretap codebooks are stacked into a vector codeword and transmitted using linear precoding without additional binning.Each message is beamformed along its corresponding rank-one direction.
  • Achievable rates: The achievable secrecy sum-rate is the sum of the simultaneously achievable secrecy rates for all users.Theorem 1 obtains this rate by combining the code-construction and single-user secrecy-rate results.
  • Achievable rates: Each user’s achievable secrecy rate is determined by the difference between intended-user and eavesdropper information rates.The resulting expression depends on the respective signal-to-interference-plus-noise ratios.
  • Design tradeoff: Linear precoder design must balance maximizing the intended-user SINR against minimizing the eavesdropper SINR.This tradeoff reflects the simultaneous effects of interference and information leakage in the multi-user setting.

C. Achievable Secrecy Sum-Rates with Regularized Channel Inversion

Regularized channel inversion improves secrecy by balancing intended signal power against interference and information leakage. The paper derives secrecy sum-rate expressions for RCI precoding using a nonnegative regularization parameter.

  • Comparison: Compared with plain channel inversion, RCI achieves better performance, particularly at low SNR.Channel inversion can cancel leaked signals but may incur a poor sum-rate.
  • RCI design: RCI precoding trades off intended-user signal power against crosstalk at the other K−1 users.The same crosstalk that causes multi-user interference can also cause information leakage when unintended users are malicious.
  • Rate derivation: The RCI construction produces received signals and corresponding intended-user and eavesdropper SINRs for each message.These quantities are substituted into the achievable secrecy-rate expression to obtain the RCI secrecy sum-rate.
  • RCI design: The regularization parameter α improves the behavior of the channel inverse but introduces non-zero crosstalk terms.Thus, α controls a secrecy-relevant balance rather than simply eliminating interference.

IV. LARGE-SYSTEM ANALYSIS

The large-system analysis lets the paper characterize RCI secrecy performance when antennas and users grow together. It derives closed-form optimal regularization and studies its SNR dependence and asymptotic behavior.

  • Asymptotic setting: The analysis considers M and K tending to infinity with a fixed ratio, focusing on the tractable and important case K=M.The normalized regularization parameter is defined as ξ=α/K.
  • Asymptotic analysis: Closed-form expressions are derived for the optimal regularization parameter and RCI secrecy sum-rate in the large-system regime.The optimized RCI precoder is also compared with other linear schemes and its secrecy-related sum-rate loss is evaluated.
  • Asymptotic analysis: As K→∞, the intended-user and eavesdropper SINRs converge to non-random functions of ξ and the noise variance, identically across messages.This permits a deterministic large-system expression for the secrecy sum-rate.
  • Parameter optimization: The optimal normalized regularization parameter ξopt is obtained by maximizing the asymptotic secrecy sum-rate.The paper derives it as a stationary point of the asymptotic rate expression and verifies the maximum is nonnegative.
  • Parameter behavior: Unlike the no-secrecy case, ξ=1/ρ is not optimal because secrecy makes crosstalk appear twice in the sum-rate expression.ξopt decreases with SNR and tends to zero as ρ→∞, while remaining upper bounded at low SNR.

C. Optimal Secrecy Sum-Rate

The paper characterizes the optimal RCI secrecy sum-rate in the large-system regime and its dependence on SNR and user count. Despite cooperating eavesdroppers, positive secrecy remains achievable, with high-SNR per-user performance matching a single-user secrecy capacity.

  • Optimal secrecy sum-rate: The optimal secrecy sum-rate depends on the SNR ρ and the number of users K.The result is obtained from the closed-form RCI secrecy-rate expression and its high-SNR asymptote.
  • Scaling: In the large-system regime, the optimal secrecy sum-rate scales logarithmically with high SNR and linearly as K/2 with the number of users.This describes the asymptotic scaling of the optimized RCI secrecy sum-rate.
  • Cooperating eavesdroppers: Positive secrecy sum-rate remains achievable as the number of cooperating eavesdroppers grows without bound.The result relies on the number of transmit antennas also growing and exceeding the number of eavesdroppers for each message.
  • High-SNR comparison: At high SNR, RCI achieves a per-user secrecy rate equal to the secrecy capacity of a single-user system.The comparison uses the per-user RCI secrecy rate and the single-user MISOME secrecy-capacity benchmark.

D. Comparison to Other Linear Schemes

RCI with the secrecy-optimized regularization outperforms CI and matched-filter precoding, while retaining the same high-SNR scaling factor as optimal RCI without secrecy requirements.

  • The CI precoder’s secrecy sum-rate grows at most sublinearly in the large-system regime.CI cancels interference and information leakage but requires K ≤ M for its inverse to exist.
  • Matched-filter precoding achieves zero secrecy sum-rate in the large-system regime.Cooperating eavesdroppers can cancel interference, while intended users suffer substantial interference.
  • RCI with ξ = 1/ρ outperforms CI and matched-filter precoding but remains suboptimal compared with ξopt.The comparison covers CI, matched-filter, and the RCI value that maximizes sum-rate without secrecy requirements.
  • The secrecy requirements preserve the high-SNR linear scaling factor K/2 of optimal RCI without secrecy requirements.The secrecy-optimized RCI incurs a power penalty rather than changing the scaling factor.
  • 3.75 dB of additional transmitted power compensates for the secrecy loss of RCI with ξopt.The loss corresponds to a power factor of 64/27 ≈ 3.75 dB.

V. POWER ALLOCATION

The paper develops power allocation for RCI at fixed regularization, reformulates the non-convex objective using bounds and logarithmic variables, and obtains a monotonically convergent local optimization procedure.

  • The proposed algorithm optimizes the power allocation vector for a fixed regularization parameter and later supports joint optimization of power and regularization.The arbitrary-power RCI formulation induces user and eavesdropper SINRs and an achievable secrecy sum-rate objective.
  • The fixed-regularization power-allocation problem is non-convex before reformulation.A bound and the transformation ep_k = log p_k are used to construct a concave objective with affine constraints.
  • The reformulated optimization problem is convex because its objective is concave and its constraints are affine.The lower-bound construction converts the original problem into a convex optimization problem.
  • Algorithm 1 increases the objective monotonically and converges to a local optimum.Feasibility is preserved across successive subproblems, while the bound makes the objective monotonically increasing.

C. Proposed Precoding Scheme

The proposed precoder jointly optimizes the RCI regularization parameter and power allocation through an iterative procedure that converges to a locally optimal pair.

  • For fixed α, optimized power allocation outperforms RCI with αLS and equal power allocation.This provides the performance comparison motivating the joint precoding scheme.
  • The joint optimization problem remains non-convex even after logarithmic transformation of the power variables.The paper therefore solves it with a dedicated iterative algorithm.
  • Algorithm 2 alternately optimizes the regularization parameter α and the power allocation vector p.Each iteration first updates α and then updates p.
  • Algorithm 2 converges monotonically to a locally optimal pair (α, p).Simulations evaluate the resulting jointly optimized precoder.

VI. NUMERICAL RESULTS

Numerical results support the large-system regularization choice in finite systems and show that optimized power allocation improves secrecy performance, especially over equal-power RCI for four users.

  • The average normalized secrecy sum-rate difference between αLS and per-channel αFS(H) is below 2.4 percent for all tested user counts.Thus αLS can replace per-realization optimization with only a small performance loss.
  • The large-system analysis is accurate at low SNR for all tested K and remains accurate at higher SNR as K increases.The comparison uses finite-user simulations averaged over 10^3 channels.
  • RCI with αLS outperforms CI and RCI with α = K/ρ, while CI suffers a large performance loss for large K.CI cancels information leakage but pays for secrecy with a poor sum-rate.
  • At K = 32 and ρ = 25 dB, the simulated per-antenna secrecy loss is 0.59 bits, close to the analytically predicted 0.62 bits.The secrecy loss is nearly constant at high SNR for large K.
  • The jointly optimized precoder differs negligibly from RCI using αLS and optimized power, enabling a lower-complexity near-optimal implementation.The paper therefore supports fixing α to αLS and optimizing the power vector separately.
  • For K = 4, optimized power allocation improves secrecy performance over equal-power RCI by up to 20 percent.The gain persists at high SNR, where equal power is suboptimal and waterfilling is optimal for the CI-like regime.
  • For ρ ≥ 15 dB, RCI with power allocation achieves a per-user secrecy rate higher than optimal RCI-EP without secrecy requirements.At high SNR, its per-user secrecy rate can also become as large as CMISOME.

VII. CONCLUSIONS

The paper proposes an RCI-based linear precoder for secret communication under cooperating-user eavesdropping, with regularization and power allocation optimized for secrecy sum-rate. RCI with optimal regularization outperforms several linear schemes, while power allocation compensates secrecy-related loss; the analysis is mainly focused on K = M and leaves broader settings for future work.

  • VII. CONCLUSIONS: The proposed precoder combines RCI with a regularization parameter and power allocation vector chosen to maximize achievable secrecy sum-rate.The design addresses secret communication in a multi-user MIMO downlink with potentially malicious users.
  • VII. CONCLUSIONS: RCI with the optimal regularization parameter outperforms several other linear precoding schemes.The comparison concerns RCI with equal power allocation and the optimal regularization parameter.
  • VII. CONCLUSIONS: Its secrecy sum-rate has the same scaling factor as the optimum RCI precoder without secrecy requirements.The paper also notes that secrecy requirements cause a sum-rate loss relative to the non-secrecy setting.
  • VII. CONCLUSIONS: The proposed power allocation scheme compensates the secrecy-related sum-rate loss and increases secrecy sum-rate compared with RCI-EP.This extends the equal-power design with an allocation strategy tailored to secrecy performance.
  • VII. CONCLUSIONS: The analysis primarily considers K = M, while generalizing to arbitrary K and M remains ongoing; when K > M, secrecy sum-rate degrades due to increased interference and information leakage.User scheduling may improve SINR, but discarded users remain able to eavesdrop; artificial noise must remain harmless to intended receivers.
  • VII. CONCLUSIONS: The worst-case model assumes users cooperate and jointly eavesdrop, whereas noncooperating or partially eavesdropping users are left for future analysis.The transmitter is assumed unable to predict whether users are eavesdropping.
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