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Per-antenna Constant Envelope Precoding for Large Multi-User MIMO Systems

Saif Khan Mohammed, Erik G. Larsson

arXiv:1201.1634v2cs.IT

TL;DR

The paper studies whether multi-user interference suppression and array power gain are possible in a multi-user MIMO broadcast channel when every antenna must transmit a constant-envelope signal. It develops a low-complexity CE precoder and shows that, under mild channel conditions, fixed-user systems can achieve array gain as antennas increase, with performance close to an average-only power-constrained scheme. The paper also notes a practical amplifier-power comparison in which CE operation requires 4.0 dB less total transmit power in the stated sufficiently-large-N setting.

  • Problem

    The paper asks whether array power gain remains achievable under the stringent per-antenna constant-envelope constraint in a multi-user MIMO broadcast channel.

  • Method

    The paper proposes a low-complexity CE precoding scheme that selects per-antenna constant-envelope signals for multi-user transmission.

  • Results

    The proposed CE precoder requires usually less than 2 dB extra total transmit power versus an APC-only constrained broadcast channel for a given per-user ergodic information rate.

  • Takeaways & Limitations

    Under certain channel conditions, array power gain can be achieved despite the per-antenna CE constraint, while CE inputs support power-efficient amplifiers.

Abstract

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We consider the multi-user MIMO broadcast channel with $M$ single-antenna users and $N$ transmit antennas under the constraint that each antenna emits signals having constant envelope (CE). The motivation for this is that CE signals facilitate the use of power-efficient RF power amplifiers. Analytical and numerical results show that, under certain mild conditions on the channel gains, for a fixed $M$, array gain is achievable even under the stringent per-antenna CE constraint (essentially, for a fixed $M$, at sufficiently large $N$ the total transmitted power can be reduced with increasing $N$ while maintaining a fixed information rate to each user). Simulations for the i.i.d. Rayleigh fading channel show that the total transmit power can be reduced linearly with increasing $N$ (i.e., an O(N) array gain). We also propose a precoding scheme which finds near-optimal CE signals to be transmitted, and has O(MN) complexity. Also, in terms of the total transmit power required to achieve a fixed desired information sum-rate, despite the stringent per-antenna CE constraint, the proposed CE precoding scheme performs close to the sum-capacity achieving scheme for an average-only total transmit power constrained channel.

I. INTRODUCTION

The paper asks whether multi-user interference suppression and array power gain remain achievable under per-antenna constant-envelope transmission. It derives MUI expressions, proposes low-complexity CE precoding, and reports analytical and simulated power gains for fixed M and increasing N.

  • Motivation and problem: Per-antenna CE constrains every antenna to transmit a constant-amplitude signal independent of the channel realization.The constraint is motivated by the use of power-efficient nonlinear RF components.
  • Motivation and problem: The central question is whether MUI suppression and array power gain remain achievable under the more restrictive per-antenna CE constraint.The introduction states that no reported work had addressed this question.
  • Contributions: The paper derives MUI expressions and proposes a low-complexity CE precoder that minimizes MUI energy for each user.For each information-symbol vector, the scheme selects per-antenna CE transmit signals so that each user's MUI energy is small.
  • Analytical results: For fixed M and fixed user alphabets, sufficiently large N can guarantee arbitrarily low MUI energy at every user.The analytical result holds under certain mild channel conditions, including i.i.d. fading.
  • Numerical results: Fixed SINR can be maintained while reducing total transmitted power as N increases, and simulations for i.i.d. Rayleigh fading show linear power reduction.The simulated scaling corresponds to an O(N) array power gain under the per-antenna CE constraint.
  • Numerical results: The proposed CE precoder requires roughly 2.0 dB more total transmit power than the optimal APC sum-capacity scheme for sufficiently large N.This comparison is made at a given desired ergodic information sum-rate.

II. SYSTEM MODEL

The system is a Gaussian broadcast channel with an N-antenna base station serving M single-antenna users. Each antenna transmits a fixed-amplitude signal represented by a phase, while the received signals include multi-user transmission and AWGN.

  • Channel model: The channel gain from antenna i to user k is h_k,i, and the channel matrix H contains these gains.The user-k channel vector is h_k = (h_k,1, ..., h_k,N)^T.
  • Transmission constraint: The per-antenna CE constraint fixes each transmitted symbol's power at |x_i|^2 = P_T/N.This is more stringent than an average-only total-power constraint.
  • Transmission constraint: Under CE transmission, each antenna signal is parameterized by a phase angle θ_i, collected in the vector Θ.The received symbol at user k is the sum of channel-weighted phase-only signals plus AWGN.
  • Received signal: The received noise at each user is circularly symmetric complex Gaussian AWGN with variance σ^2.The paper analyzes operation in regimes where MUI is not critically limiting relative to receiver noise.
  • Information symbols: The model communicates scaled information symbols u_k from fixed unit-average-energy alphabets U_k to the M users.The analysis keeps M and the user alphabets fixed while allowing N to increase.

III. MUI ANALYSIS AND THE PROPOSED CE PRECODER

The section analyzes how increasing the number of antennas can reduce multi-user interference under per-antenna constant-envelope transmission. Under mild channel conditions and fixed user count, sufficiently large arrays can approximate desired received signal vectors arbitrarily closely.

  • CE signal construction: Constant-envelope transmission uses per-antenna phase angles to produce a known scaled version of each desired information symbol at the users.Each antenna transmits a constant-amplitude signal, with phases selected so the noise-free received signal matches the desired symbol up to a known constant.
  • MUI analysis: For fixed M, the achievable noise-free received-signal region expands and becomes denser as N increases.The region is described as a direct sum of N/M topologically similar sets under statistically identical antenna channels.
  • MUI analysis: For fixed M and finite information alphabets, sufficiently large N permits phase vectors whose per-user MUI energy is at most 2∆2 for any ∆>0.The result holds for channel sequences satisfying the stated mild conditions.
  • Proof strategy: The proof treats transmit phases as independent uniform random variables and uses convergence of the received-signal variables to a Gaussian limit.This probabilistic argument establishes positive probability of reaching arbitrarily small neighborhoods around every desired symbol vector.
  • Implication: Consequently, for fixed M and fixed information-symbol energies, per-user MUI energy can be made arbitrarily small by increasing N sufficiently.The theorem motivates practical CE precoders targeting low MUI levels.

B. Proposed CE Precoding Scheme

The proposed CE precoder formulates phase selection as a non-linear least-squares problem minimizing multi-user interference. Its coordinate-wise iterative solver achieves fast convergence and O(MN) complexity, while simulations show interference decreases with array size.

  • Optimization objective: The precoder chooses phase angles by minimizing the sum of users’ MUI energies for each desired information vector.This is posed as a non-linear least-squares problem over the transmit phases.
  • Optimization caveat: The non-convex optimization has multiple local minima, but as N/M increases, local minima have been observed to yield small objective values.This observation supports using gradient-based or related iterative approaches, while the paper proposes a faster alternative.
  • Iterative solver: The proposed iterative method is experimentally observed to converge significantly faster than gradient-descent-based methods with similar performance.The algorithm terminates after a predefined number of iterations.
  • Iterative solver: Each coordinate update modifies one phase angle while keeping all other phase angles fixed.The algorithm cycles through the N antenna phases within each iteration.
  • Simulation evidence: For fixed M, alphabets, and symbol energy, simulated ergodic per-user MUI energy decreases as N increases for both 16-QAM and Gaussian alphabets.The simulations use an i.i.d. Rayleigh fading channel.
  • Complexity: O(MN) complexity results from L N sub-iterations when L is treated as a constant.The paper reports minimal incremental objective reduction beyond a fixed iteration count for sufficiently large N/M.
  • Array-gain implication: For fixed M and MUI level, the achievable information-symbol energy increases linearly with N, implying an O(N) array power gain in i.i.d. Rayleigh fading.The paper connects this scaling to linearly reducing total transmit power while maintaining fixed SINR.

IV. ACHIEVABLE INFORMATION SUM RATE

The paper derives an achievable ergodic information sum-rate for the CE-constrained broadcast channel and evaluates it numerically for i.i.d. Rayleigh fading. The results show linear power reduction with array size and a small power gap to average-only constrained transmission.

  • Model scope: The analysis restricts the multi-user study to Gaussian information alphabets because the optimal alphabet is difficult to derive in this setting.The paper notes that Gaussian alphabets need not be optimal for maximum sum-rate.
  • Achievable rate: The paper derives an achievable ergodic information sum-rate for the broadcast channel under per-antenna CE transmission.The rate expression is obtained from the per-user achievable information-rate bound.
  • Power scaling: For fixed M and target per-user rate 2 bpcu, required PT/σ2 decreases by roughly 3 dB whenever N doubles at sufficiently large N.This corresponds to linear reduction of required total transmit power with increasing antenna count.
  • Power scaling: An O(N) array power gain is observed under the stringent per-antenna CE constraint in the i.i.d. Rayleigh fading channel.The conclusion is supported by the simulated achievable sum-rate curves.
  • Comparison: At 2 bpcu per user, the CE precoder requires only 1.7 dB more total transmit power than the average-only constrained broadcast channel.The comparison concerns the extra power required by the per-antenna CE constraint.
  • Comparison: The proposed CE precoder requires roughly 3 dB less PT/σ2 than the ZF phase-only precoder when N=100 and M=40.At very large N/M, the two precoders have similar performance.
  • Finite-array behavior: Achieving 95% of the limiting per-user information rate requires at least N=96 antennas for M=12 and N=192 antennas for M=24.These examples correspond to roughly eight times as many antennas as users.

VI. CONCLUSION

The paper shows that per-antenna constant-envelope transmission can achieve array power gain under mild channel conditions and proposes a low-complexity CE precoder. Simulations indicate that its power requirement is close to that of an average-only constrained broadcast channel, while CE amplifiers can offer a net power advantage.

  • VI. CONCLUSION: Per-antenna CE transmission achieves array power gain under certain mild channel conditions.The conclusion frames this as possible despite the stringent per-antenna constraint.
  • VI. CONCLUSION: The proposed CE precoding scheme has low complexity and is evaluated through exhaustive simulations for the i.i.d. Rayleigh fading channel.The conclusion introduces the scheme and identifies the channel model used for the simulations.
  • VI. CONCLUSION: The proposed CE precoder usually requires less than 2 dB extra transmit power relative to an APC-only constrained GBC for a given per-user ergodic information rate.The comparison is stated for the simulated scenarios summarized in the conclusion.
  • VI. CONCLUSION: 4.0 dB lesser total transmit power is obtained with CE inputs and power-efficient amplifiers than with highly linear amplifiers using high-PAPR inputs.This conclusion combines 4−6 times amplifier power efficiency with an extra 2 dB transmit-power requirement for CE signals.

CONVERGENCE (IN DISTRIBUTION) OF THE SEQUENCE

This section establishes convergence in distribution for normalized random variables and then uses that result to characterize the limiting behavior of the CE-precoding construction. The proof applies the Lyapunov central limit theorem and Slutsky’s theorem under the stated channel conditions.

  • CONVERGENCE (IN DISTRIBUTION) OF THE SEQUENCE: The proof reduces multivariate convergence to convergence of scalar projections for every vector Λ.The multivariate CLT result requires the projected cumulative distribution functions to converge for each Λ.
  • CONVERGENCE (IN DISTRIBUTION) OF THE SEQUENCE: Theorem 2 states that, with fixed M, the random-vector sequence z_N converges in distribution as N increases for channel sequences satisfying the conditions in (6).Its limit is a multivariate 2M-dimensional real Gaussian random vector.
  • CONVERGENCE (IN DISTRIBUTION) OF THE SEQUENCE: The phase-angle randomness makes ζ_N a sum of N independent random variables, enabling application of the Lyapunov-CLT.The proof explicitly identifies the independent phase angles as the source of randomness.
  • CONVERGENCE (IN DISTRIBUTION) OF THE SEQUENCE: The Lyapunov-CLT yields a zero-mean unit-variance Gaussian limit for ζ_N/C_N.The argument verifies the Lyapunov conditions and then applies the theorem to the normalized variable.
  • CONVERGENCE (IN DISTRIBUTION) OF THE SEQUENCE: Slutsky’s theorem converts the normalized convergence into convergence of ζ_N to a zero-mean Gaussian variable with the limiting variance.This uses the convergence of C_N established in the proof.

PROBABILITY OF THE BOX EVENT

The paper characterizes box-event probabilities for multivariate random variables and uses Gaussian convergence to establish positivity for sufficiently large antenna counts. Numerical examples examine CE precoding under Rayleigh fading.

  • A box C(∆, α) contains vectors whose coordinates lie within ∆ of the center α.The upper and lower coordinate limits are α_k + ∆ and α_k − ∆.
  • The probability of a box event is expanded by partitioning cases according to coordinates below their lower limits.The construction enumerates subsets of coordinates satisfying lower-limit conditions while the others satisfy upper-limit conditions.
  • As N →∞, the distribution function of z_N converges to that of a 2M-dimensional Gaussian vector Y.Y has independent zero-mean components with paired variances ck/2.
  • For any δ > 0 and ∆ > 0, sufficiently large N makes the relevant box-event probabilities strictly positive.The proof selects a common finite antenna threshold over the finitely many coordinate-subset cases.
  • Under i.i.d. CN(0, 1) Rayleigh fading, ergodic per-user MUI energy decreases as N increases for fixed user alphabets and symbol energy.The numerical setting includes fixed user count and unit symbol energy.
  • For fixed M = 12 and a fixed desired ergodic MUI energy level, the study evaluates the required E⋆ as N varies.Additional experiments report information-rate and power behavior under the same Rayleigh-fading setting.
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