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Energy-Efficient Precoding for Multiple-Antenna Terminals

E. V. Belmega, S. Lasaulce

arXiv:1011.4597v1cs.ITmath.IT

TL;DR

The paper studies how to choose energy-efficient precoding for MIMO channels, where slow fading makes the optimum generally nontrivial. It defines rate-to-power efficiency and examines the goodput-to-power ratio for slow fading; static and fast-fading efficiency is maximized at low power, while the slow-fading optimization remains open generally and is treated in special cases.

Abstract

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The problem of energy-efficient precoding is investigated when the terminals in the system are equipped with multiple antennas. Considering static and fast-fading multiple-input multiple-output (MIMO) channels, the energy-efficiency is defined as the transmission rate to power ratio and shown to be maximized at low transmit power. The most interesting case is the one of slow fading MIMO channels. For this type of channels, the optimal precoding scheme is generally not trivial. Furthermore, using all the available transmit power is not always optimal in the sense of energy-efficiency (which, in this case, corresponds to the communication-theoretic definition of the goodput-to-power (GPR) ratio). Finding the optimal precoding matrices is shown to be a new open problem and is solved in several special cases: 1. when there is only one receive antenna; 2. in the low or high signal-to-noise ratio regime; 3. when uniform power allocation and the regime of large numbers of antennas are assumed. A complete numerical analysis is provided to illustrate the derived results and stated conjectures. In particular, the impact of the number of antennas on the energy-efficiency is assessed and shown to be significant.

I. INTRODUCTION

The introduction frames energy efficiency as information delivered per power consumed and asks how this objective changes for MIMO channels. It presents slow-fading goodput-to-power as a link between outage-based communication and capacity-per-unit-cost thinking.

I. INTRODUCTION

  • The paper motivates rate-to-power as a natural efficiency measure because transmission rate is a benefit and transmit power is a cost.It frames the objective as maximizing information conveyed per Joule consumed.Source: §I · pp. 1–2
  • In static and SISO settings, energy efficiency is maximized at low transmit power, but the introduction asks whether that result carries over to MIMO.For slow fading, outage events create a tradeoff between successfully transmitted bits or blocks and power consumption.Source: §I · p. 2
  • The proposed goodput-to-power ratio links a pragmatic success-based energy-efficiency measure to an information-theoretic counterpart of capacity per unit cost for slow fading.The paper considers MIMO channels with channel distribution information at the transmitter and channel state information at the receiver.Source: §I · p. 3
  • Prior pragmatic SISO work uses a benefit function based on success probability, while this paper considers its complementary-outage choice for slow-fading GPR.The introduction also distinguishes related MIMO work that did not use the proposed outage-based measure.Source: §I · pp. 2–3

II. GENERAL SYSTEM MODEL

The system model is a point-to-point MIMO link in which the transmit covariance represents the power-allocation policy. It distinguishes channel regimes by how quickly the channel varies.

II. GENERAL SYSTEM MODEL

  • The received signal uses an n_r × n_t channel matrix with i.i.d. zero-mean, unit-variance complex Gaussian entries and complex Gaussian white noise.The channel matrix is perfectly known at the receiver, while the transmitter has only its statistics.Source: §II · pp. 3–4
  • The model represents power allocation across transmit antennas with the input covariance matrix Q, subject to a total power constraint.Q is defined as the expected outer product of the transmitted-symbol vector.Source: §II · p. 4
  • The model separates static, fast-fading, and slow-fading channels according to the rate at which the channel matrix varies.The time index is suppressed for clarity.Source: §II · p. 4

A. Case of static channels

For static MIMO channels, the paper identifies energy efficiency with achievable rate per unit cost and finds its maximum at zero transmit covariance. This low-power optimum can come with a rate too small for some applications.

A. Case of static channels

  • The static-channel energy-efficiency corresponds to an achievable rate per unit cost for the MIMO channel.The paper relates this formulation to the capacity-per-unit-cost framework.Source: A. Case of static channels · pp. 4–5
  • For static MIMO channels, energy efficiency is maximized when Q = 0, corresponding to transmission at very low power.The energy-efficiency function is linked to capacity per unit cost, with transmitted-symbol power serving as the cost.Source: A. Case of static channels · p. 5
  • The low-power optimum may yield an insufficient rate, making rate-constrained transmit-power minimization more relevant in some applications.The passage gives ocean sensors measuring a slowly varying temperature field as an example.Source: A. Case of static channels · p. 5

B. Case of fast fading channels

In fast-fading MIMO, each codeword experiences many channel realizations, enabling an ergodic transmission rate. The energy-efficiency optimum is again at zero transmit covariance.

B. Case of fast fading channels

  • Because the channel varies over a codeword, self-averaging allows the ergodic mutual information to be achieved on each codeword.The rate is associated with the chosen precoding strategy Q.Source: B. Case of fast fading channels · p. 5
  • The transmitter can maximize the fast-fading rate using channel statistics, while the receiver requires instantaneous channel knowledge.The stated transmitter information includes path loss, power-delay profile, and antenna-correlation profile.Source: B. Case of fast fading channels · p. 6
  • Fast-fading MIMO energy efficiency is maximized when Q = 0, so the optimum is at low transmit power.The paper contrasts this result with slow-fading MIMO, where outage events can make the answer different.Source: B. Case of fast fading channels · p. 6

A. General MIMO channels

For slow-fading MIMO channels, energy efficiency is the goodput-to-power ratio, and its maximizing precoder is generally difficult to determine. The paper proposes a threshold-based characterization: below the threshold GPR maximization matches outage minimization, while above it uniform power allocation is optimal and using all available power is generally suboptimal.

A. General MIMO channels

  • For slow-fading channels, outage probability measures the chance that a target transmission rate is not reached, and achievable rates lie below ergodic capacity.The channel remains constant over a codeword, so the averaging effect associated with fast fading does not occur.Source: A. General MIMO channels · pp. 6–7
  • The GPR is generally maximized at a nonzero covariance, unlike energy-efficiency in static and fast-fading channels.In slow fading, GPR is expected throughput divided by average consumed transmit power.Source: A. General MIMO channels · p. 7
  • Above a power threshold, uniform power allocation is conjectured optimal for GPR, and using all available power is generally suboptimal.Below the threshold, maximizing GPR is equivalent to minimizing outage probability; the conjecture is validated in the paper’s solved special cases.Source: A. General MIMO channels · p. 7
  • Restricting the search to diagonal precoding matrices loses no optimality, although GPR is generally not concave in the covariance.Under uniform power allocation, quasi-concavity is presented as a conjecture; the paper relates its resolution to Telatar’s outage-minimization conjecture [18].Source: A. General MIMO channels · p. 8

B. MISO channels

For MISO channels, the paper characterizes energy-efficient power allocation across transmit antennas and studies uniform allocation. The optimum spreads power uniformly over a subset of antennas, with the selected subset size depending on available power; in some regimes, using all available power is suboptimal.

B. MISO channels

  • The paper gives an optimal-precoding characterization for MISO channels using threshold values cℓ and νnt.The proposition defines cℓ through an equation involving i.i.d. Gaussian variables and νnt as the solution of another equation.Source: B. MISO channels · p. 10
  • The GPR-maximizing MISO allocation spreads available power uniformly over a subset of antennas, whose size increases with transmit power.With i.i.d. channel entries, the identity of the selected antennas does not matter; above a threshold, saturating the power constraint is suboptimal.Source: B. MISO channels · p. 10
  • Under uniform power allocation, the MISO GPR is quasi-concave and has a unique maximum.The proof uses the sigmoidal goodput function, which has a unique inflection point.Source: B. MISO channels · p. 11
  • In the SISO case, the unique GPR-maximizing power is p*=c=σ2(2R−1).The paper contrasts its statistical-CSI setting with, which assumes instantaneous CSI at transmitters; its power-control interpretation adapts to path loss, not fast fading.Source: B. MISO channels · p. 12

V. SLOW FADING MIMO CHANNELS IN ASYMPTOTIC REGIMES

The paper studies finite MIMO systems in low- and high-SNR regimes, then considers uniform power allocation in large-antenna regimes.

V. SLOW FADING MIMO CHANNELS IN ASYMPTOTIC REGIMES

  • For uniform power allocation, the paper proves quasi-concavity of the GPR in asymptotic regimes where at least one antenna dimension is large and SNR is finite.It also analyzes finite systems in low- and high-SNR operating regimes.Source: §V · p. 12

A. Extreme SNR regimes

The section fixes channel parameters and considers the GPR and optimal power allocation in the low- and high-SNR limits.

A. Extreme SNR regimes

  • With channel parameters fixed, the low-SNR limit is σ2→+∞ and the high-SNR limit is σ2→0.The section considers both the GPR and the optimal power allocation problem in these regimes.Source: A. Extreme SNR regimes · p. 12

1) Low SNR regime:

The low-SNR proposition is introduced after prior results connecting outage-probability behavior to beamforming. At high SNR, uniform power allocation maximizes energy efficiency, though using all available power is not always energy-efficient.

1) Low SNR regime:

  • At low SNR, prior results show that outage probability is Schur-concave in power allocation, implying beamforming maximizes outage probability.Source: 1) Low SNR regime · p. 12
  • The text introduces a low-SNR energy-efficiency proposition but does not provide its complete conclusion here.Source: Appendix E · p. 13

2) High SNR regime:

  • At high SNR, energy efficiency is maximized by uniform power allocation because the function is Schur-convex in the power allocation vector.Source: 2) High SNR regime · p. 13
  • The high-SNR optimal covariance structure matches the one that minimizes outage probability.Source: 2) High SNR regime · p. 13

B. Large MIMO channels

This section examines energy-efficiency in large MIMO settings, covering the goodput-to-power ratio under uniform power allocation in three asymptotic regimes and special-case optimality results for static MIMO. It notes that optimal power and limiting energy-efficiency depend on the growing antenna dimension, while static-case outcomes depend on antenna configuration and SNR.

B. Large MIMO channels

  • With fixed n_t and n_r→∞, the optimal power tends to zero and Γ_UPA tends to 1, yielding infinite energy-efficiency.This is attributed to channel hardening.Source: B. Large MIMO channels · p. 14
  • With n_t→∞ and fixed n_r, the large-system numerator is sigmoidal, so the ratio is quasi-concave but not generally concave; optimal power approaches the threshold p̃_b.As n_t grows, Q(α_b(p)) approaches a step function around p̃_b.Source: B. Large MIMO channels · pp. 14–15
  • When first-order terms can be neglected, the solution is given by μI(p)=0; because μI(0)=0 and μI increases with p, the unique solution is p̃c=0, yielding the trivial solution p*.Source: B. Large MIMO channels · p. 15

PROOF OF PROPOSITION 3.1

PROOF OF PROPOSITION 4.1

PROOF PROPOSITION 4.4

  • In Proposition 4.4, each gℓ increases then decreases, with its maximum at xℓ, determined by the unique positive solution yℓ of φℓ(y) = 0.The proof also states yℓ ≤ ℓ and xℓ ≥ c.Source: PROOF PROPOSITION 4.4 · p. 24

COUNTER-EXAMPLE, TISO

  • For nt = 2 and nr = 1, beamforming power allocation maximizes energy-efficiency under power constraint c1, but ΓTISO is not quasi-concave in (p1, p2).The proof identifies nonconvex upper-level sets as the basis for the non-quasi-concavity result.Source: Appendix D · pp. 24–25

EXTREME SNR CASES, GPR

  • At high SNR, uniform power allocation (UPA) is optimal; at low SNR, the optimal power allocation is beamforming.The low-SNR result is stated for the GPR energy-efficiency setting.Source: Appendix E · pp. 25–26
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