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PAR-Aware Large-Scale Multi-User MIMO-OFDM Downlink

Christoph Studer, Erik G. Larsson

arXiv:1202.4034v3cs.IT

TL;DR

OFDM creates high PAR, increasing RF linearity, cost, and power-efficiency requirements in large-scale MU-MIMO downlinks. The paper introduces PMP, jointly optimizes precoding, modulation, and PAR reduction, and develops FITRA. Experiments show more than 11 dB PAR reduction versus conventional precoding without significant out-of-band interference.

  • Problem

    OFDM-based large-scale MU-MIMO downlinks face high PAR and therefore require costly, power-inefficient linear RF components.

  • Method

    PMP jointly performs MU precoding, OFDM modulation, and PAR reduction through convex optimization, with FITRA solving the large problem efficiently.

  • Results

    More than 11 dB PAR reduction versus conventional precoding was achieved without significant out-of-band interference.

  • Takeaways & Limitations

    PMP substantially alleviates RF linearity requirements and affects only base-station processing, supporting deployment in existing MIMO-OFDM systems.

  • Takeaways & Limitations

    Analytical PAR guarantees, imperfect-channel-state-information analysis, and further FITRA complexity reductions are left for future work.

Abstract

from arXiv · show

We investigate an orthogonal frequency-division multiplexing (OFDM)-based downlink transmission scheme for large-scale multi-user (MU) multiple-input multiple-output (MIMO) wireless systems. The use of OFDM causes a high peak-to-average (power) ratio (PAR), which necessitates expensive and power-inefficient radio-frequency (RF) components at the base station. In this paper, we present a novel downlink transmission scheme, which exploits the massive degrees-of-freedom available in large-scale MU-MIMO-OFDM systems to achieve low PAR. Specifically, we propose to jointly perform MU precoding, OFDM modulation, and PAR reduction by solving a convex optimization problem. We develop a corresponding fast iterative truncation algorithm (FITRA) and show numerical results to demonstrate tremendous PAR-reduction capabilities. The significantly reduced linearity requirements eventually enable the use of low-cost RF components for the large-scale MU-MIMO-OFDM downlink.

I. INTRODUCTION

Large-scale MU-MIMO can serve many users with excess BS antennas, but practical OFDM downlinks face high PAR and costly RF linearity requirements. The paper proposes PMP, a convex-optimization framework implemented efficiently by FITRA at the BS.

  • I. INTRODUCTION: Large-scale MIMO uses many BS antennas to serve multiple users concurrently, with the antenna count exceeding the number of users.This architecture can also reduce transmitter power consumption and simplify MUI suppression.
  • I. INTRODUCTION: Practical large-scale MIMO needs low-cost, low-power RF components, but existing constant-envelope precoding fixes PAR at unity despite trade-offs with error rate and power-amplifier efficiency.The stated trade-off motivates alternatives that do not force the minimum PAR in every operating condition.
  • I. INTRODUCTION: OFDM supports frequency-selective channels and several spectral-management functions, but its high PAR requires linear RF components that are costlier and less power efficient.These requirements become especially consequential for base stations with hundreds of antennas.
  • I. INTRODUCTION: PMP jointly performs MU precoding, OFDM modulation, and PAR reduction as a convex optimization problem at the base station.The scheme leaves terminal processing untouched.
  • I. INTRODUCTION: FITRA is an optimization algorithm designed to solve PMP efficiently for the large problem dimensions of large-scale MU-MIMO-OFDM systems.The paper also evaluates PAR, error-rate performance, out-of-band radiation, and comparisons with conventional precoding.

II. PRELIMINARIES

The system is a frequency-selective OFDM MU-MIMO downlink with many BS antennas serving single-antenna users. Precoding produces antenna-oriented signals, which are transformed by IDFT and cyclic-prefix insertion before transmission.

  • II. PRELIMINARIES: The base station has N transmit antennas and serves M much smaller than N independent single-antenna users over W OFDM tones.User symbols occupy designated data tones, while unused tones are set to zero for spectrum shaping.
  • II. PRELIMINARIES: Precoding generates transmit vectors x_w from user symbols to remove multi-user interference, followed by power normalization before transmission.The normalization ensures unit transmit power and is used in the simulations.
  • II. PRELIMINARIES: The tone-indexed transmit vectors are reordered into one frequency-domain signal per antenna, then transformed to time-domain samples using the IDFT.Parallel-to-serial conversion and cyclic-prefix insertion follow the IDFT; the prefix avoids inter-symbol interference.
  • II. PRELIMINARIES: On each OFDM tone, the received vector follows y_w = H_w x_w + n_w, with channel matrix H_w and additive Gaussian noise.Each terminal performs OFDM demodulation to recover its received frequency-domain signals.
  • II. PRELIMINARIES: The depicted downlink connects a multi-antenna base station to independent single-antenna terminals through the proposed processing chain.The BS-side chain includes MU precoding, OFDM modulation, and PAR reduction.

B. MU Precoding Schemes

MU precoding maps user symbols to antenna transmissions while suppressing inter-user interference. The section introduces linear schemes, especially LS precoding, as a convex-optimization baseline and motivates the proposed formulation.

  • B. MU Precoding Schemes: Perfect transmit-side channel knowledge enables linear precoding of each tone as x_w = G_w s_w.The precoding matrix G_w is selected using the channel matrix H_w.
  • B. MU Precoding Schemes: LS precoding uses the channel pseudoinverse and perfectly removes MUI, yielding M independent single-stream systems.Its received relation becomes y_w = s_w + n_w.
  • B. MU Precoding Schemes: LS precoding is equivalently the minimum-ℓ2-norm transmit vector satisfying the per-tone signal constraint.This convex formulation inspires the proposed MU-MIMO-OFDM optimization problem.
  • B. MU Precoding Schemes: MF precoding is another linear baseline, but it generally cannot remove MUI because its effective channel is not diagonal.It can nevertheless be competitive in some large-scale-MIMO regimes and remove MUI in the large-antenna limit.
  • B. MU Precoding Schemes: The transmitter-side IDFT creates large dynamic-range OFDM signals susceptible to saturation and clipping, motivating PAR reduction and linear RF components.The defined PAR lies between 1 and 2W, with values near one preferred to reduce hardware nonlinearity distortion.

2) PAR-Reduction Schemes for OFDM:

OFDM PAR reduction must coexist with MU interference removal and conventional receiver processing. The proposed approach exploits excess BS-side degrees of freedom and relaxes a nonconvex dynamic-range problem into a convex one.

  • 2) PAR-Reduction Schemes for OFDM:: Existing single-user PAR-reduction methods do not transfer straightforwardly to MU-MIMO because MU systems require precoding for MUI removal.Earlier MU schemes based on Tomlinson-Harashima precoding require specialized terminal processing.
  • 2) PAR-Reduction Schemes for OFDM:: The proposed scheme reduces PAR using excess BS transmit antennas while remaining transparent to receivers and compatible with existing MIMO-OFDM systems.It requires channel-state information at the transmitter but no special signal processing at mobile terminals.
  • 2) PAR-Reduction Schemes for OFDM:: PMP jointly performs MU precoding, OFDM modulation, and PAR reduction by exploiting the available degrees of freedom in large-scale MU-MIMO systems.The construction is developed from a simplified narrowband system before extension to the full MU-MIMO-OFDM downlink.
  • 2) PAR-Reduction Schemes for OFDM:: In the simplified system, the underdetermined constraint s = Hx provides infinitely many transmit vectors that remove MUI when M < N.This solution multiplicity supplies the degrees of freedom for selecting hardware-friendlier transmissions.
  • 2) PAR-Reduction Schemes for OFDM:: The dynamic-range formulation is nonconvex because it imposes both lower and upper magnitude bounds on every transmit entry.Replacing the lower-and-upper bound with only an upper bound yields a convex problem minimizing the largest entry magnitude.
  • 2) PAR-Reduction Schemes for OFDM:: The convex ℓ∞-magnitude formulation cannot produce higher PAR than LS precoding, and simulations confirm substantially lower PAR than LS.The comparison follows from minimizing the largest transmit-vector entry while satisfying the signal constraint.

2) Benefits of Large-Scale MIMO:

With more transmit antennas than users, the underdetermined MU-precoding constraints offer many transmit vectors, enabling low-PAR solutions. The joint PMP formulation preserves conventional OFDM demodulation while targeting low-PAR time-domain samples.

  • Benefits of Large-Scale MIMO:: Proposition 1 states that the P-INF solution generally has N−M+1 entries attaining its maximum magnitude.The remaining M−1 entries may have smaller magnitude.
  • Benefits of Large-Scale MIMO:: For a fixed number of users and N→∞, the P-INF PAR bound approaches 1, yielding nearly constant-envelope signals.This applies when the base station has substantially more transmit antennas than users.
  • Benefits of Large-Scale MIMO:: Applying P-INF independently to time-domain samples would prevent conventional OFDM demodulation from equalizing intersymbol interference.Such an approach would require sophisticated equalization at each terminal.
  • Benefits of Large-Scale MIMO:: The cited proposition excludes certain channel matrices, including instances with collinear columns.This is an explicit condition on the applicability of the stated result.
  • Benefits of Large-Scale MIMO:: PMP jointly performs MU precoding, OFDM modulation, and PAR reduction through a convex optimization problem.The optimization minimizes the ℓ∞-norm of time-domain samples while enforcing precoding and shaping constraints.

C. Relaxation of (PMP)

The high-dimensional PMP problem is relaxed into a regularized least-squares formulation to enable efficient optimization. The regularization parameter controls the trade-off between constraint fidelity and PAR reduction.

  • C. Relaxation of (PMP): PMP is transformed into a compact linear system b=Ca, where C includes the constraint structure and inverse Fourier transforms.The vector b concatenates transmitted symbols and zero vectors for inactive tones.
  • C. Relaxation of (PMP): Relaxing b=Ca supports efficient algorithm development and is motivated by comparable thermal-noise, MUI, and out-of-band-interference effects at medium-to-low SNR.Small relaxation values η are stated not to significantly degrade performance in that regime.
  • C. Relaxation of (PMP): PMP-L is an ℓ∞-norm-regularized least-squares problem whose λ parameter trades constraint fidelity against PAR reduction.The paper investigates the associated trade-offs in its simulations.
  • C. Relaxation of (PMP): The formulation is converted to real-valued variables, with structural details of the matrix C omitted for simplicity.The paper notes that the complex-to-real transformation is straightforward.

D. Extensions of PMP

PMP extends beyond its original MU-MIMO-OFDM setting through alternative precoders, power constraints, tone reservation, and point-to-point MIMO configurations. These extensions alter available degrees of freedom or trade interference suppression against noise enhancement.

  • D. Extensions of PMP: Replacing PMP’s precoding constraints with matrices Pw generalizes the method to various linear precoders.This can incorporate imperfect transmitter channel-state information, for example through an MMSE precoder.
  • D. Extensions of PMP: The generalized precoder choice trades MUI removal against noise enhancement and can account for imperfect channel-state information.The passage specifically identifies MMSE precoding as one possible implementation.
  • D. Extensions of PMP: Adding ||a||2≤Pmax imposes a predefined transmit-power upper bound while preserving PMP’s convexity.The existing algorithm cannot straightforwardly handle this peak-power constraint.
  • D. Extensions of PMP: Combining PMP with tone reservation reserves non-data tones for PAR reduction, providing additional degrees of freedom expected to improve PAR reduction.Only a subset Td of tones carries data, while the remaining tones are reserved.
  • D. Extensions of PMP: In point-to-point MIMO, PMP can use more flexible precoding matrices because the detector separates streams and MUI need not be removed.The extension applies when channel-state information is available at the transmitter.

5) Application to Single-Carrier Systems:

The PMP idea can be adapted to single-carrier large-scale MIMO systems with intersymbol interference by replacing frequency-domain constraints with block channel constraints. The resulting vector contains PAR-reduced time-domain samples, while detailed investigation is deferred.

  • 5) Application to Single-Carrier Systems:: For single-carrier systems with ISI, PMP can replace P-INF constraints with block-structured channel constraints.The channel matrices correspond to delay taps, and the construction depends on the block-transmission pre- and post-ambles.
  • 5) Application to Single-Carrier Systems:: The block formulation uses delay-indexed channel matrices bHt, information symbols ŝq, and Q≥D transmitted symbols per block.Here D denotes the number of channel taps represented in the block structure.
  • 5) Application to Single-Carrier Systems:: The optimization minimizes the ℓ∞-norm of x, whose entries are the time-domain samples transmitted over a block.The vector x aggregates Q PAR-reduced samples.
  • 5) Application to Single-Carrier Systems:: The proposed algorithmic setting is motivated by the high computational complexity of interior-point methods for large PMP problems.The paper instead develops efficient methods for the associated regularized optimization.
  • 5) Application to Single-Carrier Systems:: ISTA provides low-complexity iterations based on matrix-vector multiplications and simple shrinkage operations, enabled by structure in C and C^H.The cited framework is used as the basis for the algorithm for PMP-L.

2) Fast Version of ISTA:

The paper develops FITRA by adapting accelerated first-order methods to efficiently solve the ℓ∞-norm-regularized optimization underlying PMP-L.

  • Fast Version of ISTA:: FISTA improves ISTA’s convergence by evaluating the proximal map using a linear combination of the previous two iterates.This accelerates the convergence rate from approximately O(1/k) to O(1/k^2).
  • Fast Version of ISTA:: FITRA solves PMP-L through a first-order algorithm based on an efficiently evaluated proximal map.The derivation uses the Lagrangian variant of P-INF and computes the Lipschitz constant for the smooth term.
  • Fast Version of ISTA:: Algorithm 1 combines the truncation procedure with convergence-improving techniques from FISTA and is called the fast iterative truncation algorithm.The algorithm returns the estimate xK after at most K iterations.
  • Fast Version of ISTA:: The proximal map for P-INF-L lacks a simple closed-form solution, so FITRA uses scalar optimization followed by element-wise truncation.The truncation clips each element to the interval [−α, α].

2) Convergence Rate:

FITRA inherits an analytical convergence guarantee from ISTA/FISTA results, while continuation strategies remain a possible route to lower computational complexity.

  • Convergence Rate:: FITRA’s convergence rate is characterized analytically as a consequence of established ISTA/FISTA convergence results.The result applies to FITRA as specified in Algorithm 1.
  • Convergence Rate:: The objective function is F(x) = λ∥x∥∞ + ∥s − Hx∥2, with xk denoting the FITRA estimate and x∗ the solution to P-INF-L.The bound is expressed in terms of the iteration index k and the initial point x0.
  • Convergence Rate:: Continuation strategies may reduce FITRA’s computational complexity, but investigating them is left for future work.The paper identifies this as an unaddressed algorithmic improvement.

C. Related Work

The simulations compare PMP with conventional precoding under a large-scale MU-MIMO-OFDM setup, emphasizing PAR, SER, OBR, and tunable trade-offs.

  • C. Related Work: PMP, LS, MF, and LS+clip are compared using PAR, SER, and OBR-related measures.PAR performance is the maximum PAR met for 99% of transmitted OFDM symbols, while the SNR operating point is the minimum SNR achieving 1% SER.
  • C. Related Work: Simulations use N = 100 BS antennas, M = 10 single-antenna terminals, W = 128 OFDM tones, and a 4-tap frequency-selective channel.The channel matrices have i.i.d. circularly symmetric Gaussian entries, and 108 tones carry data.
  • C. Related Work: With λ = 0.25 and target PAR 4 dB for LS+clip, PMP achieves PAR = 1.9 dB versus 10.4 dB for LS and 10.1 dB for MF.The corresponding OBR values are −52.9 dB for PMP, −∞ dB for LS and MF, and −11.9 dB for LS+clip.
  • C. Related Work: PMP incurs a 1 dB SNR-performance loss compared with LS precoding, while MF and LS+clip also lose performance because of residual MUI and normalization.PMP’s PAR reduction therefore involves a PAR/SER trade-off.
  • C. Related Work: LS and MF have OBR = 0 because they operate independently on each OFDM tone, whereas PMP and LS+clip generally have OBR > 0.The simulations distinguish spectral preservation from PAR reduction when comparing the schemes.
  • C. Related Work: Increasing λ reduces PAR while gracefully degrading the SNR operating point, whereas decreasing λ approaches LS performance.LS+clip can have a similar or better PAR/SNR trade-off at less aggressive target PAR values, but its OBR increases substantially as PAR is reduced.

C. Summary of PMP Properties

PMP jointly reduces PAR while exposing tunable trade-offs with SNR and out-of-band radiation. Its PAR performance improves with more transmit antennas and channel taps, while FITRA provides an efficient numerical solution to the underlying convex problem.

  • PMP performance: PMP reduces PAR by more than 11 dB compared to LS and MF precoding at CCDF(PAR) = 1%.LS+clip achieves 4 dB PAR deterministically but does not preserve the same spectral constraints.
  • Trade-offs: Increasing λ reduces PAR but causes a graceful degradation of the SNR operating point for K = 2000 FITRA iterations.Decreasing λ approaches LS-precoding performance.
  • Trade-offs: LS+clip can outperform PMP in the high-PAR regime, but it creates substantial out-of-band interference by ignoring shaping constraints.PMP has significantly lower OBR and degrades more gracefully as PAR is reduced.
  • Trade-offs: The regularization parameter λ and maximum FITRA iterations K jointly determine PMP's PAR, OBR, and SNR performance.Reducing K increases OBR, while K = 2000 makes FITRA one-to-two orders of magnitude more complex than LS precoding.
  • Antenna and channel configuration: More transmit antennas and more non-zero channel taps improve PMP's PAR performance by providing additional exploitable degrees of freedom.LS-precoding PAR is virtually independent of the number of channel taps.
  • Framework and implementation: PMP is formulated as a convex optimization problem and solved numerically with the fast iterative truncation algorithm FITRA.The framework jointly performs precoding, OFDM modulation, and PAR reduction, and can yield constant-envelope OFDM signals as N →∞.
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