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The PAPR Problem in OFDM Transmission: New Directions for a Long-Lasting Problem

Gerhard Wunder, Robert F. H. Fischer, Holger Boche, Simon Litsyn, Jong-Seon No

arXiv:1212.2865v2cs.ITmath.MGmath.PR

TL;DR

The paper addresses the persistent PAPR problem in OFDM as energy-efficiency requirements and system complexity increase. It develops a broader framework combining alternative metrics, large-deviation theory, derandomization, and Banach-space geometry. The paper outlines theoretical limits, design directions, and practical challenges including rate loss and noisy compressed-sensing recovery.

  • Problem

    The PAPR problem remains a major multicarrier-communications problem because stricter energy-efficiency requirements and MIMO or multiuser settings impose unresolved design challenges.

  • Method

    The paper combines alternative performance metrics with large-deviation theory, derandomization, and selected Banach-space and compressed-sensing concepts.

  • Results

    The paper presents a unified perspective on PAPR-related metrics, theoretical foundations, current limits, and future directions rather than a single numerical benchmark result.

  • Takeaways & Limitations

    PAPR research should be considered within broader metric, system-design, and mathematical frameworks that include practical constraints such as MIMO, multiuser operation, and capacity.

Abstract

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Peak power control for multicarrier communications has been a long-lasting problem in signal processing and communications. However, industry and academia are confronted with new challenges regarding energy efficient system design. Particularly, the envisioned boost in network energy efficiency (e.g. at least by a factor of 1000 in the Green Touch consortium) will tighten the requirements on component level so that the efficiency gap with respect to single-carrier transmission must considerably diminish. This paper reflects these challenges together with a unified framework and new directions in this field. The combination of large deviation theory, de-randomization and selected elements of Banach space geometry will offer a novel approach and will provide ideas and concepts for researchers with a background in industry as well as those from academia.

I. ENERGY EFFICIENCY IN MOBILE COMMUNICATION NETWORKS: A DRIVING SOURCE FOR INNOVATION

Energy efficiency pressures are reshaping the PAPR problem in OFDM, especially as HPA efficiency, MIMO, and multiuser constraints become more demanding. The paper proposes reviewing PAPR through broader metrics and combining large-deviation, derandomization, compressed-sensing, and Banach-space perspectives.

  • 50% of network operational cost can come from energy use, making energy efficiency a major driver for future mobile communications.
  • HPA efficiency is directly tied to input-signal PAPR, which creates serious difficulties for OFDM and constrains uplink adoption and downlink coverage.
  • A projected 1000-fold increase in network energy efficiency would tighten component-level requirements and reduce the acceptable efficiency gap with single-carrier transmission.
  • MIMO and multiuser systems add parallel-signal and side-constraint challenges that broaden peak-power control beyond the conventional single-signal setting.
  • The paper argues that PAPR should be reviewed alongside alternative metrics that can guide algorithm design and affect higher-layer parameters such as resource allocation.
  • Compressed sensing exploits sparsity in clipped OFDM signals, while Banach-space geometry is presented as a route to understanding limits and developing algorithms.
  • The article organizes these issues around fundamentals, challenges, trends, potential solutions, and a unified framework based on derandomization.

II. THE DESIGN CHALLENGE

The OFDM design challenge is to control large envelope fluctuations while jointly balancing HPA efficiency, nonlinear distortion, signal quality, and spectral constraints. The section motivates replacing or supplementing PAPR with application-specific metrics and highlights contrasting scaling, modeling, and capacity results.

  • II. THE DESIGN CHALLENGE: Large subcarrier sums force linear HPA operation over a range N times the average power, wasting supply power when practical subcarrier counts are large.
  • II. THE DESIGN CHALLENGE: HPA backoff, predistortion, and peak-reduction processing form a joint optimization problem because power efficiency and nonlinear distortion must be traded against one another.
  • Alternative metrics: The conventional PAPR objective may need supplementation when energy consumption, low-precision amplification, capacity, or out-of-band power are the relevant design concerns.
  • Alternative metrics: The clipping level can follow a log log(N) law for practically almost constant clipped energy, contrasting with the log(N) PAPR scaling.
  • Alternative metrics: SDR and EVM measure in-band nonlinear distortion, while CM and AOM target amplifier-specific distortion behavior.
  • Alternative metrics: Strong channel coding makes SDR and EVM unsuitable performance characteristics, shifting attention toward end-to-end OFDM capacity.
  • Alternative metrics: Capacity remains unresolved for continuous-time peak-power-limited AWGN channels and OFDM over such channels, while statistical nonlinear-device models provide only lower bounds.
  • Modeling considerations: Gaussian-process theory matches simulated clip-duration distributions relatively well for large clipping levels, but EVM and SER do not match well under the same approximation.

IV. APPROACHING THE log (N) BARRIER: DERANDOMIZATION

Large deviation theory establishes log(N) as the fundamental PAPR scaling barrier for independent-subcarrier OFDM and supports derandomized peak-power control near that barrier.

  • Derandomization: Derandomization uses the LDP and conditional-expectation ideas to construct low-PAPR choices without relying on full randomness.The framework is presented as a route toward provably more efficient peak-power-control algorithms.
  • The log(N) barrier: PAPR of multicarrier signals with statistically independent subcarriers concentrates around log(N) with very high probability.This concentration establishes the log(N) barrier and a theoretical scaling law.
  • Illustration: Figure 3 compares the PAPR CCDFs of uncoded data, SLM, and derandomization.The figure illustrates the LDP’s relevance to peak-power-control performance.
  • The LDP: The LDP describes PAPR tails whose logarithm decreases linearly beyond a cutoff near log(N), up to O(log[log(N)]) terms.The formulation uses [x]−:=min(0,x) to express the tail behavior.
  • The LDP: The LDP can assess peak-power control schemes by analyzing successive estimates of PAPR for independently randomized data sequences.The paper connects this analysis to measure concentration through martingale and bounded-difference arguments.

B. Multiple signal representation and partitioning

Multiple signal representation and partitioning generate alternative transmit sequences carrying the same information, enabling selection or mapping toward lower PAPR while exposing complexity and independence constraints.

  • Multiple signal representation: SLM and PTS generate multiple redundant candidates and transmit the candidate with the best metric.Suitable transforms or mappings aim to make the candidates’ metrics statistically independent, and alternative metrics can replace PAPR.
  • Multiple signal representation: SLM’s LDP analysis assumes independent PAPR among U alternatives, but this assumption becomes unreliable when the number of alternatives is large.PTS is more critically affected because its transforms operate on data subsets; side information is another unresolved issue.
  • Partitioning: Complete partitioning maps transmit sequences between equal-information cells, with side information treated as part of the transmitted sequence and protected by embedded coding.The mapping should move high-PAPR sets into marked subsets containing at least one sequence below the threshold.
  • Partitioning: Sequence balancing encodes side information in BPSK sign vectors, inserting binary correlation into the transmitted stream.The method combines modified information and side-information sequences into the transmitted codeword.
  • Partitioning: Code strength measures the ability to realize many sign changes over any subvector and is related to dual distance.Binary codes with this property can provide a substantial fraction of the theoretically possible performance gain.
  • Complexity: Full cell-selection search is too complex in many situations, motivating the subsequent lower-complexity approach.

C. Derandomization of choices

Derandomization converts probabilistic cell or sign-vector selection into constructive algorithms that approach the log(N) barrier, while performance can involve rate loss and remains open to improvement.

  • Derandomization: Derandomization uses a probability model for cell selection to avoid full search while achieving PAPR reduction close to the log(N) barrier.The approach is based on the probabilistic method associated with Spencer.
  • Derandomization: The binary-correlation example progressively fixes random sign changes using conditional expectations for a given information sequence.This procedure reduces randomness while retaining the desired bound on the resulting PAPR.
  • Derandomization: For sufficiently large N, the LDP yields y0 ≤ log(N), while Chernoff and moment bounds provide alternative complexity-performance choices.Chernoff bounds offer good performance and low complexity, whereas moment bounds have better tail properties but higher complexity.
  • Performance: In a 128-subcarrier system, sequence-balancing simulations match the predicted cutoff log(128) ≈ 6.8 dB.The comparison includes unrestricted sign changes and a strength-10 dual BCH code with 18 redundant subcarriers.
  • Performance: At 10−3 outage probability, the HPA-backoff result matches the LDP analysis but costs 1 bit/dimension of rate loss.The paper identifies the rate–PAPR trade-off as insufficiently investigated.
  • Extensions: Combining derandomization with partitioning improves standard methods and extends to alternative metrics such as SER.Reported asymptotic results include a clipping level of log log(N) for zero clipped energy.
  • Open issues: Further work includes modeling correlations between sampling points, incorporating other metrics, and reducing the rate loss imposed by current methods.

V. ADDITIONAL RESOURCES: MIMO AND MULTIUSER SYSTEMS

MIMO and multiuser transmission add antenna-level and receiver-coordination constraints to PAPR control, but multiple antennas also provide degrees of freedom for redistributing peaks.

  • MIMO: In MIMO, the worst-case antenna candidate typically determines the PAPR metric, unlike single-antenna transmission.This makes parallel single-antenna reduction inadequate for the multi-antenna setting.
  • MIMO: MIMO transmitters can redistribute peak power across antennas, potentially increasing the CCDF slope and lowering the probability of large peaks.The paper states that the full potential of these additional degrees of freedom remains unexplored.
  • System settings: Point-to-point MIMO permits joint processing at both ends, whereas multi-user downlink permits joint signal processing only at the transmitter.This distinction restricts which PAPR-reduction schemes can be applied.
  • Point-to-point MIMO: Point-to-point schemes include coupled, conventional, and directed SLM variants tailored to antenna coordination or selective candidate testing.Directed SLM invests complexity only where peak-to-average reduction is needed.
  • Complexity control: Sequential candidate assessment can stop once a tolerable PAPR limit set by the radio frontend is reached.The average number of assessed candidates is determined by the inverse CDF of the underlying original OFDM PAPR.
  • Multi-user downlink: Multi-user downlink is more challenging because receiver-side joint processing is unavailable and candidate operations must be individually reversible.Within the SLM family, only simplified SLM is applicable in this setting.

VI. GOING BEYOND: OFDM CAPACITY FUNDAMENTALS

The OFDM peak-power-constrained channel has unresolved capacity questions despite exponentially many constant-PAPR signals, while clipping and signal shaping offer practical routes toward capacity-oriented designs.

  • Capacity fundamentals: The capacity of the OFDM peak-power-constraint channel remains open, and no practical encoding scheme approaches the theoretical constant-PAPR result.At least (2 − δ_K)^N BPSK signals can have PAPR not exceeding K, but generating many such signals for a given K remains unresolved.
  • Clipping model: Clipping maps each frequency-domain OFDM vector C deterministically to a clipped vector Z through IDFT, nonlinear amplification, and DFT.The mapping is one-to-one between unclipped and clipped symbol vectors, with additive noise and possible carrier-wise fading at reception.
  • Clipping model: For N = 3 with 2-PAM per carrier, clipping deforms the possible-frame hypercube, but reduced output energy dominates the deformation.This motivates average power at the power-amplifier output as a capacity-oriented metric.
  • Capacity-oriented shaping: Signal shaping can predistort constellation points so the clipped frequency-domain vectors approximately recover a hypercube while retaining energy near the initial constellation.Active constellation extension is identified as an early implementation of this strategy.

VII. EMERGING SOLUTIONS: AN OPEN FIELD

Low-PAPR design spans code and sequence construction, code-based candidate selection, and constellation shaping, but rate and implementation complexity remain central constraints.

  • A. New Trends in Code Design: Constant-PAPR BPSK signals are exponentially numerous, with at least (2 − δ_K)^N signals having PAPR not exceeding K.The constant δ_K depends on K and tends to zero as K grows, while efficient generation for a specified K remains open.
  • A. New Trends in Code Design: Rudin–Shapiro and Golay complementary sequences provide constructions with PAPR at most 2, although sub-2 PAPR is unknown for BPSK signals.Rudin–Shapiro sequences apply at lengths that are powers of 2.
  • A. New Trends in Code Design: PAPR bounds have been studied for codes using size, minimum Euclidean distance, distance distributions, and algebraic-code structure, while LDPC-code analysis remains open.Computing a code’s PAPR is computationally demanding, though simple maximum-likelihood decoding can enable efficient determination.
  • A. New Trends in Code Design: Off-the-shelf Reed–Solomon and Simplex codes can generate candidates across OFDM frames, after which the best candidate is selected using any chosen optimality criterion.The approach also applies to jointly treated temporal blocks of consecutive OFDM frames.
  • A. New Trends in Code Design: Constellation shaping seeks a frequency-domain constellation whose time-domain shaping region has low PAPR and a reasonably simple encoder and decoder.Hadamard-transform and matrix-decomposition approaches are promising in simulation, but their implementation complexity remains unaffordable.

B. Banach space geometry

Banach space geometry reframes PAPR through norm and large-deviation questions, showing limits of alternative orthonormal systems and revealing modulation-dependent behavior in single-carrier signals.

  • B. Banach space geometry: Banach space geometry relates norms and metrics across finite-dimensional spaces, providing a framework for studying PAPR and developing algorithmic solutions.The paper uses relations between unit norm balls and their projections to connect geometric properties with signal behavior.
  • B. Banach space geometry: For any finite-dimensional orthonormal system, expected PAPR remains of order log(N), so changing the signaling system does not improve average PAPR.Worst-case PAPR is of order N regardless of the signaling system.
  • B. Banach space geometry: Single-carrier band-limited signals can have bounded samples at integer sampling points while their within-interval worst-case PAPR grows without bound linearly in N.This contrasts sampling-point behavior with the potentially large peaks occurring between samples.
  • B. Banach space geometry: Large-deviation results prove that the feared single-carrier PAPR behavior cannot occur arbitrarily, establishing a constant c0 > 0 under the stated setting.The passage presents this as the exact resolution of the previously unexplored problem.
  • B. Banach space geometry: Higher modulation sizes can make data influence dominant as the distribution becomes Gaussian-like, bringing PAPR behavior back toward log(N).The passage identifies this modulation regime as the principal qualification to the favorable result.
  • B. Banach space geometry: Hilbert-transform PAPR results are fragile for wideband signals around zero frequency, and some band-limited examples have unbounded transform-domain PAPR.The transform domain may therefore require careful signal shaping for certain single-carrier analytic modulation schemes.

3) Overcomplete expansions with uniformly bounded PAPR:

Overcomplete frame expansions offer a route to uniformly bounded PAPR, unlike the discouraging efficiency behavior established for tone reservation and natural orthogonal signaling families.

  • Overcomplete expansions: Frames are overcomplete vector systems U ∈ R^n×N with N ≥ n, and tight frames satisfy U^T U = I_n.Kashin representations interpret the mapping as an embedding between l∞ and l2 spaces, with level λ := N/n.
  • Uniformly bounded PAPR: Kashin proved that an overcomplete frame-generated subspace can keep K(λ) uniformly bounded in n when fixed λ > 1.The constant c1 > 0 has no good known estimates.
  • Implications for OFDM: Using the matrix U as an OFDM precoder can achieve uniformly bounded PAPR, although the optimal subspace construction remains unknown.Random partial Fourier matrices provide Kashin representations exploiting the uncertainty principle.
  • Tone reservation: Tone reservation seeks reserved subcarriers and compensation values that maximize PAPR reduction for each transmit sequence, but general achievability and limits remain unknown.For arbitrary fixed compensation sets, maintaining a peak constraint independent of subcarriers forces the information-to-compensation cardinality ratio toward zero.
  • Orthogonal signaling: Extended analyses of Walsh and other orthogonal signaling families report similarly discouraging system-efficiency behavior, motivating a conjecture covering all natural orthogonal families.The conjecture is stated in, not established as a theorem in the passage.

C. Compressed sensing

Compressed sensing exploits sparse clipping noise to reduce PAPR-related distortion, but recovery remains sensitive to noise; OFDM schemes use reserved or data tones as measurements for cancellation.

  • C. Compressed sensing: Compressed sensing recovers sparse signals from fewer linear measurements than their ambient dimension using a sensing matrix with suitable properties such as RIP.Basis pursuit provides l1-minimization recovery, while greedy algorithms reduce computational complexity.
  • C. Compressed sensing: Communication systems recover corrupted measurements g′ = g + z, but existing recovery algorithms may perform inadequately at the very low error rates required in severely noisy wireless environments.This is an explicit limitation of compressed-sensing recovery for communications.
  • Clipping-noise cancellation: Clipping noise is sparse, enabling compressed sensing to recover and cancel the distortion introduced when OFDM amplitudes are clipped.Earlier impulse-noise cancellation work established the use of unused tones as time-domain noise measurements.
  • Clipping-noise cancellation: Schemes in and [94] reserve M tones before clipping as measurements, causing data-rate loss; [96] instead uses clipping noise mixed into data tones and avoids that loss.The [96] method can adjust the number of measurements according to received-data reliability and AWGN level.
  • Performance: More than 3 dB BER gain at error probability 10^-6 is reported when AWGN is not too strong, relative to the original unclipped OFDM baseline.The comparison covers compressed-sensing cancellation schemes using reserved-tone and data-tone measurements.
  • OFDMA extensions: OFDMA extensions decompose the FFT into smaller blocks and use selected rows of a small DFT matrix as a sensing matrix for sparse clipping-noise recovery.The passage describes this as a proposed compressed-sensing approach for OFDMA systems.

VIII. CONCLUSIONS

The article revisits the persistent PAPR problem through alternative metrics, theoretical foundations, current limits, and future designs, including MIMO and multiuser systems.

  • VIII. CONCLUSIONS: The article presents a fresh perspective on PAPR using alternative metrics, theoretical foundations, and related designs.MIMO and multiuser systems are included as special cases.
  • VIII. CONCLUSIONS: It discusses current limits and new future directions after more than two decades of intensive PAPR research.The conclusion emphasizes the problem’s continuing practical impact in multicarrier theory.

X. BIOGRAPHIES

The biographies identify the article’s authors and summarize their academic training, appointments, awards, and research interests in communications and signal processing.

  • X. BIOGRAPHIES: Gerhard Wunder’s background includes electrical engineering degrees, a PhD on the OFDM PAPR problem, habilitation, visiting professorships, and Bell Labs consulting.His current interests include information theory, coded modulation, digital communications, and signal processing.
  • X. BIOGRAPHIES: Dr. Fischer is described as an award-winning author whose interests include precoding and shaping techniques for high-rate transmission schemes.The biography also cites his textbook on precoding and signal shaping for digital transmission.
  • X. BIOGRAPHIES: Holger Boche’s biography records electrical-engineering and mathematics degrees, doctoral training in pure mathematics, and postgraduate mathematical studies.The supplied passage ends before describing his full career.
  • X. BIOGRAPHIES: Simon Litsyn is a professor at Tel-Aviv University and has worked as chief scientist at SanDisk since 2005.His listed degrees are in electrical engineering from Perm Polytechnical Institute and Leningrad Electrotechnical Institute.
  • X. BIOGRAPHIES: Jong-Seon No’s biography lists engineering degrees from Seoul National University and a PhD from the University of Southern California, followed by industry and university appointments.The supplied passage ends during his career history.
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