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Peak Reduction and Clipping Mitigation by Compressive Sensing

Ebrahim B. Al-Safadi, Tareq Y. Al-Naffouri

arXiv:1101.4335v1cs.ITmath.ITmath.STstat.TH

TL;DR

OFDM has high PAPR because superposed subcarriers create large temporal envelope fluctuations, while conventional tone-reservation methods require transmitter-side optimization to keep clippers spectrally separate from data. The system applies sparse clipping at the transmitter and uses reserved-tone compressive sensing at the receiver, enhanced with data-based weighting, phase augmentation, and Bayesian estimation. The evaluated clipping-mitigation methods show that combining support and phase augmentation with LASSO can approach the support-oracle SER and improve performance at low clipping thresholds.

  • Problem

    OFDM has high PAPR because superposed subcarriers create large temporal envelope fluctuations, while conventional tone-reservation methods require transmitter-side optimization to keep clippers spectrally separate from data.

  • Method

    The system applies sparse clipping at the transmitter and uses reserved-tone compressive sensing at the receiver, enhanced with data-based weighting, phase augmentation, and Bayesian estimation.

  • Results

    The evaluated clipping-mitigation methods show that combining support and phase augmentation with LASSO can approach the support-oracle SER and improve performance at low clipping thresholds.

  • Takeaways & Limitations

    Receiver-side clipping mitigation can shift processing complexity away from the transmitter while improving the capacity–PAPR trade-off through modest algorithmic augmentation.

Abstract

from arXiv · show

This work establishes the design, analysis, and fine-tuning of a Peak-to-Average-Power-Ratio (PAPR) reducing system, based on compressed sensing at the receiver of a peak-reducing sparse clipper applied to an OFDM signal at the transmitter. By exploiting the sparsity of the OFDM signal in the time domain relative to a pre-defined clipping threshold, the method depends on partially observing the frequency content of extremely simple sparse clippers to recover the locations, magnitudes, and phases of the clipped coefficients of the peak-reduced signal. We claim that in the absence of optimization algorithms at the transmitter that confine the frequency support of clippers to a predefined set of reserved-tones, no other tone-reservation method can reliably recover the original OFDM signal with such low complexity. Afterwards we focus on designing different clipping signals that can embed a priori information regarding the support and phase of the peak-reducing signal to the receiver, followed by modified compressive sensing techniques for enhanced recovery. This includes data-based weighted {\ell} 1 minimization for enhanced support recovery and phase-augmention for homogeneous clippers followed by Bayesian techniques. We show that using such techniques for a typical OFDM signal of 256 subcarriers and 20% reserved tones, the PAPR can be reduced by approximately 4.5 dB with a significant increase in capacity compared to a system which uses all its tones for data transmission and clips to such levels. The design is hence appealing from both capacity and PAPR reduction aspects.

I. INTRODUCTION

OFDM’s high PAPR arises from large envelope fluctuations, motivating many reduction techniques. This paper introduces receiver-side compressive sensing for sparse clipping, shifting key processing away from the transmitter.

  • Motivation: High PAPR in OFDM results from superposed subcarriers producing large temporal envelope fluctuations.The resulting power-ratio problem affects high-frequency power amplifiers.
  • Prior approaches: Existing PAPR-reduction approaches include coding, selective mapping, partial transmit sequences, tone injection, tone reservation, and companding.
  • Contribution: The paper designs a tone-reservation system that uses reserved tones for receiver-side reconstruction rather than conventional transmitter-side peak-signal optimization.
  • Contribution: Signal-processing complexity is shifted from the transmitter to the receiver, targeting communication models where transmitter complexity is a bottleneck.
  • Method: The method assumes clipping events are sparse in time and uses null tones to estimate their locations and complex coefficients through compressive sensing.It applies sparse-signal recovery from incomplete frequency information to OFDM clipping mitigation.

III. BASIC PAPR REDUCTION DESIGN

The basic design clips sparse time-domain peaks at the transmitter and recovers the resulting clipper from reserved frequency measurements at the receiver. Compressive sensing estimates support and amplitudes, subject to sparsity and clipping constraints.

  • Basic design: The transmitter adds a peak-reducing signal to the high-PAPR OFDM waveform, while the receiver estimates and subtracts that signal after demodulation.
  • Measurement model: The clipper is sparse in time but dense in frequency, so reserved tones provide compressed measurements rather than a disjoint spectral support.
  • Measurement model: Reserved tones are selected from the subcarriers excluded from data transmission, with random assignment described as near-optimal for estimating the clipper.
  • Measurement model: The receiver projects the frequency-domain observation onto the complement of the data subspace, producing m measurements of an N-dimensional sparse clipper corrupted by noise.Here m equals the number of reserved tones.
  • Scope and constraints: Reliable recovery requires clipping below bounds determined by the number of reserved tones and the clipper sparsity, although data augmentation can relax these generic limits.
  • Recovery: Convex compressive-sensing recovery using basis pursuit or LASSO estimates nonzero positions and approximates their amplitudes, after which support-conditioned refinement can improve coefficients.

IV. COMPARISON WITH TYPICAL TONE-RESERVATION PAPR REDUCTION TECHNIQUES

Typical tone-reservation methods optimize a peak-reducing signal at the transmitter and confine it to tones disjoint from data. The proposed design instead accepts arbitrary spectral support and reconstructs clipping from reserved measurements at the receiver.

  • Conventional tone reservation: Conventional tone reservation searches for a PAPR-reducing signal confined to reserved tones so it remains orthogonal to data-carrying tones.
  • Comparison: For the same number of reserved tones, conventional optimization can reduce more peaks because the proposed clipping design restricts the number of clipped peaks to s < m.
  • Complexity comparison: The principal optimization complexity in conventional techniques lies at the transmitter, where the peak-reducing signal must be found and spectrally separated from data.

V. ENHANCED PAPR REDUCTION BY DATA-INDUCED WEIGHTED AND PHASE-AUGMENTED ℓ1 MINIMIZATION

The paper enhances compressive-sensing recovery by exploiting information about the clipping signal in both time and frequency representations. It identifies a trade-off between support recovery and coefficient estimation when designing clipping schemes.

  • The baseline model assumes no prior knowledge of the sparse clipper’s locations, magnitudes, or phases beyond reserved-tone observations.
  • Weighted, constrained, or rotated frequency-domain searches use data-derived information from the time domain to improve clipping-signal estimation.
  • Clipping design cannot simultaneously optimize support recovery and coefficient estimation, so the two objectives require a compromise.

A. Homogeneous Clipping Techniques

The paper introduces simple homogeneous clipping techniques without transmitter-side optimization or spectral confinement. These techniques motivate subsequent compressive-sensing enhancements based on clipping structure.

  • A. Homogeneous Clipping Techniques: The section defines two simple clipping techniques that require neither optimization nor spectral confinement.
  • A. Homogeneous Clipping Techniques: The methods initially focus on deterministic compressive-sensing enhancements, postponing Bayesian compressive estimation to a later section.
  • A. Homogeneous Clipping Techniques: Peak suppression reduces coefficients whose envelopes exceed γ to γ while preserving their angles, producing |x_i + c_i| = γ.

1) Peak Suppression to γ (PS):

Peak Suppression clips oversized OFDM coefficients to a fixed threshold γ while preserving their angles, enabling support information but creating difficult coefficient-recovery conditions.

  • 1) Peak Suppression to γ (PS):: Peak Suppression maps coefficients with |x_i| ≥ γ to magnitude γ while preserving their angles, thereby reducing peak magnitudes.
  • 1) Peak Suppression to γ (PS):: The clipping scheme can reveal likely support locations from the distance between estimated coefficient magnitudes and γ.
  • 1) Peak Suppression to γ (PS):: Unenhanced Peak Suppression requires more measurements for the same sparsity level and SER than other clipping techniques.
  • 1) Peak Suppression to γ (PS):: The clipping signal’s average sparsity is the expectation of the corresponding Binomial sparsity level.
  • 1) Peak Suppression to γ (PS):: Random clipping magnitudes approach zero separation from γ at the minimum, creating a critical compressive-sensing bottleneck that increased CNR cannot completely compensate.

2) Digital-Magnitude Clipping (DMC):

Digital-Magnitude Clipping replaces random clipping magnitudes with fixed values, reducing coefficient-estimation degrees of freedom while preserving anti-phase structure. Its magnitude parameter improves support detection but increases the consequences of detection errors.

  • 2) Digital-Magnitude Clipping (DMC):: With all active magnitudes equal to ζ, the unknown clipping signal is reduced to its support and phase, while retaining the anti-phase property.
  • 2) Digital-Magnitude Clipping (DMC):: Digital-Magnitude Clipping restricts clipping magnitudes to a finite set, with this section focusing on the binary space |c| ∈ {0, ζ}.
  • 2) Digital-Magnitude Clipping (DMC):: Equal nonzero coefficient magnitudes are identified as favorable for compressive estimation, and DMC can be recast as sparse-lattice detection with regularized sphere decoding.
  • 2) Digital-Magnitude Clipping (DMC):: Increasing ζ raises CNR and eases support detection, but faulty support detection causes dramatically larger system error and more complex subsequent oversampling.
  • 2) Digital-Magnitude Clipping (DMC):: The clipping magnitude must be bounded below so clipped coefficients remain at or below the desired threshold γ.

B. Externally Weighted ℓ1 Minimization

Externally weighted ℓ1 minimization uses data-derived information about likely clipping locations to guide sparse recovery, while accounting for numerical stability and limitations of the clipping-based prior.

  • B. Externally Weighted ℓ1 Minimization: A one-shot external weighting scheme based on the estimated data vector avoids the computational expense and sensitivity of repeated internally weighted CS.Internal weighting repeats CS and depends on the first unguided estimate, whereas the external approach minimally increases ordinary LASSO complexity.
  • B. Externally Weighted ℓ1 Minimization: Data-based weighting uses the estimated data envelope’s distance from the clipping threshold to penalize less likely clipping locations more strongly.The weighting vector can be defined from d(i) = ||x̂(i)| − γ| or from the posterior probability of no clipping.
  • B. Externally Weighted ℓ1 Minimization: The weighting construction assumes a Gaussian least-squares envelope error, yielding Rayleigh models for the relevant error and observed-envelope densities.These distributions support the posterior-probability weighting formulation.
  • B. Externally Weighted ℓ1 Minimization: The stabilization parameter ε > 0 is introduced to improve numerical stability in the weighted ℓ1 procedure.The procedure is referred to as internally weighted ℓ1 minimization in the cited formulation.
  • B. Externally Weighted ℓ1 Minimization: Peak suppression provides useful probabilistic exclusion of false positives, but natural proximity of coefficients to γ can still bias candidate locations even at infinite CNR.Suppressing to the envelope mean would instead make many locations appear plausible clipping positions.

C. Phase-Augmented CS for Homogenous Clippers

Phase-augmented compressed sensing incorporates estimated phase information for homogeneous clippers, rotating the sensing problem so recovery can exploit a real-valued sparse representation. The reliability of this prior depends on the quality of the receiver’s data estimate and the operating SNR.

  • C. Phase-Augmented CS for Homogenous Clippers: For homogeneous clipping, the transmitter phase equals the original signal phase, but the receiver only has an estimate whose usefulness depends on data-estimation quality measured by SNR.The method therefore treats SNR as the parameter governing the quality of the available phase prior.
  • C. Phase-Augmented CS for Homogenous Clippers: In practical regimes, estimated-data phase information is expected to be more reliable than CS-only phase information, although an additional phase replacement step showed no significant improvement.The comparison follows the stated CNR–SNR behavior as ζ increases.
  • C. Phase-Augmented CS for Homogenous Clippers: Phase augmentation uses estimated data phases to realign clipping coefficients and reduce recovery to a real sparse-vector problem involving locations and magnitudes.With known clipping phases, the measurement matrix can absorb those phases; when unknown, the receiver uses the estimated data phases.
  • C. Phase-Augmented CS for Homogenous Clippers: Sense then Rotate first recovers a complex clipping estimate with standard or weighted CS, then rotates its nonzero coefficients using estimated directions from the data vector.This is the alternative phase-augmentation ordering to Rotate then Sense.
  • C. Phase-Augmented CS for Homogenous Clippers: Rotate then Sense supplies estimated phases before CS, producing 2m real observations for estimating a real vector.The transformed sensing matrix is defined as Ψ̃c = ΨΘc.

VI. BAYESIAN ESTIMATION OF SPARSE CLIPPING SIGNALS

The Bayesian estimation section combines support priors, truncated searches, and conditional amplitude estimation to reduce the cost of sparse clipping recovery. Its β-FBMP procedure reduces executions by 60–80% under the stated practical parameters.

  • VI. BAYESIAN ESTIMATION OF SPARSE CLIPPING SIGNALS: LMMSE improves amplitude estimation over least squares when the coefficient distribution is Gaussian, but it does not incorporate statistical information into support estimation.This limitation motivates broader MMSE treatment of support and amplitudes.
  • VI. BAYESIAN ESTIMATION OF SPARSE CLIPPING SIGNALS: MMSE estimation requires selecting candidate support vectors and evaluating their priors, likelihoods, and conditional coefficient expectations.The exact estimator sums over 2^N terms, motivating a truncated candidate set J* and normalized weighted estimates.
  • VI. BAYESIAN ESTIMATION OF SPARSE CLIPPING SIGNALS: The estimated data vector identifies likely clipping locations because coefficients near the clipping threshold receive higher support probability.For γ = 2σ|X|, the supplied discussion states that 70% of N indices could be excluded as low-probability clipping locations.

VII. PERFORMANCE ANALYSIS AND SIMULATIONS

Simulations evaluate sparse-clipping recovery across error, SER, complexity, PAPR, and capacity measures, showing that data-augmented CS improves the capacity–peak-reduction trade-off. The experiments use 256 subcarriers with 20% measurement tones and compare reserved-tone estimation against naive clipping.

  • Experimental setup: The simulations use N = 256 subcarriers, m = 0.2N randomly dispersed measurement tones, 32-QAM data, a 32-tap Rayleigh channel, and 30 dB SNR, evaluating SER, complexity, PAPR reduction, and capacity.The clipping threshold controls both PAPR reduction and the sparsity and CNR conditions governing CS recovery.
  • SER: Weighting and phase augmentation both improve LASSO SER, with their combination approaching the support-oracle performance and even exceeding it at low thresholds when s > 0.55 m.Weighting alone is more effective than phase augmentation in the reported comparison.
  • Digital clipping: Embedding phase information directly into LASSO is more effective than rotating post-CS estimates and remains close to a phase oracle at practical ζ, but diverges as ζ increases and phase information becomes less accurate.Forcing estimate magnitudes is generally ineffective except in very sparse cases for the basic method.
  • Complexity: LASSO-based methods require less than 12% of Tellado’s primary QCQP execution time on average, while β-FBMP requires less than 2%.Execution times were collected over 2000 runs and normalized by the maximum observed time.
  • Capacity: Reserving 20% of tones for data-based weighted and phase-augmented LASSO significantly outperforms naive all-tone transmission in capacity, while reducing γ from 2.5 σ|X| to 2 σ|X| costs less than 1 bit/s per transmitted tone.Typical LASSO remains effective at clipping thresholds as low as 1.9 σ|X|.
  • Capacity: Increasing SNR benefits the reserved-tone system more than naive clipping at γ = 2.3 σ|X| because improved noise conditions enhance CS estimation, whereas naive capacity saturates after 35 dB.The reserved-tone system exhibits a semi-linear capacity relation with SNR.

VIII. CONCLUSION

The paper establishes receiver-side compressive sensing as a general approach to OFDM clipping mitigation, using reserved subcarriers to estimate sparse clipping events rather than optimize transmitter-side peak-reducing signals. Data-derived support, magnitude, and phase information further strengthens recovery and capacity at low clipping thresholds.

  • Core framework: Receiver-side CS estimates the locations and amplitudes of sparse clipped portions from reserved subcarriers, shifting signal-processing complexity away from the transmitter.This framework uses reserved tones for estimation rather than as the spectral support of transmitter-optimized peak-reducing signals.
  • Enhanced recovery: With marginal added complexity, data-derived clipping locations, magnitudes, and phases augment standard ℓ1 minimization and enable recovery beyond generic CS conditions, including sparsity above 55% of m.The augmentation significantly boosts capacity at low clipping thresholds, creating a compromise between capacity and peak reduction.
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