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Joint PAPR and OOBE Reduction for AFDM via Chirp Parameter Tuning
Vincent Savaux, Hyeon Seok Rou, Zeping Sui, Giuseppe Thadeu Freitas de Abreu, Zilong Liu
TL;DR
AFDM suffers from high PAPR and significant OOBE, while existing approaches often optimize these metrics separately. The paper selects c2 from a finite candidate set using a weighted cost function, and simulations show simultaneous reductions in both metrics with performance close to single-objective methods.
Problem
AFDM suffers from high PAPR and significant OOBE, while existing approaches often focus exclusively on one metric.
Method
The method normalizes PAPR and OOBE, computes a weighted Euclidean distance, and selects c2 by minimizing it over a discrete candidate set.
Results
The proposed method simultaneously reduces PAPR and OOBE versus conventional AFDM, with slight losses compared with single-objective minimization.
Takeaways & Limitations
Equal weighting provides substantial gains for both metrics while allowing the trade-off to be adjusted through ρ.
Abstract
from arXiv · showhide
This paper addresses the joint reduction of the peak-to-average power ratio (PAPR) and out-of-band emissions (OOBE) in affine frequency division multiplexing (AFDM) systems by selecting the pre-chirp parameter c2. While existing approaches typically optimize either PAPR or OOBE independently, the proposed method jointly considers both metrics. To this end, a weighted cost function combining PAPR and OOBE is introduced to evaluate the trade-off between the two objectives. A pre-chirp selection scheme, inspired by the selected mapping (SLM) technique, is then employed to identify the optimal c2 value from a finite set of candidates, yielding a Pareto-optimal operating point within a discrete set. Simulation results demonstrate that the proposed approach simultaneously reduces both PAPR and OOBE compared with conventional AFDM. Moreover, its performance remains close to that of methods specifically optimized for a single objective, with only about a 1 dB degradation in PAPR reduction and a 2-3 dB degradation in OOBE suppression.
I. INTRODUCTION
AFDM is positioned for high-mobility communications but, like OFDM, suffers from high PAPR and significant OOBE. The paper extends pre-chirp selection to jointly reduce both metrics through weighted optimization.
- AFDM targets high-mobility wireless systems while retaining compatibility with existing multicarrier processing.
- High PAPR and significant OOBE remain key drawbacks of AFDM, motivating mitigation methods.
- The proposed method jointly minimizes PAPR and OOBE by selecting the pre-chirp parameter c2 with a weighted cost function.
- The weighting factor provides a flexible trade-off by assigning different priorities to PAPR and OOBE.
- Performance approaches that of pre-chirp selection optimized separately for PAPR or OOBE, and improves as the candidate-set size increases.
II. SYSTEM MODEL
The system model describes AFDM transmission with chirp-modulated subcarriers, a chirp-periodic prefix, oversampling, and a multipath Doppler channel. The pre-chirp parameter c2 remains flexible and is used to improve waveform properties.
- The SISO AFDM signal comprises N orthogonal chirp subcarriers and is represented at the Nyquist sampling rate.
- The AFDM waveform uses post-chirp parameter c1 and flexible pre-chirp parameter c2 through diagonal chirp matrices.
- A chirp-periodic prefix is appended to each AFDM symbol to avoid inter-symbol interference from the multipath channel.
- Oversampling produces x′ with N′ > N using an upsampling matrix that appends zero subcarriers at the frequency-domain edges.
- The channel model includes P paths with path coefficients, delays, Doppler shifts, a cyclic-shift matrix, and CPP effects.
- At the receiver, channel estimation and equalization recover the data, but equalization is not further detailed because the paper focuses on the transmitter.
- Choosing c2 offers flexibility for improving secondary AFDM features, including physical-layer security and sensing.
III. PRE-CHIRP PARAMETER-BASED JOINT PAPR AND OOBE REDUCTION
The paper formulates joint PAPR and OOBE reduction as a Pareto-like optimization over a discrete set of candidate solutions.
- Joint PAPR and OOBE reduction is based on a Pareto-like optimum obtained from a discrete set.
A. PAPR Definition
PAPR is characterized through the complementary cumulative distribution function (CCDF), using the signal's peak power relative to its average power and accounting for oversampling.
- A. PAPR Definition: PAPR is the maximum instantaneous signal power divided by the average signal power.The denominator is constant and equals unit signal power when the signal is normalized.
- A. PAPR Definition: The CCDF describes the probability that PAPR exceeds a threshold λ.For an oversampled signal x′, the CCDF is modeled as 1 − (1 − e^-λ)^αN with α = 2.8.
B. OOBE Definition
OOBE is quantified from the instantaneous power spectral density of the oversampled AFDM signal by integrating spectral power outside the signal bandwidth.
- B. OOBE Definition: The instantaneous power spectral density Sx(f) is defined from the DTFT X′(f) of the oversampled CPP-AFDM signal x′.The PSD is obtained from the power of the signal's discrete-time Fourier transform.
- B. OOBE Definition: OOBE β is the integral of the power spectral density over frequencies outside the signal bandwidth.The out-of-band frequency region is denoted by Ωf.
C. Joint PAPR and OOBE Reduction
The proposed method jointly reduces PAPR and OOBE by selecting c2 from a finite candidate set using a normalized weighted-distance criterion, approximating a Pareto optimum.
- C. Joint PAPR and OOBE Reduction: Pre-chirp selection chooses c2 from the finite set Ωc to minimize a metric representing either PAPR or OOBE.The method is analogous to selected mapping in OFDM and can target either metric through the choice of cost function.
- C. Joint PAPR and OOBE Reduction: The method approximates the continuous Pareto front by searching for an optimum over the discrete candidate set Ωc.The candidate-set cardinality is Mc.
- C. Joint PAPR and OOBE Reduction: The weighted Euclidean distance combines normalized PAPR γ̄ and OOBE β̄ to select the operating point.Normalization uses the expected values of the two metrics.
- C. Joint PAPR and OOBE Reduction: The weighting factor ρ selects the objective emphasis: ρ = 1 targets PAPR, ρ = 0 targets OOBE, and ρ = 0.5 weights both equally.Thus, the same framework supports single-objective and joint optimization settings.
- C. Joint PAPR and OOBE Reduction: The proposed PSM requires side information to transmit the selected c2 value, although c2 could alternatively be estimated.The alternative estimation approach is not further addressed in the paper.
IV. SIMULATION RESULTS
Simulations evaluate joint PAPR and OOBE reduction using CCDF and PSD measurements across candidate-set sizes and weighting factors. Equal weighting substantially improves both metrics, while larger candidate sets improve PAPR reduction.
- Simulation setup: The simulations use CCDF of PAPR and PSD to evaluate the proposed joint reduction method, with Mc ∈ {8, 32}.The AFDM waveform uses N = 256 subcarriers, QPSK data, fourfold oversampling, and 10^4 Monte Carlo runs.
- PAPR results: At CCDF = 10^-3 with Mc = 8, ρ = 1 and ρ = 0.5 achieve PAPR gains of about 3 dB and 2 dB over conventional AFDM.The equal-weighting case loses only 1 dB relative to PAPR-only optimization, while ρ = 0 matches conventional AFDM.
- PAPR results: With Mc = 32, ρ = 1 and ρ = 0.5 achieve PAPR gains of about 3.5 dB and 2.5 dB over conventional AFDM.The results indicate better PAPR reduction as the cardinality of the candidate set increases.
- OOBE results: For Mc = 8, OOBE reduction is about 4.5 dB with ρ = 0.5 and 6 dB with ρ = 1 at f = ±0.5.The PSD for ρ = 1 matches conventional AFDM because PAPR-only optimization does not affect OOBE.
- OOBE results: For Mc = 32, OOBE reduction reaches up to 6 dB with ρ = 0.5 and 9 dB with ρ = 1.The simulations conclude that equal weighting yields substantial gains for both metrics with slight losses relative to single-metric optimization.
- Trade-off: Optimizing only PAPR or only OOBE does not improve or worsen the other metric, whereas ρ = 0.5 jointly improves both.The paper attributes this independence to the two metrics being uncorrelated.
V. CONCLUSION
The paper proposes selecting AFDM’s chirp parameter c2 with a weighted PAPR–OOBE distance and evaluates the resulting discrete Pareto optimum. Simulations show simultaneous reduction of both metrics with performance close to single-objective methods.
- Conclusion: The method selects chirp parameter c2 by minimizing a weighted sum of PAPR and OOBE over a discrete set of possible solutions.This minimization is equivalent to finding a Pareto optimum in a two-dimensional problem.
- Conclusion: Simulation results show that both PAPR and OOBE can be reduced, with performance close to methods optimized for a single objective.The conclusion also identifies joint optimization with additional objective functions as a direction for further study.