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Effects of Residual Chirp Rate on DAFT-s-AFDM Uplink Performance

Henrique L. Senger, Gustavo P. Gonçalves, Bruno S. Chang, Hyeon Seok Rou, Giuseppe Thadeu Freitas de Abreu

arXiv:2610.09851v1eess.SP

TL;DR

High-mobility AFDM uplinks must address doubly dispersive channels and transmit-envelope constraints, while the effects of independently chosen precoder and modulator chirps have not been explained across their joint design space. The paper reduces their effect to a residual chirp rate and analyzes its transmitter and MMSE-receiver consequences. At BER 10^-3, a generic residual rate reduces required SNR by 4.4 dB relative to β = 0, practically overlapping AFDM.

  • Problem

    The joint effect of DAFT-s-AFDM precoder and modulator chirp rates has been studied for only a few combinations, leaving intermediate tradeoffs and their mechanism unexplained.

    Source: §I · p. 1

  • Method

    For a single-user localized allocation, the paper analyzes the difference between the precoder and modulator chirp rates as a residual rate acting through a unitary circulant kernel.

    Source: §I · p. 2

  • Results

    At BER 10^-3, β = π mod 1 reduces required SNR by 4.4 dB relative to β = 0, practically overlapping AFDM.

    Source: A. BER Results · p. 5

  • Takeaways & Limitations

    Varying the residual rate provides operating points between DFT-s-OFDM-like and AFDM-like behavior, allowing the PAPR–BER balance to be adjusted to system requirements.

    Source: §I · p. 2

Abstract

from arXiv · show

Affine frequency division multiplexing (AFDM) is robust to doubly dispersive channels in high-mobility 6G systems but, like orthogonal frequency division multiplexing (OFDM), suffers from a high peak-to-average power ratio (PAPR). Spreading with a discrete affine Fourier transform (DAFT) yields DAFT-s-AFDM, whose precoder and modulator chirps can be chosen independently. For a single user with zero-offset localized allocation, we show that these two chirps merge into a single residual chirp whose rate, with the other two chirp rates fixed, alone determines their effect. At the transmitter, this rate defines a circulant spreading kernel whose fourth moment quantifies the envelope fluctuation; symmetries reduce its characterization to rates in [0, 1/4]. At a linear minimum mean-square error (MMSE) receiver, the residual chirp redistributes the mean- square error across symbols without changing its total; under a Gaussian approximation, a more uniform per-symbol error lowers the uncoded bit error rate (BER). In uncoded QPSK simulations, a zero rate yields the lowest PAPR, whereas a generic rate reduces the signal-to-noise ratio required for a BER of $10^{-3}$ by 4.4 dB at a 3.33 dB PAPR cost (exceedance probability $10^{-3}$);a rate of 1/3 captures 2.8 dB of this gain for only 0.88 dB of that cost.

I. INTRODUCTION

The paper studies how the residual chirp rate in DAFT-s-AFDM affects the tradeoff between transmit-envelope behavior and BER. It analyzes the mechanism and evaluates operating points for a single-user, localized-allocation uplink.

I. INTRODUCTION

  • In high-mobility uplinks, reliable transmission over doubly dispersive channels and a well-behaved envelope for power-amplifier-constrained devices are simultaneous challenges.AFDM's configurable chirps provide a structured channel representation that enables full-diversity detection,.Source: §I · p. 1
  • Prior work examined only a few precoder–modulator chirp combinations, leaving intermediate tradeoffs and their mechanism unexplained.The paper focuses on the residual chirp rate for a single-user DAFT-s-AFDM uplink with localized allocation and linear MMSE reception.Source: §I · pp. 1–2
  • For zero-offset localized allocation, the precoder and modulator chirps affect the transmit signal through their difference, the residual rate, acting as a unitary circulant kernel.For an even number of precoded symbols, kernel symmetries reduce effective-spread characterization to rates in [0, 1/4], without similarly reducing the PAPR–BER design space.Source: §I · p. 2
  • The kernel's effective spread sets the fourth moment tracking bulk PAPR, while residual chirp redistributes per-symbol MSE without changing total MSE; under a Gaussian approximation, more uniform errors lower uncoded BER.The precoder input chirp does not change per-symbol MSE.Source: §I · p. 2
  • Simulations map PAPR–BER operating points against DFT-s-OFDM and AFDM, with zero residual rate matching the former's PAPR and generic rates performing like AFDM.Small-denominator rates such as 1/3 provide intermediate operating points.Source: §I · p. 2

II. SYSTEM MODEL

The system model composes DAFT spreading, localized mapping, and an inverse-DAFT modulator for a single-user uplink. In this setting, the two intervening chirp rates combine into a residual chirp parameter.

II. SYSTEM MODEL

  • The single user sends M independent, equiprobable, unit-energy uncoded QPSK symbols, with perfect synchronization and perfect channel knowledge assumed at the receiver.The transmitter is assumed linear.Source: §II · p. 2
  • The transmitter uses a DAFT precoder followed by localized mapping and an inverse-DAFT modulator, with N > M.For a single user, the mapping matrix is Γ0 = [IM 0]T.Source: §II · p. 2
  • With localized mapping, the precoder and modulator chirp diagonals become adjacent and combine into a residual chirp rate β = c2 − δ2.The other chirp rates have distinct roles: c1 separates delay–Doppler paths at the modulator, and δ1 sets the precoder phase progression.Source: §II · pp. 2–3
  • When β = 0, adjacent chirp matrices cancel and the transmitted block has the envelope of localized DFT-s-OFDM, while a unit-modulus chirp sweeps the occupied band across the block.This is the matched case c2 = δ2.Source: §II · p. 3

A. Doubly Dispersive Uplink Channel

The channel model represents a doubly dispersive uplink as a sum of paths with gains, delays, and Doppler shifts. The analysis assumes blockwise constant path parameters and specifies delay, prefix, and path-separation conditions.

A. Doubly Dispersive Uplink Channel

  • The channel is modeled using P resolvable paths with complex gains, delays, and Doppler shifts that remain constant over one block period.Source: A. Doubly Dispersive Uplink Channel · p. 3
  • Path delays are integer sample delays bounded by lmax < N, while Doppler indices may be fractional; a chirp-periodic prefix of length at least lmax is appended and removed.The resulting channel is represented by a circular channel matrix.Source: A. Doubly Dispersive Uplink Channel · p. 3
  • When the stated chirp-integrality conditions hold, the prefix phase matrix is the identity and the chirp-periodic prefix reduces to a cyclic prefix.For integer Doppler indices, the passage gives delay–Doppler path-separation conditions and a minimal choice of c1.Source: A. Doubly Dispersive Uplink Channel · p. 3

B. Affine-Domain MMSE Receiver

The receiver removes the prefix, applies a DAFT, and uses linear MMSE equalization before deprecoding. The residual chirp preserves total block MSE but can redistribute the individual symbol errors.

B. Affine-Domain MMSE Receiver

  • After prefix removal and the receiver DAFT, the receiver equalizes the precoded symbols using Heff and then deprecodes.The noise is modeled as AWGN with covariance N0IN.Source: B. Affine-Domain MMSE Receiver · p. 3
  • For unit-energy QPSK symbols, γ = Es/N0 = 1/N0, and the SNR per useful sample is (M/N)γ.For M = N/2, this useful-sample SNR is 3 dB below γ.Source: B. Affine-Domain MMSE Receiver · p. 3
  • For fixed channel, SNR, c1, and allocation, the baseline error covariance C0 is independent of β and δ1, and the covariance transformation is a unitary similarity.Source: B. Affine-Domain MMSE Receiver · p. 3
  • The residual chirp preserves total block MSE but can change individual symbol errors and off-diagonal error correlations.Source: B. Affine-Domain MMSE Receiver · p. 3

III. RESIDUAL CHIRP ANALYSIS

Both transmitter and receiver metrics depend on the residual chirp rate β through a single matrix. Its unitary action preserves total energy and total MSE while redistributing each across symbols or positions.

III. RESIDUAL CHIRP ANALYSIS

  • Both metrics depend on β through a single matrix, Ξβ.The proof identifies the matrix as unitary, preserving the kernel's total tap energy and the total MSE.Source: §III · p. 3
  • At the transmitter, the residual chirp redistributes energy across the precoded symbols; at the receiver, it redistributes error among symbols.Each spread symbol combines data symbols, while each symbol's error combines the β = 0 errors at several time positions.Source: §III · p. 3
  • The unitary matrix preserves total transmit power and total MSE, even as their distributions change.Source: §III · p. 3

A. Effects on PAPR

The residual-chirp kernel acts as a circular filter whose spread determines the fourth moment of the transmit symbols and characterizes envelope fluctuation. Kernel symmetries, rational rates, and implementation details qualify how that spread and PAPR are assessed.

A. Effects on PAPR

  • PAPRJ uses J uniform samples of the Fourier interpolation; the simulations use J = 4N.This reconstruction corresponds to an analog-chirp implementation; applying the chirp digitally before interpolation can increase peak values, while other pulses require individual attention.Source: A. Effects on PAPR · p. 4
  • Each spread symbol is a weighted sum of circularly shifted precoded symbols, with constant magnitude only for a single-tap kernel.The matrix Ξβ acts as a circular filter across the precoded symbols.Source: A. Effects on PAPR · p. 4
  • For QPSK, E{|v[r]|^4} = 2 - 1/Seff(β), linking the kernel's effective spread to the fourth moment.Source: A. Effects on PAPR · p. 4
  • Kernel symmetries reduce Seff characterization to β ∈ [0, 1/4]; under the stated condition, a half-period shift leaves the PAPR distribution unchanged.The half-period shift permutes per-symbol MSEs, and the PAPR invariance holds when 4Mδ1 is an integer.Source: A. Effects on PAPR · p. 4
  • For β = u/q with q | M, the kernel has equal-magnitude taps at S positions; odd q gives q taps and even q gives q/2 taps.Source: A. Effects on PAPR · p. 4
  • When q does not divide M, cancellation is incomplete; irrational β near zero can still yield a near-one-tap kernel.Thus, the block kernel, not β's numerical size or rationality alone, determines spread and the general PAPR trend.Source: A. Effects on PAPR · p. 4

B. Effects on BER

The residual chirp redistributes per-symbol MMSE without changing its total, and under a Gaussian approximation a more uniform error distribution lowers uncoded BER. The redistribution averages errors across time positions, with the resulting BER–PAPR tradeoff depending on β.

B. Effects on BER

  • For β = u/q with q | M, the kernel places equal-weight copies at positions spaced N/S samples apart.The per-symbol MSE includes a weighted average of the β = 0 errors at those copy positions, plus a correlation term.Source: B. Effects on BER · p. 4
  • The averaging term is never less uniform than the β = 0 errors, while the correlation term sums to zero.Source: B. Effects on BER · p. 4
  • Under a proper-complex-Gaussian approximation for residual interference, the analysis evaluates QPSK BER.Source: B. Effects on BER · pp. 4–5
  • β preserves total MSE while changing individual symbol errors; per-symbol MSE, SINR, and BER depend on β but not δ1.Source: B. Effects on BER · pp. 4–5
  • Among per-symbol MSE vectors with the same total, a more uniform vector yields lower average BER, and equal errors yield the lowest BER.The residual chirp can therefore lower uncoded BER without reducing total error.Source: B. Effects on BER · p. 5
  • At β = 0, PAPR is lowest and error is most uneven; generic β nearly fully redistributes both, while small-denominator β gives intermediate operating points.The transmit envelope fluctuates more as spreading increases, while the per-symbol error fluctuates less.Source: B. Effects on BER · p. 5

IV. SIMULATION RESULTS

The simulations compare DAFT-s-AFDM residual-chirp settings with AFDM and DFT-s-OFDM under a shared single-user uplink setup. BER and PAPR are assessed in separate figures.

IV. SIMULATION RESULTS

  • M = 64 QPSK symbols occupy the first 64 of N = 128 bins, over three Rayleigh paths with delays up to 3 samples and maximum normalized Doppler 3.The channel model corresponds to about 500 km/h maximum speed and a 3 µs delay spread.Source: §IV · p. 5
  • All schemes use the same localized mapping, channel and noise realizations, useful block energy, and prefix length; receivers have perfect channel knowledge and use linear MMSE equalization.DAFT-s-AFDM and AFDM use the modulation and prefix of Section II, while DFT-s-OFDM uses DFT precoding, an N-point IDFT, and a conventional cyclic prefix.Source: §IV · p. 5
  • Figures 2 and 3 plot simulated BER versus γ and the simulated CCDF of PAPR, respectively.The evaluated residual chirp rates are β ∈ {0, 1/3, 1/16} and β = π mod 1.Source: Figure 2 · p. 5, Figure 3 · p. 5, §IV · p. 5

A. BER Results

Residual chirp reduces required SNR for a BER target as its kernel spread grows, with gains saturating beyond an effective spread of about 8. The results connect intermediate operating points to small-denominator rates.

A. BER Results

  • 4.4 dB lower required SNR at BER 10^-3 is achieved by β = π mod 1 relative to β = 0, while β = 1/3 and β = 1/16 reduce it by 2.8 and 4.0 dB.β = 0 provides approximately 2 dB gain over DFT-s-OFDM while remaining in the low-PAPR operating region.Source: A. BER Results · p. 5
  • Beyond S_eff ≈ 8, BER gain saturates: β = 1/16 is within 0.4 dB of the generic value despite having far fewer kernel taps.The gain grows with kernel spread before reaching this saturation.Source: A. BER Results · p. 5
  • Small-denominator β values provide intermediate operating points that were not evaluated in the earlier study, whose unequal settings produced generic, AFDM-like rates.The equal setting in corresponds to β = 0; its unequal settings had S_eff between 165 and 290 at M = 512.Source: A. BER Results · p. 5

B. PAPR Results

The PAPR tail and median rise as residual-chirp spreading increases, while small-denominator rates offer BER gains at lower PAPR cost than a generic rate.

B. PAPR Results

  • At CCDF 10^-3, β = 0 has PAPR close to DFT-s-OFDM at 7.68 dB, while β = 1/3, 1/16, π mod 1, and AFDM require 8.56, 10.30, 11.01, and 11.06 dB, respectively.The complete range from DFT-s-OFDM-like to AFDM-like operation costs about 3.38 dB.Source: B. PAPR Results · p. 6
  • 2.8 dB of the generic rate's 4.4 dB SNR gain is attained by β = 1/3 for 0.88 dB of its 3.33 dB PAPR cost, measured at BER and CCDF 10^-3.β = 1/16 comes within 0.4 dB of the generic SNR gain at a 0.7 dB lower PAPR.Source: B. PAPR Results · p. 6
  • The median PAPR rises from 5.7 to 7.9 dB as 2 − S_eff^-1 increases from 1 to 1.98; β = 1/4 is dominated by β = 1/3, while β = 1/6 matches it within 0.05 dB in both metrics.The fourth moment approaches the Gaussian value 2 as more symbols are combined and tracks the bulk of the PAPR distribution.Source: B. PAPR Results · p. 6

V. CONCLUSION

For a single-user zero-offset localized allocation, the difference between precoder and modulator chirp rates determines DAFT-s-AFDM behavior through a residual-chirp kernel. Its spreading trades PAPR against uncoded BER while conserving transmit power and total MSE.

V. CONCLUSION

  • The residual-chirp kernel conserves transmit power and total MSE while redistributing both across symbols.Its effective spread raises the fourth moment, which tracks bulk PAPR, and makes per-symbol MSE more uniform, which lowers uncoded BER.Source: §V · p. 6
  • Zero residual chirp matches DFT-s-OFDM PAPR, while generic rates perform like AFDM; β = 1/3 captures about 60% of the SNR gain for about one quarter of the PAPR cost.The stated scope is a single user with zero-offset localized allocation.Source: §V · p. 6
  • Future work includes coded transmission, higher-order constellations, multi-user allocation, imperfect channel knowledge, and interference-cancellation receivers.Source: §V · p. 6
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