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Affine Frequency Division Multiplexing for Next Generation Wireless Communications

Ali Bemani, Nassar Ksairi, Marios Kountouris

arXiv:2204.12798v2cs.ITeess.SP

TL;DR

The paper introduces AFDM as a new waveform and derives its DAFT-domain input-output relation. It establishes full delay-Doppler representation and diversity, proposes low-complexity detection and channel estimation, and reports that AFDM outperforms OFDM and other DAFT-based schemes.

  • Problem

    AFDM is introduced as a promising new waveform for time-varying channels.

  • Method

    The paper derives AFDM's DAFT-domain input-output relation, tunes its pulse parameters, and uses zero-padding for low-complexity detection and channel estimation.

  • Results

    AFDM's DAFT-domain impulse response provides a full delay-Doppler representation, enabling full diversity; simulations show it outperforms OFDM and other DAFT-based multicarrier schemes.

  • Takeaways & Limitations

    AFDM combines full delay-Doppler representation with low-complexity detection and channel estimation, while reducing pilot and user multiplexing overhead relative to OTFS.

Abstract

from arXiv · show

Affine Frequency Division Multiplexing (AFDM), a new chirp-based multicarrier waveform for high mobility communications, is introduced here. AFDM is based on discrete affine Fourier transform (DAFT), a generalization of discrete Fourier transform, which is characterized by two parameters that can be adapted to better cope with doubly dispersive channels. First, we derive the explicit input-output relation in the DAFT domain showing the effect of AFDM parameters in the input-output relation. Second, we show how the DAFT parameters underlying AFDM have to be set so that the resulting DAFT domain impulse response conveys a full delay-Doppler representation of the channel. Then, we show analytically that AFDM can achieve full diversity in doubly dispersive channels, where full diversity refers to the number of multipath components separable in either the delay or the Doppler domain, due to its full delay-Doppler representation. Furthermore, we present a low complexity detection method taking advantage of zero-padding. We also propose an embedded pilot-aided channel estimation scheme for AFDM, in which both channel estimation and data detection are performed within the same AFDM frame. Finally, simulations corroborate the validity of our analytical results and show the significant performance gains of AFDM over state-of-the-art multicarrier schemes in high mobility scenarios.

I. INTRODUCTION

High mobility creates Doppler shifts and inter-carrier interference that challenge conventional multicarrier waveforms. The introduction motivates chirp-based alternatives and presents AFDM as a DAFT-based scheme designed for full delay-Doppler separability and diversity.

  • High mobility produces large Doppler shifts that reduce OFDM performance by causing inter-carrier interference and loss of orthogonality.
  • Shortening OFDM symbols can limit channel variation within each symbol but significantly reduces spectral efficiency because of the cyclic prefix.
  • Finding an orthonormal basis for general time-varying channels is difficult, motivating polynomial-phase and chirp bases as alternatives to complex exponentials.
  • Prior DAFT-OFDM and OCDM schemes reduce interference or improve performance in some settings but can have low or non-full diversity and require channel-path information for parameter tuning.
  • AFDM is introduced as a DAFT-based waveform using orthogonal chirp signals, with information multiplexed so paths separate and symbols experience all channel paths.
  • The paper derives AFDM’s DAFT-domain input-output relation, proves full diversity under suitable parameter tuning, and develops low-complexity detection and embedded channel estimation.
  • The proposed detector has linear complexity in subcarriers and paths, similar performance to LMMSE detection, while embedded channel estimation causes only marginal degradation relative to perfect channel knowledge.

II. AFFINE FOURIER TRANSFORM

The Affine Fourier Transform is a four-parameter linear integral transform that generalizes several established transforms and provides added flexibility for time-frequency analysis and communications.

  • The Affine Fourier Transform is also known as the Linear Canonical Transform and is defined as a four-parameter class of linear integral transforms.
  • AFT generalizes the Fourier, Laplace, fractional Fourier, Fresnel, and scaling transforms through different parameter choices.
  • AFT flexibility has been used in filter design, time-frequency analysis, phase retrieval, and communication multiplexing.
  • The transform twists the Wigner distribution of the input signal, providing a time-frequency interpretation of its operation.

B. Discrete Affine Frequency Transform

The discrete affine Fourier transform adapts the affine transform to sampled signals using a unitary matrix built from chirp diagonal matrices and the DFT, forming AFDM’s transform basis.

  • Discrete AFT has two essentially equivalent types, differing in parameterization and whether the input is continuous before sampling or purely discrete.
  • The transform is reversible under the stated parameter condition, and its inverse is the Hermitian transpose when the matrix is unitary.
  • Sampling imposes periodicity across domains, which determines the prefix structure required by DAFT-based multicarrier signaling.
  • For M=N, DAFT maps s to S through A=Λ_c2 F Λ_c1, where F is the DFT matrix and Λ_c is a chirp diagonal matrix.
  • AFDM uses IDAFT to map data symbols into time-domain samples and DAFT at the receiver to obtain the affine Fourier-domain channel response.

A. Modulation

AFDM modulates QAM information symbols in the discrete affine Fourier domain, uses a chirp-periodic prefix against multipath, and models reception through an effective channel matrix.

  • AFDM represents information symbols as a vector x in the discrete affine Fourier domain, with QAM symbols used as the modulation alphabet.
  • A chirp-periodic prefix replaces the OFDM cyclic prefix because AFDM has a different signal periodicity.
  • The prefix length must cover the channel’s maximum delay spread, with L_cp at least the required delay-dependent length.
  • The channel model includes multipath gains, Doppler shifts, integer delays, and paths sharing delays while having distinct Doppler frequencies.
  • After prefix removal and DAFT processing, received symbols are related to transmitted symbols through H_eff, with transformed noise retaining the same statistics under the unitary transform.
  • The effective channel has a sparse structure controlled by AFDM parameters, which influences diversity, detection complexity, and channel estimation.

1) Integer Doppler Shifts:

AFDM parameter selection separates channel paths in the DAFT domain: integer Doppler produces one nonzero per row, while fractional Doppler produces a finite clustered support, enabling a full delay-Doppler representation.

  • 1) Integer Doppler Shifts:: Integer Doppler shifts yield exactly one nonzero element in each row of H_i, located at q=(p+loc_i)_N.
  • 1) Integer Doppler Shifts:: Fractional Doppler shifts spread each row across 2k_ν+1 nonzero elements centered at q=(p+loc_i)_N.
  • 1) Integer Doppler Shifts:: The finite-support approximation retains entries within k_ν of the peak, where k_ν is selected using a sensitivity threshold.
  • E. AFDM Parameters: AFDM parameters are chosen so paths with different delays or Doppler shifts have non-overlapping DAFT-domain supports.
  • E. AFDM Parameters: The fractional-Doppler design introduces flexibility through ξ_ν, reducing pilot overhead at the expense of H_eff no longer being strictly circulant.
  • E. AFDM Parameters: Under underspread-channel conditions, bounds on maximum delay and Doppler prevent modular overlap between extreme-path responses.
  • E. AFDM Parameters: Separated path supports make the delay-Doppler profile recoverable from nonzero entries in a row of H_eff, a property unavailable to DAFT-OFDM and OCDM.
  • E. AFDM Parameters: This full delay-Doppler representation enables AFDM to achieve full diversity in linear time-varying channels.

IV. DIVERSITY ANALYSIS

The diversity analysis relates AFDM diversity to the rank of a matrix formed from pairwise symbol differences. It establishes that suitable AFDM parameter selection achieves full diversity under the stated channel conditions.

  • AFDM diversity is determined by the minimum rank r of the matrix Φ(δ(m,n)) across distinct symbol pairs.At high SNR, the pairwise error probability has an SNR exponent equal to this rank.
  • A necessary condition for full diversity is that the channel paths occupy distinct locations in the DAFT-domain channel representation.The condition can hold for AFDM but not for OCDM or DAFT-OFDM.
  • When two paths share a channel-matrix location, a possible difference vector can make the corresponding matrix columns dependent, preventing full rank.Thus, the rank cannot reach the number of paths under that overlapping-location condition.
  • Full-diversity analysis therefore reduces to showing that tuning c2 makes Φ(δ) full rank.
  • AFDM with c1 satisfying (42) achieves full diversity, ρ = P, for a linear time-varying channel with maximum delay lmax and normalized Doppler shift αmax.The theorem statement covers the specified channel setting, and the analysis also addresses fractional Doppler shifts.

V. LOW-COMPLEXITY WEIGHTED MRC-BASED DFE DETECTION

The paper proposes a low-complexity weighted MRC-based decision-feedback detector that exploits AFDM’s sparse effective channel representation and zero padding. Iterative interference cancellation combines multiple impaired copies of each data symbol.

  • Zero-padding permits treating a truncated part of the effective channel as a band matrix and removes the need for modular operations.The null symbols do not add overhead because they can also support embedded pilot-aided channel estimation.
  • The proposed detector replaces potentially prohibitive O(N^3) LMMSE equalization with weighted MRC-based DFE detection.It exploits the sparse channel representation provided by AFDM.
  • Each effective-channel column has L nonzero entries, with L = P for integer Doppler and L = (2ξν + 1)P for fractional Doppler.Each nonzero entry provides a received copy of the corresponding data symbol.
  • The detector estimates each symbol by weighted maximum-ratio combining of its L channel-impaired copies.Inter-symbol interference is canceled iteratively in the branches selected for combining.
  • The detector’s total complexity is niter(5L + 1)(N − Q).The per-symbol processing uses scalar operations based on the sparse effective channel.
  • The algorithm updates residual received samples after symbol estimation and continues until reaching niter or a change below ϵ.

1) Convergence:

The convergence analysis expresses the iterative detector in matrix form and applies an iterative-method criterion. Convergence follows when the relevant Hermitian matrix is positive definite.

  • The weighted MRC-based DFE iteration is written using S, R, and b derived from the effective-channel Gram matrix and received vector.
  • Convergence requires the spectral radius ρ(−S^-1(R − S)) to be strictly smaller than one.This is the stated convergence criterion for the iteration.
  • Theorem 2 states that the iteration converges if R = HH is a positive definite Hermitian matrix.
  • Under this convergence condition, the weighted MRC-based DFE converges to the LMMSE estimate.

2) Relation to the Gauss-Seidel method:

The paper connects the detector iteration to Gauss-Seidel methods and develops an embedded pilot-aided channel-estimation procedure. A pilot surrounded by null guards separates estimation from data interference within the same AFDM frame.

  • 1) Convergence:: The detector iteration is analyzed using the properties of iterative methods for linear systems, specifically the Gauss-Seidel method.
  • 2) Relation to the Gauss-Seidel method:: The embedded estimation design places one pilot symbol in each AFDM frame with Q null guard samples on each side.The guards separate data symbols from the pilot, enabling interference-free channel estimation.
  • 2) Relation to the Gauss-Seidel method:: The channel estimator recovers each path’s delay, Doppler shift, and complex gain, totaling 3P unknown parameters.It uses the received portion associated with the pilot symbol.
  • 2) Relation to the Gauss-Seidel method:: Pilot-related transmit, receive, and effective-channel components are extracted using selection matrices Tt,E and Tr,E.
  • 2) Relation to the Gauss-Seidel method:: For fixed delays and Doppler shifts, estimating each complex path gain reduces to solving a linear system because the log-likelihood is quadratic in the gain.
  • 2) Relation to the Gauss-Seidel method:: The low-complexity estimator avoids brute-force search over the 3P-dimensional continuous parameter space.It reduces delay-and-Doppler estimation to locating the largest entries of the pilot-related received vector in the integer-Doppler case.

B. Fractional Doppler Case

The fractional-Doppler analysis develops approximate path identification and evaluates AFDM through simulations. AFDM achieves full diversity and matches OTFS BER in the reported settings, while fractional Doppler increases overhead and pilot-power requirements.

  • Fractional-Doppler channel estimation: For fractional Doppler, the channel-estimation procedure approximates negligible inter-path interference to identify delay and integer Doppler components before estimating fractional parts and complex gains.The method searches delay–integer-Doppler combinations, then refines fractional Doppler shifts and solves for complex gains.
  • Simulation results: AFDM achieves the full diversity of each simulated channel, with slope-reference curves indicating diversity orders without serving as upper bounds.The simulations use multiple channel realizations and evaluate AFDM under high-mobility conditions.
  • Comparison with multicarrier schemes: AFDM achieves full diversity through path separation, whereas OCDM can have diversity one like OFDM when overlapping paths add destructively.OCDM performance depends on the channel delay–Doppler profile, while AFDM separates paths by tuning c1 and c2.
  • Comparison with multicarrier schemes: AFDM has the same BER performance as OTFS in the reported comparison and outperforms OFDM and OCDM under LMMSE detection.The comparison uses equal AFDM and OTFS resources and practical QPSK configurations.
  • Pilot overhead and spectral efficiency: OTFS requires twice the pilot overhead of AFDM, giving AFDM a spectral-efficiency advantage when channel estimation is included.The reported embedded-pilot occupancy differs because OTFS uses a two-dimensional underlying transform.
  • Fractional-Doppler effects: Fractional Doppler causes more overlap and overhead, so larger pilot SNR is needed; increasing ξν reduces overlap and improves estimation and error performance.With SNRp = 40 dB, embedded-pilot estimation approaches perfect-channel performance in the fractional-Doppler case.

VIII. CONCLUSION

The conclusion presents AFDM as a DAFT-based chirp waveform whose parameters encode the channel’s delay–Doppler structure. Analytical and simulation results support full diversity, low-complexity processing, and favorable comparisons with existing multicarrier schemes.

  • Contributions: AFDM uses multiple discrete-time orthogonal chirp signals generated through the discrete affine Fourier transform.DAFT is the transform underlying AFDM’s waveform construction.
  • Contributions: Deriving the DAFT-domain input–output relation reveals how AFDM parameters can represent the channel’s delay–Doppler profile fully.The parameters are tuned so the DAFT-domain channel impulse response becomes a full delay–Doppler representation.
  • Contributions: AFDM can achieve full diversity in doubly dispersive channels by properly tuning its pulse parameters.The diversity result is established analytically from the full delay–Doppler representation.
  • Contributions: Zero-padding in the DAFT domain enables low-complexity AFDM detection and channel-estimation algorithms.The proposed processing exploits null symbols inserted into the AFDM frame.
  • Results: Simulations show that AFDM outperforms OFDM and other DAFT-based multicarrier schemes while offering advantages over OTFS in pilot and user multiplexing overhead.The conclusion identifies high-mobility communications as the target application.

APPENDIX A

The appendix proves rank and convergence properties used by the AFDM analysis. It establishes linear independence of path-related columns under the parameter construction and bounds the relevant iteration eigenvalues.

  • Rank proof: The proof establishes that the P columns of Φ(δ) are linearly independent, so rank(Φ(δ)) = P.The argument assumes a linear dependence and derives that every coefficient must be zero.
  • Rank proof: Choosing an irrational c2 prevents the phase relation required for a nontrivial linear dependence among the path columns.The contradiction follows because one side becomes irrational while the other is integer-valued.
  • Convergence proof: The convergence proof reduces to showing that every eigenvalue of −S^-1(R − S) has magnitude below one.The appendix concludes |λ(−S^-1(R − S))| < 1.
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