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AFDM: A Full Diversity Next Generation Waveform for High Mobility Communications

Ali Bemani, Nassar Ksairi, Marios Kountouris

arXiv:2104.11331v1cs.ITcs.NI

TL;DR

High mobility and doubly dispersive channels undermine conventional multicarrier orthogonality and motivate a waveform adapted to delay-Doppler structure. The paper proposes AFDM, a DAFT-based multi-chirp scheme whose parameters separate channel paths in the DAFT domain. AFDM achieves full diversity in LTV channels and matches OTFS BER while reducing pilot and user-multiplexing overhead.

  • Problem

    High Doppler shifts destroy subcarrier orthogonality, while general LTV channels lack an easily constructed orthonormal eigenbasis for reliable multicarrier transmission.

  • Method

    AFDM uses DAFT-generated multi-chirp signals and tunes its two parameters so distinct delay-Doppler paths do not overlap in the DAFT domain.

  • Results

    AFDM achieves full diversity in doubly dispersive LTV channels, with BER matching OTFS and outperforming OFDM and other DAFT-based multicarrier schemes.

  • Takeaways & Limitations

    AFDM provides a promising high-mobility waveform with lower pilot and user-multiplexing overhead than OTFS.

Abstract

from arXiv · show

We present Affine Frequency Division Multiplexing (AFDM), a new chirp-based multicarrier transceiver scheme for high mobility communications in next-generation wireless systems. AFDM is based on discrete affine Fourier transform (DAFT), a generalization of discrete Fourier transform characterized with two parameters that can be adapted to better cope with doubly dispersive channels. Based on the derived input-output relation, the DAFT parameters underlying AFDM are set in such a way to avoid that time domain channel paths with distinct delays or Doppler frequency shifts overlap in the DAFT domain. The resulting DAFT domain impulse response thus conveys a full delay-Doppler representation of the channel. We show that AFDM can achieve the full diversity of linear time-varying (LTV) channels. Our analytical results are validated through numerical simulations, which evince that AFDM outperforms state-of-the-art multicarrier schemes in terms of bit error rate (BER) in doubly dispersive channels.

I. INTRODUCTION

High mobility creates doubly dispersive channels that disrupt conventional multicarrier orthogonality. AFDM addresses this challenge with an adaptively parameterized chirp-based waveform designed to preserve channel-path separability and achieve full diversity.

  • Large Doppler shifts in high-mobility environments destroy OFDM and SC-FDMA subcarrier orthogonality.
  • Orthogonal eigenfunctions are optimal for time-varying multipath channels, but finding them for general LTV channels is difficult.
  • Chirp bases provide an adjustable alternative for time-varying channels, although they are not generally optimal.
  • AFDM uses multi-chirp signals and tunes DAFT parameters to prevent distinct-delay or distinct-Doppler paths from overlapping in the DAFT domain.
  • AFDM is analytically shown to achieve full LTV-channel diversity, with BER comparable to OTFS at lower complexity.

II. AFFINE FOURIER TRANSFORM

The affine Fourier transform generalizes standard time-frequency transforms, while its discrete form provides the transform framework used by AFDM. Sampling and periodicity constraints determine the discrete implementation and prefix requirements.

  • The AFT is a continuous transformation that maps a continuous-time signal into a transformed representation and underlies AFDM.
  • The AFT generalizes several standard transforms, including the Fourier, Laplace, and fractional Fourier transforms.
  • The DAFT is a discretization of the AFT whose periodicity constraints govern sampling and the prefix used by DAFT-based multicarrier symbols.
  • DAFT signals are represented with samples s(nT) and S(nF) arranged as vectors over the period [0, N).
  • The discrete transform includes a quadratic phase factor through Λc = diag(e−j2πcn2, n = 0, 1, . . . , N −1).

III. AFFINE FREQUENCY DIVISION MULTIPLEXING

AFDM is a DAFT-based multicarrier transceiver: inverse DAFT maps information symbols into the time domain, and receiver-side DAFT produces the effective affine-Fourier-domain channel response.

  • AFDM uses inverse DAFT for transmission and DAFT at reception to obtain the effective discrete affine Fourier domain channel response.
  • The resulting receiver representation describes how the channel acts on the transmitted data in the affine Fourier domain.

A. Modulation

AFDM modulation maps QAM information symbols through the DAFT framework and uses a chirp-periodic prefix to accommodate multipath propagation under the waveform’s periodicity.

  • Modulation: The input vector x contains QAM information symbols in the discrete affine Fourier domain.
  • Modulation: AFDM modulation maps the affine-Fourier-domain symbols into a time-domain signal using the DAFT-based construction.
  • Modulation: AFDM requires a chirp-periodic prefix instead of an OFDM cyclic prefix because its signal periodicity differs.
  • Modulation: The prefix length L must be at least the channel’s maximum delay spread measured in samples.
  • Modulation: The chirp-periodic prefix samples are formed from the end of the time-domain signal with a phase factor involving c1, N, and the prefix index.
  • Modulation: The chirp-periodic prefix reduces to a cyclic prefix when 2Nc1 is an integer and N is even.

B. Channel

The channel model represents a doubly dispersive channel as multiple delayed and Doppler-shifted paths, with assumptions that simplify diversity analysis and cyclic-prefix processing.

  • Channel model: The received-sample model describes P channel paths through complex gains, Doppler shifts, and integer delays.Paths may share a delay while having different Doppler shifts, allowing Doppler spreads on individual delay taps.
  • Channel model: The Doppler shift is normalized by subcarrier spacing and decomposed into integer and fractional parts.The normalized shift is ν_i = α_i + a_i, with bounded integer and fractional components.
  • Analysis assumptions: The analysis assumes zero fractional Doppler parts, maximum delay l_max < N, and a cyclic-prefix length exceeding l_max − 1.Neglecting fractional Doppler is stated to be reasonable for large N.
  • Matrix model: After cyclic-prefix removal, the received signal is written in matrix form with an N × N channel matrix and circular Gaussian noise.The noise vector has covariance N_0I.
  • Matrix model: When 2Nc1 is an integer and N is even, the cyclic-prefix-related matrix Γ_CPP_i becomes the identity.This condition simplifies the channel representation after cyclic-prefix processing.

C. Demodulation

AFDM demodulation applies the DAFT to received samples after IDAFT-based transmission, producing effective DAFT-domain symbols and noise with unchanged covariance.

  • Receiver processing: The receiver obtains DAFT-domain output symbols by applying the DAFT after transmission through the channel.This operation extracts the effective affine-Fourier-domain channel response.
  • Effective channel: In matrix form, the demodulated output is expressed through an effective channel matrix Heff.The effective channel is formed by DAFT and inverse-DAFT-related transformations around the time-domain channel matrix.
  • Noise transformation: The transformed noise is ew = Λc2FΛc1w, and its covariance matches that of the original noise because the transform is unitary.Unitary processing preserves the noise covariance.

D. Input-Output Relation

The input-output relation shows how each channel path maps into the DAFT domain, with parameter choices determining the location of its nonzero response.

  • Path representation: The channel-path matrix Ai places the delayed contribution at a column indexed modulo N by the path delay.The row-dependent location uses the modulo-N operation.
  • Path representation: For integer Doppler shifts and suitable c1, each path matrix Hi has one nonzero element in every row.The nonzero location is shifted by loci = αi + 2Nc1li.
  • Input-output mapping: The nonzero entry in row p of Hi occurs at column (p + loci)N.This location determines how the path appears in the DAFT-domain input-output relation.

IV. AFDM PARAMETERS

AFDM selects DAFT parameters so distinct delay-Doppler paths occupy separate DAFT-domain positions, yielding a full delay-Doppler channel representation and full LTV-channel diversity.

  • Parameter design: AFDM chooses c1 and c2 so its DAFT-domain impulse response forms a full delay-Doppler representation, unlike the stated limitations of OCDM and DAFT-OFDM.The parameter choice is designed to support full diversity in LTV channels.
  • Path separation: Distinct path responses avoid overlap when the corresponding ranges of loci and locj have an empty intersection.The separation condition is derived from the unique nonzero entry associated with each path.
  • Path separation: Paths with equal delays but different Doppler shifts always occupy distinct DAFT-domain positions.For unequal delays, an additional constraint is required to maintain separation.
  • Parameter design: With no time-domain impulse-response sparsity, c1 must satisfy the minimum-delay-separation constraint implied by adjacent delays.This setting determines the chirp time-frequency representation shown in Fig. 2.
  • Path separation: The condition 2αmaxlmax + 2αmax + lmax < N prevents modular overlap between paths at the minimum and maximum delays.For underspread channels, the paper states that this can hold with moderate N.
  • Effective-channel structure: Under these settings, Heff separates paths by delay or Doppler, so their delay-Doppler profile is recoverable from nonzero-entry positions.The resulting effective-channel structure is illustrated in Fig. 3.

V. DIVERSITY ANALYSIS OF AFDM

The diversity analysis characterizes AFDM through the rank of a matrix built from pairwise symbol differences and shows that suitable DAFT parameters make this rank full for sufficiently large N.

  • Diversity metric: AFDM diversity is determined by the minimum rank of Φ(δ(m,n)) over distinct transmitted symbol vectors.At high SNR, the BER is dominated by pairwise error events with the minimum rank.
  • Full-diversity result: Theorem 1 states that AFDM achieves full diversity, ρ = P, for sufficiently large N when c1 satisfies the required channel-dependent condition.Here P is the number of channel paths, and ρ is the minimum rank across distinct symbol pairs.
  • Proof strategy: The proof establishes full rank by showing that the P columns of Φ(δ) are linearly independent under the required parameter conditions.The general proof follows the two-path construction presented in detail.
  • Parameter condition: Choosing c2 as an arbitrary irrational number or sufficiently small rational number ensures the key inequality and makes Φ(δ) full rank.This condition is established after excluding the zero-difference case and using the nonzero structure of δ.

VI. SIMULATION RESULTS

Simulations compare AFDM with DAFT-OFDM, OCDM, and OTFS under two doubly dispersive channel settings, showing that path separation avoids destructive overlap and matches OTFS BER performance.

  • Experimental setup: AFDM, DAFT-OFDM, OCDM, and OTFS are compared using 10^6 independent channel realizations with Gaussian path gains.The two-path experiment uses BPSK and ML detection; the 21-path experiment uses QPSK and MMSE detection.
  • Two-path comparison: DAFT-OFDM has diversity order one, while OCDM can also fall to diversity one when overlapping paths destructively add.OCDM does not achieve full diversity even when its effective channel has two nonzero elements per row.
  • AFDM performance: AFDM achieves full diversity through path separation, avoiding the destructive addition observed for OCDM.The result is reported for both the two-path and 21-path LTV channel experiments.
  • Pilot overhead: AFDM requires fewer pilot guard symbols than OTFS, producing a throughput gap that increases with the number of required orthogonal pilot transmissions.The reported guard-symbol requirements are (2lmax + 2)(2αmax + 1) − 2 for AFDM and (2lmax + 1)(4αmax + 1) − 1 for OTFS.

VII. CONCLUSIONS

AFDM is a DAFT-based waveform using orthogonal chirps whose parameters represent the channel’s delay-Doppler profile without path overlap. The analysis shows full diversity in doubly dispersive channels, with advantages over conventional multicarrier schemes and OTFS overhead.

  • VII. CONCLUSIONS: AFDM uses multiple discrete-time orthogonal chirp signals generated by the discrete affine Fourier transform (DAFT).The DAFT is characterized by two parameters.
  • VII. CONCLUSIONS: AFDM parameters are set so the DAFT-domain channel impulse response fully represents the channel’s delay-Doppler profile.This design avoids overlap among paths with distinct delays or Doppler shifts.
  • VII. CONCLUSIONS: AFDM always achieves full diversity in doubly dispersive channels.This is an analytical result for the proposed waveform.
  • VII. CONCLUSIONS: AFDM outperforms OFDM and other DAFT-based multicarrier schemes in high-mobility communications.The conclusion also identifies advantages over OTFS in pilot and user multiplexing overhead.
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