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AFDM-Enabled Integrated Sensing and Communication: Theoretical Framework and Pilot Design

Fan Zhang, Zhaocheng Wang, Tianqi Mao, Tianyu Jiao, Yinxiao Zhuo, Miaowen Wen, Wei Xiang, Sheng Chen, George K. Karagiannidis

arXiv:2502.14203v2eess.SP

TL;DR

Doubly dispersive channels challenge classical ISAC waveforms, while AFDM-enabled ISAC still lacks theoretical sensing analysis and faces pilot-spacing tradeoffs. The paper derives sensing bounds and ambiguity properties and proposes a pilot design; its analysis and numerical results support improved dual-functional performance and relief from spacing-related range limits.

  • Problem

    Classical ISAC waveforms face challenges in doubly dispersive channels, while AFDM-enabled ISAC investigations lack theoretical sensing analysis and existing pilot schemes face sensing-range and delay-spread limits from pilot spacing.

    Source: A. Related Works on AFDM-Enabled ISAC · p. 2

  • Method

    The paper derives closed-form distance and velocity CRBs, formulates pilot-assisted AFDM ambiguity properties, and proposes a superimposed-pilot design for sensing and channel estimation.

    Source: B. Motivation and Our Contributions · p. 2, §VI · p. 11

  • Results

    The proposed pilot design is analytically shown to achieve an ideal ambiguity function and optimal channel estimation simultaneously, and numerical results demonstrate its dual-functional superiority.

    Source: B. Motivation and Our Contributions · p. 2, §VI · p. 11

  • Takeaways & Limitations

    The proposed design overcomes sensing-range and delay-spread limitations imposed by pilot spacing while targeting accurate channel estimation and stable sensing.

    Source: B. Motivation and Our Contributions · p. 2, §VI · p. 11

Abstract

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The integrated sensing and communication (ISAC) has been envisioned as one representative usage scenario of sixth-generation (6G) network. However, the unprecedented characteristics of 6G, especially the doubly dispersive channel, make classical ISAC waveforms rather challenging to guarantee a desirable performance level. The recently proposed affine frequency division multiplexing (AFDM) can attain full diversity even under doubly dispersive effects, thus becoming a competitive candidate for next-generation ISAC waveforms. Relevant investigations are still at an early stage, which involve only straightforward design lacking explicit theoretical analysis. This paper provides an in-depth investigation on AFDM waveform design for ISAC applications. Specifically, the closed-form Cr\'{a}mer-Rao bounds of target detection for AFDM are derived, followed by a demonstration on its merits over existing counterparts. Furthermore, we formulate the ambiguity function of the pilot-assisted AFDM waveform for the first time, revealing conditions for stable sensing performance. To further enhance both the communication and sensing performance of the AFDM waveform, we propose a novel pilot design by exploiting the characteristics of AFDM signals. The proposed design is analytically validated to be capable of optimizing the ambiguity function property and channel estimation accuracy simultaneously as well as overcoming the sensing and channel estimation range limitation originated from the pilot spacing. Numerical results have verified the superiority of the proposed pilot design in terms of dual-functional performance.

I. INTRODUCTION

The introduction motivates AFDM-enabled ISAC by the difficulty of balancing communication and sensing in doubly dispersive channels and identifies limited sensing theory and pilot-spacing tradeoffs. It frames a theoretical analysis and pilot design, then specifies the monostatic system and AFDM signal model.

I. INTRODUCTION

  • In doubly dispersive channels, OFDM inter-carrier interference degrades communication and destroyed subcarrier orthogonality impairs both communication and sensing.ISAC shares hardware and spectrum for both functions, with resource use and information exchange described as potential benefits.Source: §I · p. 1

A. Related Works on AFDM-Enabled ISAC

B. Motivation and Our Contributions

  • The paper derives distance and velocity CRBs, analyzes pilot-assisted AFDM ambiguity properties, and proposes a pilot design targeting stable sensing and accurate channel estimation.The design incorporates DAFT parameter setting, pilot spacing, and a sequence that exploits AFDM multipath separation; simulations report estimation errors close to the derived CRBs.Source: B. Motivation and Our Contributions · pp. 2–3

II. SYSTEM MODEL

  • The modeled monostatic BS transmits data to a UE and uses target echoes to estimate target distance and radial velocity, assuming independent channels and perfect self-interference cancellation.The sensing and communication channels are modeled as independent and uncorrelated.Source: §II · p. 3

A. Signal Model

  • The signal model uses a superimposed pilot scheme and constrains AFDM parameters to prevent effective-path overlap and delay ambiguity in the DAFT domain.A chirp-periodic prefix is inserted to combat multipath-induced inter-symbol interference.Source: A. Signal Model · p. 3

B. Channel Estimation and Data Demodulation

The communication channel is represented using multipath components in the DAFT domain, with integer Doppler paths forming a basis. Pilots support effective-channel estimation, and channel estimation and data demodulation can be iterated to reduce mutual interference.

B. Channel Estimation and Data Demodulation

  • Each path’s DAFT-domain channel matrix incorporates its delay, Doppler shift, cyclic prefix, and Doppler-shift matrix.The passage defines normalized delay and Doppler and identifies the Doppler-shift matrix.Source: B. Channel Estimation and Data… · p. 4
  • Integer-Doppler paths form a basis for representing the time-varying channel, with at most Lm = (2νm + 1)(τm + 1) possible paths.The normalized delay and Doppler ranges determine the path count.Source: B. Channel Estimation and Data… · p. 4
  • Pilots are used to estimate the effective channel by obtaining the complex gain of each possible path, with MMSE yielding estimated gains.A path indicator marks whether each path exists, and a predefined threshold is used to form the estimated effective channel.Source: B. Channel Estimation and Data… · pp. 4–5
  • Channel estimation and data demodulation can be iterated to reduce pilot–data mutual interference and enhance channel estimation accuracy and data detection performance.Source: B. Channel Estimation and Data… · p. 5

C. Target Sensing

The sensing model treats the received signal as echoes from a point target with distance and radial velocity, sampled after propagation through noise and clutter. A Doppler-compensated correlation produces a range-Doppler function for target detection and parameter estimation.

C. Target Sensing

  • The target model assumes reflection from a point target, with distance and relative radial velocity encoded in the received echo.The received signal is sampled after analog-to-digital conversion; noise includes thermal noise and clutter.Source: C. Target Sensing · p. 5
  • A Doppler-compensation-based correlation method forms the range-Doppler function (RDF), which is used to detect a target and obtain its distance and velocity.Detection uses a hypothesis test at candidate delay-Doppler points.Source: C. Target Sensing · p. 5
  • Delay resolution can be reduced by oversampling the received and transmitted signals, while Doppler resolution can be reduced through finer-grained Doppler compensation.Source: C. Target Sensing · p. 5

III. THEORETICAL FRAMEWORK FOR AFDM-ENABLED ISAC

The paper develops a theoretical framework for AFDM-enabled ISAC by deriving sensing CRBs for distance and velocity and comparing AFDM with OFDM and OCDM. Its analysis finds that AFDM sensing is more robust to subcarrier power variation.

III. THEORETICAL FRAMEWORK FOR AFDM-ENABLED ISAC

  • The framework derives CRBs for target distance and velocity from the received echo model and Fisher information matrix.The bounds apply to full-data and pilot-assisted AFDM signals using their respective subcarrier-power expressions.Source: §III · p. 5, A. Sensing CRBs Analysis · pp. 5–6

A. Sensing CRBs Analysis

  • The CRB expressions also cover OFDM and OCDM by selecting c1 = 0 and c1 = 1/(2Nc), respectively.Source: A. Sensing CRBs Analysis · p. 6
  • AFDM achieves the most balanced subcarrier sensing weights, closest to zero, while OFDM has the largest weight fluctuations, making AFDM’s delay CRB more stable under flexible power allocation.The comparison uses Nc = 16 subcarriers and motivates robustness in settings such as multi-user scenarios with dynamic demands.Source: A. Sensing CRBs Analysis · p. 6
  • AFDM has a negligible delay-CRB PDF tail, whereas OFDM has the highest tail probability because of its largest variance, despite similar mean CRBs across the three waveforms.The figure compares the PDFs of CRB¯τ for AFDM, OCDM, and OFDM.Source: A. Sensing CRBs Analysis · pp. 6–7, Figure 4 · p. 6

B. Ambiguity Function Analysis

The paper formulates the pilot-assisted AFDM ambiguity function and analyzes its statistical properties under stated modulation and symbol assumptions. The results relate ambiguity behavior to modulation modulus and subcarrier count, with implications for sensing stability.

B. Ambiguity Function Analysis

  • The pilot-assisted AFDM ambiguity function is formulated and analyzed through its statistical properties, with the discrete function defined over delay-Doppler indices.The paper relates the range-Doppler function to a shifted and scaled ambiguity function and specifies the region relevant to sensing.Source: B. Ambiguity Function Analysis · p. 7
  • The statistical analysis assumes constellation points with at least three distinct phase values, symmetric about the center, and independent, identically distributed symbols across subcarriers.The text notes that QPSK and 16-QAM satisfy the constellation assumption, unlike BPSK.Source: B. Ambiguity Function Analysis · p. 7
  • For arbitrary delay-Doppler indices, the ambiguity function of constant-modulus modulation schemes such as QPSK is no larger than that of nonconstant-modulus schemes.Source: B. Ambiguity Function Analysis · pp. 7–8
  • The ambiguity-function quantity χ(τ,ν) converges to zero as the number of subcarriers approaches infinity.Source: B. Ambiguity Function Analysis · p. 8
  • Data randomness fluctuates ambiguity-function mainlobes and sidelobes, potentially causing missed detections or false alarms; constant-modulus modulation and more subcarriers are proposed to reduce σ²χ(τ,ν).The stated mitigation follows from the constant-modulus and large-subcarrier results.Source: B. Ambiguity Function Analysis · p. 8

IV. PILOT DESIGN FOR SENSING AND COMMUNICATION PERFORMANCE ENHANCEMENT

The pilot design shapes AFDM pilots to achieve an ideal ambiguity function, supporting sensing and channel estimation together. Its delay-independent pilot spacing addresses the range-versus-pilot-count trade-off, while phase-adjusted CAZAC sequences preserve autocorrelation under Doppler.

IV. PILOT DESIGN FOR SENSING AND COMMUNICATION PERFORMANCE ENHANCEMENT

  • When 2c1Nc is an integer, a pilot with an ideal ambiguity function achieves optimal channel estimation by minimizing mean squared error.The expected ambiguity function of the pilot-assisted signal equals total signal power at (0, 0), while other grid values depend on the pilot ambiguity function.Source: §IV · p. 8
  • Conventional guard spacing must grow with delay spread, limiting sensing and estimation range at fixed spacing and reducing nonzero pilot count when spacing increases.The resulting trade-off makes pilot assignment challenging.Source: §IV · p. 8
  • Theorem 5 specifies AFDM parameters, pilot spacing, and a CAZAC-based sequence that yield an ideal ambiguity function.The spacing Q is independent of maximum delay τm.Source: §IV · p. 8
  • Because Q is independent of maximum delay, the design need not reduce nonzero pilot count as τm increases, helping maintain estimation accuracy and long-range sensing.The proposed design is described as enabling simultaneous channel estimation and target sensing with superior accuracy.Source: §IV · p. 8
  • Unlike directly transforming a CAZAC sequence into the DAFT domain, the design phase-adjusts sequences at critical delay-Doppler pairs to preserve autocorrelation under substantial Doppler shifts.Direct transformation cannot guarantee an ideal ambiguity function because CAZAC autocorrelation deteriorates at large Doppler shifts.Source: §IV · p. 9

V. NUMERICAL RESULTS

The numerical evaluation considers AFDM communication and single-target sensing in small- and large-range scenarios, with maximum normalized Doppler shift set to 2.

V. NUMERICAL RESULTS

  • The simulations use 128 subcarriers, QPSK symbols, and maximum normalized Doppler frequency shift νm = 2.Both small-range (τm = 2) and large-range (τm = 15) scenarios are considered.Source: §V · p. 9
  • The communication channel has three paths with randomly selected normalized delays and Doppler shifts, and complex Gaussian path coefficients.For sensing, a single target has normalized delay and Doppler shift drawn uniformly from the stated ranges.Source: §V · p. 9
  • Table I summarizes simulation parameters, and Fig. 6 compares channel-estimation performance among pilot designs for the large-range case τm = 15.Source: Table I · p. 9, Figure 6 · p. 9

A. Communication Performance

Across the evaluated delay ranges, the proposed pilot design maintains channel-estimation performance and avoids the BER degradation seen with traditional SPI as delay spread grows. In the small-range case, both proposed and traditional SPI pilots improve MSE over a single-pilot design.

A. Communication Performance

  • The evaluation averages MSE over 10,000 simulation runs and compares a single pilot, traditional SPI with Np = 8, and proposed designs with Np = 8 or 16.All schemes use the same AFDM parameters, c1 = 4/Nc and c2 = π−3.Source: A. Communication Performance · p. 9
  • 7 dB improvement in MSE over the single-pilot scheme is achieved by both the proposed and traditional SPI designs with Np = 8 at τm = 2.Source: A. Communication Performance · p. 9
  • At τm = 15, traditional SPI with Np = 8 suffers pilot interference and BER degradation, while the proposed design maintains optimal channel estimation and avoids BER degradation.The proposed design has similar MSEs in the τm = 2 and τm = 15 scenarios.Source: A. Communication Performance · p. 10, Figure 8 · p. 10, Figure 7 · p. 10
  • For the proposed and traditional SPI designs, BER generally falls as SNRd rises below 18 dB but rises above 21 dB as channel-estimation accuracy declines.The single-pilot design has much larger BER than the other two designs.Source: A. Communication Performance · p. 10

B. Sensing Performance

The proposed pilot has a much lower missed-detection probability than the traditional SPI pilot at the same false-alarm probability in the large-range test. Sensing errors also track the derived CRB bounds, while the conclusion presents the AFDM framework and pilot design as supporting sensing and communication performance.

B. Sensing Performance

  • At τm = 15, the proposed pilot has a much lower missed-detection probability than traditional SPI at the same false-alarm probability.The proposed pilot's sensing performance is primarily determined by receive SNR rather than high sidelobes.Source: B. Sensing Performance · p. 10
  • Distance error decreases with sampling period and velocity error decreases with subcarrier spacing, while both RMSEs remain close to the square roots of their derived CRBs.The comparisons use fc = 28 GHz and τm = 15.Source: B. Sensing Performance · p. 11, Figure 10 · p. 11

VI. CONCLUSIONS

  • The paper derives closed-form CRBs for AFDM target sensing, presenting them as system-design guidelines and insight into AFDM's advantages over alternatives.The conclusion also describes a theoretical framework for AFDM-enabled ISAC.Source: §VI · p. 11
  • The proposed superimposed pilot is analytically shown to achieve an ideal ambiguity function and optimal channel estimation simultaneously, overcoming pilot-spacing range limitations.Numerical results are reported to validate the derivations and demonstrate the pilot's dual-functional superiority.Source: §VI · p. 11

APPENDIX

The appendix derives the Fisher information matrix by simplifying terms and combining its elements. It also proves the ambiguity-function decomposition for a signal containing pilot and data components.

A. Derivation of FIM

  • The FIM derivation drops ∂t when it is zero in most cases, then calculates an example entry and combines the elements to obtain the matrix.For FI[1,1], the summation simplifies because terms for distinct indices m1 and m2 vanish.Source: A. Derivation of FIM · p. 11

B. Proof of Theorem 1

  • The proof substitutes the pilot-plus-data signal into the expected squared ambiguity function and identifies the pilot and data contributions.The two terms are identified as the ambiguity functions of the pilot and data signals, respectively.Source: B. Proof of Theorem 1 · p. 12
  • Terms containing the stated data-signal expectations vanish, after which the proof substitutes intermediate expressions and evaluates terms in the resulting equation.The proof evaluates the first term and notes that the second and third terms are equal.Source: C. Derivation of σ2 · p. 12

D. Proof of Theorem 2

The proof shows that ambiguity-function variance depends on modulation only at zero delay and Doppler. There, constant-modulus modulation minimizes the variance.

D. Proof of Theorem 2

  • For nonzero delay-Doppler offsets, the ambiguity-function variance is independent of the specific modulation method.At zero offset, the variance can be reduced by minimizing the constellation's fourth amplitude moment.Source: D. Proof of Theorem 2 · p. 12
  • At zero delay and Doppler, the fourth amplitude moment is at least the square of the second moment, with equality for constant-modulus constellation points.The proof invokes the Cauchy-Schwarz inequality for this bound.Source: D. Proof of Theorem 2 · p. 12
  • Constant-modulus modulation therefore minimizes ambiguity-function variance at zero delay and Doppler.This conclusion follows from attaining the lower bound on the fourth amplitude moment.Source: D. Proof of Theorem 2 · p. 12

F. Proof of Theorem 4

The proofs connect pilot ambiguity properties to channel-estimation matrix orthogonality and show that the proposed pilot design yields an ideal ambiguity function.

F. Proof of Theorem 4

  • An ideal ambiguity function makes distinct columns of Ψp orthogonal, a condition associated with minimum channel-estimation error.The proof interprets their inner product as pilot cross-correlation at nonzero delay-Doppler offsets.Source: F. Proof of Theorem 4 · pp. 12–13
  • When 2c1Nc is an integer, ΓCPPi becomes the identity matrix in the proof's simplification.The proof also uses the stated relation between delay and Doppler operators.Source: F. Proof of Theorem 4 · p. 13

G. Proof of Theorem 5

  • The proposed pilot design satisfying (54) has χp(τ,ν) non-zero only at (τ,ν)=(0,0), so it can ensure an ideal ambiguity function for the stated design.When Q does not divide loc(τ,ν), χp(τ,ν)=0 because no k1 and k2 satisfy the stated equation.Source: G. Proof of Theorem 5 · p. 13

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