On the Robustness of AFBM Sensing to Power Amplifier Nonlinearities

Abstract: We investigate the impact of power amplifier (PA) nonlinearities on the sensing performance of affine filter bank modulation (AFBM). While AFBM offers several advantageous properties for integrated sensing and communications (ISAC) - including reduced out-of-band emission (OOBE), low peak-to-average power ratio (PAPR), and natural robustness to doubly-dispersive (DD) channel effects - mitigating waveform distortion typically requires highly linear PAs. This creates a fundamental contradiction with ISAC applications, which demand high transmit power for reliable sensing. Our analytical results reveal that the structure of the effective AFBM modulation matrix dictates how distortion propagates within the ambiguity function (AF). Furthermore, simulations demonstrate that both the AF and the overall sensing performance of AFBM remain remarkably insensitive to such nonlinearities. These findings highlight the robustness of AFBM, making it a highly viable candidate for practical ISAC deployments constrained by hardware impairments.

• The paper develops a Bussgang-based semi-analytical model showing that AFBM ambiguity sidelobes remain largely stable under severe PA distortion, including operation at 1 dB input back-off.
• Simulations show that nonlinearities mainly reduce high-Doppler sidelobe depressions while leaving delay sidelobes nearly unchanged, with the model matching empirical results except in deep sidelobes below −40 dB.
• AFBM achieves lower delay-Doppler estimation RMSE than AFDM at 2 dB input back-off, indicating that its filtered affine structure improves ISAC sensing robustness beyond what ambiguity-function analysis alone predicts.

Overview

The paper investigates the impact of power amplifier (PA) nonlinearities on the sensing performance of affine filter bank modulation (AFBM), a filtered variant of affine frequency division multiplexing (AFDM) that combines DAFT-domain spreading with per-subcarrier filtering. The motivation stems from a tension inherent to integrated sensing and communications (ISAC): sensing favors driving the PA near saturation to maximize radiated power, while waveform distortion mitigation conventionally requires large input back-off (IBO). Prior work has shown that OFDM sensing degrades markedly under PA nonlinearities, whereas AFDM's ambiguity function (AF) exhibits inherent insensitivity to such distortion. The central question addressed is whether AFBM's filtered architecture preserves or enhances this resilience (2606.11879).

System model

The AFBM transmit chain consists of a DAFT-spread AFDM block followed by a polyphase filterbank network. The single-symbol transmission matrix is $$\bar{\mathbf{G}}' = \tilde{\mathbf{G}}\mathbf{Q}_P\mathbf{C}_f$$, where $\tilde{\mathbf{G}}$ is the filtering matrix from a prototype filter of length $ON$ (overlap factor $O$), $\mathbf{Q}_P$ is an extended inverse DAFT with frequency-domain zero-padding, and $$\mathbf{C}_f = \mathbf{W}_L\mathrm{diag}\{\tilde{\mathbf{b}}\}$$ is a precoding matrix that restores complex orthogonality by compensating filter-induced interference for $O \le 1.5$. Data symbols occupy the first and last $L/4$ subcarrier positions, consistent with precoded FBMC conventions. The full $K$-block transmit signal is $\mathbf{s} = \bar{\mathbf{G}}\mathbf{x}$, with $\tilde{\mathbf{G}}$0; notably, no cyclic prefix is required given the well-localized filter structure.

The PA is modeled with the Rapp (SSPA) AM/AM characteristic, parameterized by smoothness factor $\tilde{\mathbf{G}}$1 and saturation voltage $\tilde{\mathbf{G}}$2, with distortion severity controlled via IBO $\tilde{\mathbf{G}}$3. Exploiting the central limit theorem, the Gaussian-distributed input permits Bussgang decomposition of the PA output as $\tilde{\mathbf{G}}$4, where the distortion $\tilde{\mathbf{G}}$5 is uncorrelated with $\tilde{\mathbf{G}}$6 and $\tilde{\mathbf{G}}$7 is the complex Bussgang gain.

Ambiguity function analysis

The core analytical contribution is a semi-analytical expression for the average AF power of the nonlinearly amplified AFBM signal. The AF of the input signal admits the quadratic form $\tilde{\mathbf{G}}$8 with ambiguity matrix $\tilde{\mathbf{G}}$9, where $ON$0 and $ON$1 are the delay-shift and Doppler-modulation matrices. The expected squared magnitude of the linear case follows the standard trace/Frobenius-norm decomposition involving the second and fourth moments of the constellation.

For the nonlinear case, the distortion is projected onto the modulation subspace via $ON$2, which is approximated as a zero-mean circular complex Gaussian vector with covariance $ON$3, valid when the prototype filter is well localized. Under moment-factorization assumptions (justified empirically), the resulting expression for $ON$4 comprises the scaled linear term weighted by $ON$5, signal-distortion cross-terms involving traces of $ON$6, and pure distortion terms. This formulation explicitly exposes how the structure of $ON$7 and the distortion statistics govern sidelobe behavior under nonlinear amplification.

Numerical results

Simulations use $ON$8 active subcarriers, chirp size $ON$9, filterbank DFT size $O$0, $O$1 blocks of 4-QAM symbols, and a three-path doubly dispersive channel at $O$2 GHz, with the Phydyas prototype filter at $O$3 unless stated otherwise. Three sets of results are reported.

First, the zero-Doppler and zero-delay AF cuts before and after nonlinear amplification at IBO = 1 dB — an aggressively nonlinear operating point — show that ranging sidelobes remain essentially unchanged. The Doppler cut exhibits a reduction in the depth of recurrent sidelobe depressions, but the overall sidelobe level is nearly invariant. The semi-analytical expression closely tracks the empirical AF, with mismatches up to 3 dB confined to depressions below −40 dB, where the projection-based modeling — which omits out-of-subspace distortion components — can slightly overestimate AF energy. This is an acknowledged limitation of the analytical framework.

Second, the behavior is shown to be filter-agnostic: across prototype filters with different overlap factors (truncated to a common support of $O$4 for fairness), nonlinearities leave delay sidelobes largely unaffected and mainly reduce Doppler sidelobes at high Doppler shifts.

Third, comparisons against AFDM and OFDM (with matched effective length $O$5 and AFDM chirp parameters $O$6, $O$7) show that AFBM and AFDM share similar insensitivity in the delay domain, while OFDM exhibits sidelobe regrowth near the mainlobe yet retains globally lower delay sidelobes. In the Doppler domain, AFBM achieves lower sidelobes than both AFDM and OFDM and responds less to nonlinear distortion.

For system-level sensing, a probabilistic data association (PDA)-based estimator is applied to estimate delay and Doppler of the three paths. Under severe nonlinearity (IBO = 2 dB), AFBM yields lower RMSE than AFDM across the SNR range, despite both waveforms' AFs being largely unchanged. The authors attribute this gap to the estimator's sensitivity to how nonlinear distortion projects onto the sensing dictionary — an effect not captured by ambiguity sidelobes alone. This is the paper's strongest practical claim: AFBM outperforms AFDM in radar parameter estimation accuracy under highly impaired hardware operation, indicating that the filtered affine structure confers a robustness advantage beyond what the AF reveals.

Limitations and open questions

Several assumptions bound the validity of the results. The Bussgang analysis relies on the CLT-driven Gaussian approximation of the input, the projection of distortion onto the modulation subspace, and moment factorization between data and distortion; the last is supported only empirically. The analytical AF model degrades precisely in the deep sidelobe depressions (below −40 dB), where it can overestimate energy by up to 3 dB, so the framework is least accurate where sidelobe suppression matters most for detection. The evaluation is restricted to the Rapp AM/AM model with a single prototype filter family, a three-path channel, and a PDA-based estimator; sensitivity to other AM/PM characteristics, memory effects, alternative estimators, and denser multipath remains unexamined. The paper also leaves open whether the observed RMSE advantage over AFDM persists at other IBO values and overlap factors, and how other RF impairments — phase noise, oscillator drift, I/Q imbalance — interact with the filtered affine structure.

Conclusion

The paper establishes, through a Bussgang-based semi-analytical AF characterization and system-level simulations, that AFBM sensing is fundamentally insensitive to PA nonlinearities: its ambiguity properties and RMSE performance remain largely intact even at IBO as low as 1–2 dB, and it surpasses AFDM in parameter estimation accuracy under severe nonlinearity. These findings position AFBM as a viable ISAC waveform for hardware-constrained deployments, while the analytical framework's accuracy in deep sidelobe regimes and the extension to broader impairment models remain open problems.