This paper investigates the ambiguity function (AF) of the emerging affine frequency division multiplexing (AFDM) waveform for random integrated sensing and communication (ISAC) signaling under a pulse shaping regime. Specifically, we first derive the closed-form expression of the average squared discrete period AF (DPAF) for AFDM waveform without pulse shaping, revealing that the AF depends on the parameterc1and the kurtosis of random communication data, while being independent of the parameterc2. As a step further, we conduct a comprehensive analysis on the DPAFs of various waveforms, including AFDM, orthogonal frequency division multiplexing (OFDM) and orthogonal chirp-division multiplexing (OCDM). Our results indicate that all three waveforms exhibit the same number of regular depressions in the sidelobes of their DPAFs, which incurs performance loss for detecting and estimating weak targets. However, the AFDM waveform can flexibly control the positions of depressions by adjusting the parameterc1, which motivates a novel design approach of the AFDM parameters to mitigate the adverse impact of depressions of the strong target on the weak target. Furthermore, the closed-form expressions of the average squared DPAFs for pulse-shaped AFDM, OFDM and OCDM waveforms are derived, which demonstrates that the pulse shaping filter generates the shaped mainlobe along the delay axis and the rapid roll-off sidelobes along the Doppler axis. Numerical results verify the effectiveness of our theoretical analysis and proposed design methodology for the AFDM waveform.

Affine frequency division multiplexing (AFDM) and orthogonal time frequency space (OTFS) are two promising advanced waveforms proposed for reliable communications in high-mobility scenarios. In this paper, we introduce a simple transmit diversity technique, termed cyclic delay-Doppler shift (CDDS), for these two advanced waveforms to achieve ultra-reliable communications in doubly selective channels (DSCs). Two simple CDDS schemes, named modulation-domain CDDS (MD-CDDS) and time-domain CDDS (TD-CDDS), are proposed, which perform CDDS in advance at the transmitter before and after the modulation, respectively. We demonstrate that both of the two proposed CDDS schemes can be implemented efficiently and flexibly by multiplying the transmit vector with a well-designed precoding matrix, which is nothing but a sparse phase-compensated permutation matrix. Moreover, we theoretically and numerically prove that CDDS can provide MIMO-AFDM and MIMO-OTFS with optimal transmit diversity gain when a proper CDDS step is adopted. Compared to the conventional transmit diversity techniques, the proposed CDDS scheme enjoys the advantages of lower channel estimation overhead, implementation complexity, and signal processing latency, making it particularly suitable for ultra-reliable communications in high-mobility scenarios.

This paper investigates the sensing potential of affine frequency division multiplexing (AFDM) in high-mobility integrated sensing and communication (ISAC) from the perspective of radar waveforms. We introduce an innovative parameter selection criterion that establishes a precise mathematical equivalence between AFDM subcarriers and Nyquist-sampled frequency-modulated continuous-wave (FMCW). This connection not only provides a clear physical insight into AFDM's sensing mechanism but also enables a direct mapping from the DAFT index to delay-Doppler (DD) parameters of wireless channels. Building on this, we develop a novel input-output model in a DD-parameterized DAFT (DD-DAFT) domain for AFDM, which explicitly reveals the inherent DD coupling effect arising from the chirp-channel interaction. Subsequently, we design two matched-filtering sensing algorithms. The first is performed in the time-frequency domain with low complexity, while the second is operated in the DD-DAFT domain to precisely resolve the DD coupling. Simulations show that our algorithms achieve effective pilot-free sensing and demonstrate a fundamental trade-off between sensing performance, communication overhead, and computational complexity. The proposed AFDM outperforms classical AFDM and other variants in most scenarios.

This paper investigates affine frequency division multiplexing (AFDM) with practical digital-to-analog conversion (DAC) and analog-to-digital conversion (ADC) filters over doubly-dispersive channels. Firstly, we derive the input-output relationship for AFDM with practical DAC and ADC filters, thereby characterizing the discrete affine Fourier transform (DAFT) domain channel. Subsequently, the equivalent sampled DAFT channel matrix is visualized. Additionally, an analysis of the auto-ambiguity function of the AFDM pilot signal is conducted, focusing on the impact of various DAC and ADC filters. The selection of these filters exerts a significant influence on the sidelobe behavior in the time-delay domain of the sub-ambiguity function. Simulations demonstrate that AFDM with practical DAC/ADC outperforms OFDM and OCDM in bit error rate under identical conditions, while the root-raised cosine filter enhances spectral efficiency.

This paper explores a novel integrated localization and communication (ILAC) system using the affine Fourier transform multicarrier (AFT-MC) waveform. Specifically, we consider a multiple-input multiple-output (MIMO) AFT-MC system with ILAC and derive a continuous delay and Doppler shift channel matrix model. Based on the derived signal model, we develop a two-step algorithm with low complexity for estimating channel parameters. Furthermore, we derive the Cramér-Rao lower bound (CRLB) of location estimation as the fundamental limit of localization. Finally, we provide some insights about the AFT-MC parameters by explaining the impact of the parameters on localization performance. Simulation results demonstrate that the AFT-MC waveform is able to provide significant localization performance improvement compared to orthogonal frequency division multiplexing (OFDM) while achieving the CRLB of location estimation.

Affine frequency division multiplexing (AFDM) is a promising chirp-based waveform designed for communications in high-mobility scenarios. In this paper, the pulse shaping for AFDM over doubly selective channels (DSC) is investigated. We first develop the pulse-shaped AFDM (PS-AFDM) system, where different transmit pulses and receive pulses can be used for each chirp carrier. Based on that, we formulate the impacts of pulse shaping on the input-output relationship of PS-AFDM system with fractional delay and fractional Doppler shifts. In particular, we reveal that there exists inter-pulse interference (IPI) within the pilot region and inter-region interference (IRI) between the pilot region and the data region in the AFDM/PS-AFDM received symbols. To provide an instructive guideline for interference suppression, we elaborate how the adopted transmit and receive pulses determine the IPI and IRI. Furthermore, we demonstrate that applying the pulse-shaping window with low sidelobe levels in PS-AFDM can suppress the IPI and IRI, facilitating the channel estimation and signal detection processes significantly. Simulations verify that the proposed PS-AFDM systems can achieve lower overhead and higher accuracy channel estimation compared to the conventional AFDM systems.

Analogous to orthogonal time frequency space (OTFS), affine frequency division multiplexing (AFDM) emerges as a promising solution for achieving ultra-reliable communication under time-varying channels with large Doppler shifts. To apply this new modulation technique for next-generation communications, there is an expectation that it will be easily integrated into current systems without major modifications. In this paper, we propose a low-complexity waveform called simplified-AFDM (S-AFDM), which is more compatible with existing techniques by reducing the parameter settings in AFDM. First, We provide a general framework and formulate the input-output relation of the S-AFDM system in the discrete affine Fourier transform (DAFT) domain. Furthermore, we present a detailed analysis of the diversity order of S-AFDM in single-input single-output (SISO) setting with maximum likelihood (ML) detection. Numerical results demonstrate that the proposed modulation scheme exhibits the same performance of classic AFDM with commonly used detectors, while halving its addtional modulation complexity superimposed on the orthogonal frequency division multiplexing,

On the heels of orthogonal time frequency space (OTFS) modulation, the recently discovered affine frequency division multiplexing (AFDM) is a promising waveform for the sixth-generation wireless network. In this paper, we study the widely-used embedded pilot-aided (EPA) channel estimation in multiple-input multiple-output AFDM (MIMO-AFDM) system with fractional Doppler shifts. We first formulate the vectorized input-output relationship of MIMO-AFDM, and theoretically prove that MIMO-AFDM can achieve full diversity in doubly selective channels. Then we illustrate the implementation of EPA channel estimation in MIMO-AFDM and unveil that serious inter-Doppler interference (IDoI) occurs if we try to estimate the channel gain, delay shift, and Doppler shift of each propagation path. To address this issue, the diagonal reconstructability of AFDM subchannel matrix is studied and a low-complexity embedded pilot-aided diagonal reconstruction (EPA-DR) channel estimation scheme is proposed. The EPA-DR scheme calculates the AFDM effective channel matrix directly without estimating the three channel parameters, eliminating the severe IDoI inherently. Since the effective channel matrix is necessary for MIMO-AFDM receive processing, we believe this is an important step to bring AFDM towards practical communication systems. Simulation results validate the effectiveness of the proposed EPA-DR scheme.