Augmented Affine Frequency Division Multiplexing for Both Low PAPR Signaling and Diversity Gain Protection

Technical Summary: Augmented Affine Frequency Division Multiplexing (A2FDM)

Problem Statement
Affine Frequency Division Multiplexing (AFDM) has emerged as a promising waveform for high-mobility wireless scenarios (e.g., 6G, V2X, UAVs) due to its ability to achieve full diversity gain in double-selective (time-frequency) fading channels. However, AFDM inherits the high Peak-to-Average Power Ratio (PAPR) problem characteristic of Orthogonal Frequency-Division Multiplexing (OFDM). High PAPR limits transmitter efficiency, causes nonlinear distortion, and reduces coverage. Furthermore, the performance of AFDM is critically dependent on the system parameter $c_1$. While literature provides a formula for selecting $c_1$ to achieve full diversity, the authors demonstrate that this condition is necessary but not sufficient; specific values of $c_1$ satisfying the formula can still cause multipath components to overlap in the effective channel matrix, resulting in a catastrophic loss of diversity gain. Additionally, the parameter $c_2$ in standard AFDM is shown to have a negligible impact on performance yet contributes to the PAPR issue, representing an unnecessary component in the signal structure.

Methodology
The paper proposes an enhanced scheme termed Augmented AFDM (A2FDM). The core methodology involves replacing the diagonal matrix associated with the superfluous parameter $c_2$ in the standard AFDM transmitter with a new unitary matrix. This new matrix performs two functions simultaneously:

1. Sub-block-based Discrete Fourier Transform (DFT): It processes data symbols in groups.
2. Symbol Mapping: It redistributes the processed symbols across subcarriers.

The authors introduce two specific implementations based on Single-Carrier Frequency-Division Multiple Access (SC-FDMA) principles:

• Interleaved A2FDM (IA2FDM): Uses an interleaved permutation matrix to map symbols evenly across the entire bandwidth.
• Localized A2FDM (LA2FDM): Uses a localized permutation matrix to map symbols to contiguous subcarriers.

A new system parameter, $\mu$, is defined as the number of DFT sub-blocks. The total number of subcarriers $N$ is divided into $\mu$ blocks of size $N_\mu = N/\mu$. The transmitter structure utilizes the Discrete Augmented Affine Fourier Transform (DA2FT), defined as $A = \Gamma_\mu^H F \Lambda_{c_1}$, where $\Gamma_\mu$ incorporates the block-DFT and permutation operations.

Key Contributions

1. PAPR Mitigation: By replacing the $c_2$ matrix with the block-DFT and mapping structure, A2FDM reduces the active subcarriers at any given time instant to $\mu$ (for IA2FDM) or $\mu+2$ (for LA2FDM), rather than the full $N$. Consequently, the PAPR is reduced from the order of $N$ to the order of $\mu$.
2. Robustness to Parameter $c_1$: The paper analyzes the input-output relationships and demonstrates that while standard AFDM can lose all diversity gain if $c_1$ is poorly chosen (even if it satisfies existing literature formulas), A2FDM guarantees a minimum diversity order of $\min\{N_\mu, L\}$, where $L$ is the number of channel paths. This ensures that even with a suboptimal $c_1$, the system retains diversity gain proportional to the sub-block size, rather than collapsing to a diversity order of 1.
3. Complexity Analysis: The authors show that the A2FDM transmitter incurs only a marginal increase in complexity compared to AFDM (specifically $O(N \ln(N/\mu))$ extra operations due to the sub-block FFTs), while the receiver complexity remains comparable to AFDM when using the same equalizer (e.g., MMSE).
4. Derivation of Input-Output Relationships: The paper derives the precise mathematical relationships for IA2FDM and LA2FDM over double-selective fading channels, accounting for integer and fractional Doppler shifts.

Results
Extensive simulations were conducted to validate the proposed schemes against standard AFDM:

• Diversity Gain: In scenarios where $c_1$ is set to a value that causes diversity loss in standard AFDM (specifically satisfying the condition in Eq. 53), IA2FDM and LA2FDM maintain robust Bit Error Rate (BER) performance. When $\mu$ is large (e.g., $\mu=128$), the BER performance becomes independent of the specific $c_1$ value, confirming the robustness claim.
• PAPR Reduction: The Complementary Cumulative Distribution Function (CCDF) of PAPR shows that A2FDM achieves significantly lower PAPR than AFDM. For $\mu=1$, the system behaves as a single-carrier scheme with the lowest PAPR. As $\mu$ increases, PAPR increases but remains bounded by $\mu$, offering a tunable trade-off between PAPR and implementation complexity.
• Performance Trade-offs:
  • IA2FDM: Shows BER performance nearly identical to AFDM when $c_1$ is optimal, but superior performance when $c_1$ is suboptimal.
  • LA2FDM: Exhibits a non-monotonic BER performance as $\mu$ varies. Performance initially degrades as $\mu$ increases (due to correlated fading in localized blocks) but improves as $\mu$ further increases (converging to AFDM behavior).
  • Data Rate vs. Diversity: Simulations demonstrate a trade-off where activating more symbols ($N_A$) increases Inter-Symbol Interference (ISI), degrading BER. However, A2FDM with higher $\mu$ (e.g., $\mu=4$) handles higher $N_A$ better than $\mu=1$ due to reduced ISI per subcarrier.

Significance and Claims
The paper claims that A2FDM successfully circumvents the high PAPR problem inherent in AFDM and OFDM without sacrificing the system's ability to achieve diversity gain. A primary significance highlighted is the system's resilience to the "undesirable conditions" of the parameter $c_1$. While standard AFDM is vulnerable to diversity loss if $c_1$ is not perfectly tuned to the specific channel profile, A2FDM provides a design mechanism (via $\mu$) to guarantee a minimum diversity order, thereby protecting the system against the loss of diversity gain in dynamic high-mobility environments. The authors conclude that A2FDM offers a flexible signaling waveform suitable for next-generation high-mobility communications, balancing PAPR, diversity, and implementation complexity.