Source-linked AI summary
Probabilistic Symbol-Level Precoding based Affine Frequency Division Multiplexing Transmission
Shuntian Tang, Xinyi Wang, Shuangyang Li, Tianqi Mao, Zilong Liu, Zesong Fei
TL;DR
Conventional AFDM receivers can impose high computational demands on downlink users. This paper uses uplink CSI to design probabilistic-SLP AFDM waveforms at the base station, enabling direct symbol detection without receiver channel estimation or equalization. Simulations report comparable BER to conventional methods with substantially reduced receiver complexity.
Problem
Conventional AFDM receivers have high computational complexity, challenging practical use on downlink devices with limited processing capability.
Source: §I · p. 1Method
The BS uses uplink CSI and channel reciprocity to optimize probabilistic-SLP AFDM symbols for correct decision-region reception, with the problem approximated as an SOCP.
Source: §I · p. 2, §III · p. 3, B. Problem Transformation and its… · p. 4Results
The proposed scheme achieves BER performance comparable to conventional methods under perfect and estimated CSI while significantly reducing receiver complexity.
Source: §VI · pp. 5–6Takeaways & Limitations
The framework supports direct user-side symbol detection without channel estimation or equalization and is reported as effective and practical for AFDM systems.
Source: §VI · pp. 5–6
Abstract
from arXiv · showhide
Affine frequency division multiplexing (AFDM) has recently gained significant attention due to its robustness against time-frequency doubly selective channel fading. However, the high computational complexity at the receiver poses a critical challenge for practical deployment. To overcome this issue, we propose a probabilistic symbol-level precoding (SLP)-based AFDM transmission framework, in which the processing burden in downlink transmission is shifted from the user to the base station (BS), enabling direct symbol detection without channel estimation or equalization at the receiver. In the proposed framework, the BS exploits the uplink channel state information (CSI) to design the downlink transmit waveform based on uplink-downlink channel reciprocity. In particular, we innovatively introduce a probabilistic SLP technology by explicitly characterizing the likelihood of symbol detection errors under noise perturbations. Specifically, the transmitted symbols are optimized to minimize the likelihood that the received symbols fall into erroneous decision regions, where the resulting error-probability minimization problem is subsequently approximated as a second order cone programming (SOCP) problem by exploiting the monotonicity of the objective function. Simulation results show that the proposed probabilistic SLP-based scheme achieves performance comparable to that of conventional AFDM receivers, whilst enjoying significant reduction of computational complexity at the receiver end. These results demonstrate the effectiveness and practical potential of the proposed approach.
I. INTRODUCTION
AFDM is presented as robust to doubly selective channels, but conventional AFDM receivers can be too computationally demanding for downlink user equipment. The paper proposes an SISO probabilistic-SLP AFDM framework that shifts waveform processing to the base station and reports performance comparable to conventional methods with reduced receiver complexity.
I. INTRODUCTION
- AFDM is described as robust to strong Doppler shifts and hardware impairments, with adjustable IDAFT parameters supporting multipath separation and full diversity in doubly dispersive channels.The introduction also notes AFDM applications including MIMO detection, non-orthogonal AFDM, index modulation, and integrated sensing and communication –[13].Source: §I · p. 1
- Conventional AFDM receivers are computationally demanding for downlink user equipment, while earlier SLP approaches rely on multi-antenna processing and do not directly apply to SISO.The cited alternatives include delay-Doppler alignment modulation and geometric constellation shaping –.Source: §I · p. 1
- The proposed AF-domain probabilistic SLP framework targets SISO AFDM, optimizing transmit symbols to increase the probability of correct decision-region reception.The original error-probability problem is approximated as a tractable SOCP using objective-function monotonicity.Source: §I · p. 2
- The proposed scheme is reported to achieve performance comparable to traditional MMSE with much lower receiver complexity under varied CSI conditions.The introduction characterizes the framework as user-equalization-free.Source: §I · p. 2
II. BASIC PRINCIPLES OF AFDM
AFDM maps constellation symbols from the affine-frequency domain to time samples with an IDAFT, then uses a DAFT at reception. A chirp-periodic prefix and channel-matrix structure support transmission over multipath channels with Doppler effects.
II. BASIC PRINCIPLES OF AFDM
- The N-point IDAFT transforms an N-symbol AF-domain vector into time-domain samples, and the DAFT recovers the signal in the AF domain at the receiver.The AFDM parameters c1 and c2 enable full diversity over doubly selective fading channels.Source: §II · p. 2
- A chirp-periodic prefix (CPP) is appended to maintain AF-domain periodicity, with its length chosen above the maximum channel delay spread.When 2Nc1 is an integer and N is even, the CPP reduces to a conventional cyclic prefix.Source: §II · p. 2
- The channel model assumes P propagation paths with complex gains, integer delays, Doppler shifts, and AWGN, while fractional delays are ignored.The text justifies the fractional-delay approximation by the high sampling period,.Source: §II · p. 2
- After CPP removal, the AF-domain input-output relation is expressed using path-contribution matrices, with definitions for the DFT, Doppler-shift, cyclic-permutation, and diagonal matrices.For even N and integer 2Nc1, Γi simplifies to the identity matrix.Source: §II · p. 2
- For integer normalized Doppler shifts the channel matrix is sparse, and parameter selection can make each row and column of Heff contain exactly P nonzero elements; fractional Doppler shifts have a separate expression.The normalized Doppler shift is decomposed into integer and fractional components.Source: §II · p. 2
III. SYSTEM MODEL
The framework separates uplink channel acquisition from downlink precoding, using uplink CSI and reciprocity to construct the downlink waveform. Under its stated channel assumptions, the user detects symbols directly without channel estimation or equalization.
III. SYSTEM MODEL
- The system shifts most signal processing to the BS, which acquires uplink CSI and then designs the downlink waveform using channel reciprocity.The paper assumes a neural network perfectly compensates the difference between uplink and downlink CSI.Source: §III · p. 3
- The frame begins with a pilot followed by L−1 data symbols, and its duration is assumed short enough that the effective-channel sparsity pattern stays unchanged.The pilot enables the BS to estimate the effective channel matrix and equalize the uplink data.Source: §III · p. 3
- During downlink waveform design, transmitted symbols are constructed so channel interference contributes constructively, allowing direct detection without receiver channel estimation or equalization.The design relies on the estimated effective channel matrix and uplink-downlink reciprocity.Source: §III · p. 3
- Moving processing to the BS alleviates computational demands on user devices and removes the need for frequent downlink pilot insertion relative to the conventional embedded-pilot AFDM scheme.The text connects the latter change to improved spectral efficiency.Source: §III · p. 3
IV. DOWNLINK PROBABILISTIC SLP-BASED WAVEFORM DESIGN
The section introduces probabilistic SLP waveform design for downlink transmission and frames its optimization for tractable solution.
IV. DOWNLINK PROBABILISTIC SLP-BASED WAVEFORM DESIGN
- The downlink design uses probabilistic SLP to construct the transmitted waveform.The section first introduces the principle and formulates the waveform-design problem.Source: §IV · p. 3
- The waveform-design problem is reformulated into a more tractable form for solution.The section presents this reformulation after introducing the probabilistic SLP principle and formulation.Source: §IV · p. 3
- The design discussion is scoped to the downlink phase and its corresponding waveform optimization.The section focuses on the probabilistic SLP-based design process.Source: §IV · p. 3
A. Problem Formulation
The formulation designs AF-domain transmit symbols to account for channel-induced displacement and minimize detection errors under a transmit-power constraint. It characterizes correct-detection probabilities for 4PSK symbols under noise.
A. Problem Formulation
- Probabilistic SLP precompensates transmitted symbols for multipath and Doppler displacement so received symbols fall in desired decision regions with high probability.The formulation considers correct-detection probabilities across quadrants and the error probability for a first-quadrant intended symbol.Source: A. Problem Formulation · pp. 3–4
- With AWGN, the received symbol is modeled as complex Gaussian with independently distributed real and imaginary components.This probability model supports evaluating whether a received symbol falls in its intended decision region.Source: A. Problem Formulation · p. 3
- For 4PSK, the received symbol’s real and imaginary signs determine the transmitted information, so correct detection depends on its quadrant.The formulation expresses the probability of a positive real component using the noise model and standard-normal functions.Source: A. Problem Formulation · pp. 3–4
- The BER-minimization problem optimizes the transmitted vector subject to a transmit-power budget.The budget is denoted Pm.Source: A. Problem Formulation · p. 4
B. Problem Transformation and its Solution
The transformation replaces the probability-based objective with a tractable formulation by exploiting monotonicity. Introducing an auxiliary variable yields an SOCP that can be solved with standard convex optimization solvers.
B. Problem Transformation and its Solution
- Taking the logarithm of the correct-detection probability transforms the objective into an equivalent form for optimization.T(h_nz) denotes the correct-detection probability associated with symbol n.Source: B. Problem Transformation and its… · p. 4
- Monotonicity of log(1 + x) and erf(x) permits approximating the objective by focusing on their arguments.The simplified problem uses the real- and imaginary-component terms associated with the intended symbol.Source: B. Problem Transformation and its… · p. 4
- Introducing an auxiliary variable t transforms the simplified problem into an SOCP solvable with off-the-shelf convex optimization software.The passage identifies CVX as an example solver.Source: B. Problem Transformation and its… · p. 4
- The authors state that solving the dual problem reduces computational complexity.The passage also introduces N_it as the iteration number.Source: B. Problem Transformation and its… · p. 4
V. NUMERICAL RESULTS
Simulations show comparable downlink BER to conventional schemes, including with estimated CSI, and strong performance across AFDM, OTFS, and OFDM implementations.
V. NUMERICAL RESULTS
- The simulations use randomly generated path Doppler shifts within [−αmax, αmax] and Zadoff–Chu pilots for AF-domain channel estimation.Channel estimation uses the algorithm proposed in.Source: §V · p. 5
- With perfect CSI at 20 dB, received 4PSK symbols concentrate in their correct quadrants, supporting detection without receiver channel estimation or equalization.The waveform pre-adjusts transmitted symbols to counteract channel-induced distortion.Source: §V · p. 5
- BER remains comparable to conventional schemes across SNR, with a slight high-SNR gain and only minor degradation under estimated CSI.The proposed scheme retains its advantage with estimated CSI.Source: Figure 2 · p. 5, §V · p. 5
- AFDM and OTFS implementations show almost indistinguishable BER performance, and both surpass the OFDM-based scheme.The paper attributes this gain to AFDM and OTFS harnessing diversity in doubly selective channels.Source: §V · p. 5
VI. CONCLUSION
The conclusion presents probabilistic SLP-based waveform design for AFDM to reduce receiver burden, reporting comparable BER and lower receiver complexity.
VI. CONCLUSION
- The BS uses uplink CSI to design AFDM transmit symbols, enabling direct detection without receiver channel estimation or equalization.The design minimizes the probability of erroneous detection.Source: §VI · p. 5
- The proposed scheme achieves BER performance comparable to conventional methods under both perfect and estimated CSI.The conclusion also reports significantly reduced receiver complexity.Source: §VI · pp. 5–6
- The scheme significantly reduces receiver complexity.This is reported alongside comparable BER under perfect and estimated CSI.Source: §VI · p. 6