This supplementary document is organized as follows: Section 1 shows the proof that the formula of FRFT degrades to that of FT when α = π/ 2. Section 2 shows the discrete implementation of 2D FRFT. Section 4 shows the experimental results with single branch. Section 5 shows the architecture design of SFC and example usage of SFC and MFRFC. Section 6 introduces the periodicity of FRFT. Section 7 introduces the energy distribution of FRFT.

Existing deep learning-based computer vision methods usually operate in the spatial and frequency domains, which are two orthogonal \textbf{individual} perspectives for image processing.In this paper, we introduce a new spatial-frequency analysis tool, Fractional Fourier Transform (FRFT), to provide comprehensive \textbf{unified} spatial-frequency perspectives.The FRFT is a unified continuous spatial-frequency transform that simultaneously reflects an image's spatial and frequency representations, making it optimal for processing non-stationary image signals.We explore the properties of the FRFT for image processing and present a fast implementation of the 2D FRFT, which facilitates its widespread use.Based on these explorations, we introduce a simple yet effective operator, Multi-order FRactional Fourier Convolution (MFRFC), which exhibits the remarkable merits of processing images from more perspectives in the spatial-frequency plane. Our proposed MFRFC is a general and basic operator that can be easily integrated into various tasks for performance improvement.We experimentally evaluate the MFRFC on various computer vision tasks, including object detection, image classification, guided super-resolution, denoising, dehazing, deraining, and low-light enhancement. Our proposed MFRFC consistently outperforms baseline methods by significant margins across all tasks.

Our code is released publicly at https://github.com/yuhuUSTC/FRFT.

Accurate spectrum prediction is crucial for dynamic spectrum access (DSA) and resource allocation. However, due to the unique characteristics of spectrum data, existing methods based on the time or frequency domain often struggle to separate predictable patterns from noise. To address this, we propose the Spectral Fractional Filtering and Prediction (SFFP) framework. SFFP first employs an adaptive fractional Fourier transform (FrFT) module to transform spectrum data into a suitable fractional Fourier domain, enhancing the separability of predictable trends from noise. Subsequently, an adaptive Filter module selectively suppresses noise while preserving critical predictive features within this domain. Finally, a prediction module, leveraging a complex-valued neural network, learns and forecasts these filtered trend components. Experiments on real-world spectrum data show that the SFFP outperforms leading spectrum and general forecasting methods.

Existing deep learning-based computer vision methods usually operate in the spatial and frequency domains, which are two orthogonal \textbf{individual} perspectives for image processing.In this paper, we introduce a new spatial-frequency analysis tool, Fractional Fourier Transform (FRFT), to provide comprehensive \textbf{unified} spatial-frequency perspectives.The FRFT is a unified continuous spatial-frequency transform that simultaneously reflects an image's spatial and frequency representations, making it optimal for processing non-stationary image signals.We explore the properties of the FRFT for image processing and present a fast implementation of the 2D FRFT, which facilitates its widespread use.Based on these explorations, we introduce a simple yet effective operator, Multi-order FRactional Fourier Convolution (MFRFC), which exhibits the remarkable merits of processing images from more perspectives in the spatial-frequency plane. Our proposed MFRFC is a general and basic operator that can be easily integrated into various tasks for performance improvement.We experimentally evaluate the MFRFC on various computer vision tasks, including object detection, image classification, guided super-resolution, denoising, dehazing, deraining, and low-light enhancement. Our proposed MFRFC consistently outperforms baseline methods by significant margins across all tasks.

Collecting high-quality preference datasets for reinforcement learning from human feedback (RLHF) is resource-intensive and challenging. As a result, researchers often train reward models on extensive offline datasets which aggregate diverse generation sources and scoring/alignment policies. We hypothesize that this aggregation has an averaging effect on reward model scores, which limits signal and impairs the alignment process. Inspired by the field of inverse RL, we define the 'inverse alignment problem' in language model training, where our objective is to optimize the critic's reward for a fixed actor and a fixed offline preference dataset. We hypothesize that solving the inverse alignment problem will improve reward model quality by providing clearer feedback on the policy's current behavior. To that end, we investigate whether repeatedly fine-tuning a reward model on subsets of the offline preference dataset aligned with a periodically frozen policy during RLHF improves upon vanilla RLHF. Our empirical results demonstrate that this approach facilitates superior alignment and faster convergence compared to using an unaligned or out-of-distribution reward model relative to the LLM policy.

To comprehensively assess optical fiber communication system conditions, it is essential to implement joint estimation of the following four critical impairments: nonlinear signal-to-noise ratio (SNRNL), optical signal-to-noise ratio (OSNR), chromatic dispersion (CD) and differential group delay (DGD). However, current studies only achieve identifying a limited number of impairments within a narrow range, due to limitations in network capabilities and lack of unified representation of impairments. To address these challenges, we adopt time-frequency signal processing based on fractional Fourier transform (FrFT) to achieve the unified representation of impairments, while employing a Transformer based neural networks (NN) to break through network performance limitations. To verify the effectiveness of the proposed estimation method, the numerical simulation is carried on a 5-channel polarization-division-multiplexed quadrature phase shift keying (PDM-QPSK) long haul optical transmission system with the symbol rate of 50 GBaud per channel, the mean absolute error (MAE) for SNRNL, OSNR, CD, and DGD estimation is 0.091 dB, 0.058 dB, 117 ps/nm, and 0.38 ps, and the monitoring window ranges from 0~20 dB, 10~30 dB, 0~51000 ps/nm, and 0~100 ps, respectively. Our proposed method achieves accurate estimation of linear and nonlinear impairments over a broad range, representing a significant advancement in the field of optical performance monitoring (OPM).