1D Fourier Convolution Neural Network

• 1D-FCNN is a neural architecture that leverages Fourier transform-based convolution alongside traditional 1D convolutions to efficiently extract both global and local features from sequential signals.
• It employs a dual-path design with adaptive stride strategies, combining frequency-domain and time-domain processing to overcome the limitations of conventional local receptive fields.
• The approach demonstrates significant improvements in fault diagnosis performance, particularly in limited-data scenarios, and is applicable to diverse fields such as audio, biomedical, and industrial monitoring.

A one-dimensional Fourier Convolution Neural Network (1D-FCNN) is a neural architecture distinguished by the integration of frequency-domain convolution operations into the processing of 1D sequential signals, such as vibration data, audio, or time series. Unlike conventional 1D CNNs that operate in the time domain using local convolutions, 1D-FCNN explicitly leverages the Fourier transform to perform convolution, thereby enabling efficient global feature extraction and improved robustness, especially under limited data scenarios. Recent advances in 1D-FCNN have emphasized dual-path designs combining local and global awareness, adaptive stride strategies, and fusion mechanisms to address the limitations of local receptive fields and to enhance feature diversity in sparse-data tasks, notably in industrial fault diagnosis (Sun et al., 14 Sep 2025).

1. Theoretical Foundation and Fourier Convolution Principle

The central mathematical foundation of 1D-FCNNs is the use of the Fourier transform for convolution operations. The Fourier transform, given by

$\mathcal{F}(\omega) = \int_{-\infty}^{+\∞} f(t) e^{-i\omega t} dt$

translates temporal signals $f(t)$ into their frequency representations. In 1D-FCNN, convolution in the frequency domain is achieved via element-wise multiplication of Fourier-transformed inputs and kernels, followed by inverse transformation,

$$y(t) = \mathcal{F}^{-1}\{\mathcal{F}(x(t)) \cdot \mathcal{F}(h(t))\}$$

This approach efficiently extracts global frequency patterns, addressing the inadequacy of local receptive fields in conventional CNNs. By leveraging the spectral domain, the network can learn feature relationships extending across the entire signal, which is particularly advantageous for scenarios with limited or diverse data (Sun et al., 14 Sep 2025).

2. Dual-Path Architecture: Local and Global Awareness

The architecture of advanced 1D-FCNNs incorporates a dual-path paradigm:

• Local-Aware Branch: Employs traditional 1D convolutions to capture fine-grained, local features. These conventional layers are adept at identifying short-range patterns characteristic of standard CNNs.
• Global-Aware Branch: Utilizes Fourier transform-based operations to convert input signals into the frequency domain and perform convolution. This branch features an adaptive convolutional stride strategy—starting with large stride for coarse, long-range global features and then refining with smaller stride for detail.

Inputs are typically expanded from single-channel to multiple channels using 1×1 convolution, and subsequently split between local ($X_l$) and global ($X_g$) groups. At each layer, a fusion mechanism exchanges information between the paths:

$F_g(X_g) = F_{g→g}(X_g) + F_{l→g}(X_l)$

$F_l(X_l) = F_{l→l}(X_l) + F_{g→l}(X_g)$

where $F_{g→g}$ and $F_{l→l}$ are the main extraction operations (Fourier convolution and local convolution, respectively), and $F_{l→g}, F_{g→l}$ are fusion layers facilitating cross-path information exchange (Sun et al., 14 Sep 2025).

3. Role within Data Augmentation and Contrastive Learning Frameworks

Within comprehensive frameworks such as DAC-FCF, the 1D-FCNN is deployed alongside generative data augmentation (e.g., CCLR-GAN) and contrastive learning strategies. While generative models synthesize diverse training samples and contrastive learning optimizes inter-sample relationships, the 1D-FCNN provides advanced feature extraction capabilities. Specifically, the global pathway of the 1D-FCNN leverages frequency domain information to capture long-range dependencies and overall signal patterns, which local branches may under-represent.

This combination enhances the model's capacity to distinguish subtle differences in fault types and improves generalization in regimes of scarce labeled data—a recurring problem in practical industrial condition monitoring (Sun et al., 14 Sep 2025).

4. Performance Metrics and Empirical Results

1D-FCNNs within the DAC-FCF framework have demonstrated substantial improvements in bearing fault diagnosis:

• On the CWRU dataset: Outperformed baselines by up to 32%.
• On self-collected test benches: Achieved approximately 10% improvement.

These gains are attributed to the ability of Fourier convolution operations to extract global and subtle frequency features, complementing local pattern recognition. Performance enhancement is especially pronounced when the sample size per class is small, indicating effectiveness in few-shot learning and limited-data situations (Sun et al., 14 Sep 2025).

5. Comparative Analysis: 1D-FCNN versus Conventional CNNs

A direct comparison reveals that traditional CNNs rely on stacking local convolution kernels or increasing their receptive field to approximate global context, which can lead to redundancy and overfitting—especially with limited data. In contrast, the Fourier convolution mechanism in 1D-FCNNs achieves global contextual modeling with fewer layers and parameters.

Additionally, adaptive stride strategies in the frequency domain allow the network to modulate its receptive field to the scale of input features, increasing efficiency and robustness. The fusion of local and global branches further equips the network to jointly learn high-resolution details and global signal structure, overcoming the intrinsic limitations of purely local or purely global approaches (Sun et al., 14 Sep 2025).

6. Application Domains and Extensibility

While originally validated for bearing fault diagnosis, the design principles of 1D-FCNNs generalize to multiple application domains leveraging 1D sequential signals:

• Speech and audio processing: Extraction of global spectral properties for tasks such as speech recognition and noise reduction.
• Seismology: Capturing global frequency patterns in seismic signal analysis.
• Biomedical signal analysis: Interpreting ECG and EEG data where diagnosis depends on both local and global temporal patterns.
• Industrial monitoring: Processing diverse sensor data (temperature, vibration, pressure) for anomaly detection.
• Financial time series: Joint modeling of global trends and local fluctuations for forecasting and outlier detection.

A plausible implication is that 1D-FCNN architectures can be flexibly adapted to any task where both local detail and global frequency structure of 1D signals are relevant, particularly under data scarcity conditions (Sun et al., 14 Sep 2025).

7. Significance and Future Prospects

The integration of Fourier convolution into 1D neural architectures marks a significant progression in feature extraction for time-series data, particularly in resource-constrained regimes. The dual-path (local/global) strategy, adaptive stride mechanisms, and feature fusion operations provide the basis for efficient, robust, and scalable model designs. Extensive ablation studies have confirmed the effectiveness of these components. The demonstrated improvements in industrial fault diagnosis under limited data suggest that expanding the use of 1D-FCNNs to domains with similar requirements can substantially advance the state of the art. Future research may consider further optimization of fusion mechanisms, stride scheduling, and application to multimodal signals.

In summary, 1D Fourier Convolution Neural Networks provide a rigorous, computationally efficient method for extracting both local and global features from sequential data, substantially improving robustness and accuracy in diverse signal analysis tasks, especially when labeled samples are rare (Sun et al., 14 Sep 2025).