# A Dual-Branch Speech Enhancement Model with Harmonic Repair
## 2 Proposed Methods
### Fast Fourier Convolution (FFC)  
The Fast Fourier Transform (FFT) converts time-domain signals into frequency-domain signals. Compared to the Short-Time Fourier Transform (STFT), which decomposes time-domain signals into spectral components of a series of window functions, FFT first decomposes the signal into a sum of sine and cosine functions to represent the spectrum, thus efficiently computing the Discrete Fourier Transform and obtaining the spectral information of the signal.  
For traditional fully convolutional models, the growth of the effective receptive field is too slow, and the lack of an effective context-capturing structure often results in suboptimal enhancement effects, a problem that is more prominent in wideband, long-duration audio. In the amplitude spectrogram of speech, harmonic structures often form periodic patterns, a feature that is suitable for processing with FFC, i.e., repetitive microstructures. Considering the model's aim to expand the neural network's receptive field for speech context, using FFC is more appropriate for analyzing the entire speech spectrum. It is a widely used non-global operator in the field of Computer Vision (CV) and can replace traditional convolution layers in network architectures, playing a significant role in repairing damaged periodic backgrounds.  
In computer vision research, the Fourier transform generally applies a complex two-stage method. In this work, we set the working domain of the Fourier transform as the frequency component of the feature map. The basic structure of FFC is shown in Figure 1. Specifically, the basic structure of FFC is implemented as follows.  
* Before the signal S enters the operator, it is divided along the feature channels of the feature map into the local block S\({}_{1}\) and the global block S\({}_{g}\), where the local block S\({}_{1}\) will use adjacent local blocks as learning objects, and the remaining global block S\({}_{g}\) is used to obtain speech context associations. We use parameter \(\alpha\) to control the division ratio of channels.
* To cover the entire spectrum with the receptive field of the global block, the original feature space is transformed into a specific domain on the global Fourier unit, and after the spectral data is updated, it is restored to a spatial format. Meanwhile, additional segmentation and combination are performed in the local Fourier unit to make it more sensitive to spectral detail features. Finally, the output data of the two units are connected using a residual connection.
* The results of the global and local blocks are simply connected to form the output of the complete operator. At this point, the entire FFC module is fully differentiable and can replace all traditional convolutions.  
Figure 1: The basic structure of FFC.  
We apply a real one-dimensional fast Fourier transform on the frequency dimension of the input feature map at the global component level and then concatenate the real and imaginary parts of the spectrum along the channel dimension. Next, we apply convolutional blocks on the frequency domain and finally restore the spatial structure using inverse FFT.