## 1 Introduction
### Introduction of Cross-Domain Features in DNN-Based SE Network  
Many conventional statistical model-based or novel DNN-based SE methods employ the time-frequency (T-F) spectrum (spectrogram) obtained by short-time Fourier transform (STFT) as the fundamental representation of speech signals, which is equivalent to using a fixed and known sinusoid basis set. Although sinusoids have theoretical advantages in serving as a basis for analyzing signals, they may not be the optimal choice for practical use. For example, Fourier transform (FT) is unsuitable for analyzing non-stationary signals. Thus, STFT is a compromise that applies FT to segmented and windowed signals to capture the spectral properties within a short period of a signal. Alternatively, other transformation methods might be more suitable for directly analyzing non-stationary signals, such as wavelet transformation and Hilbert Huang transformation [29] or data-driven time-domain transformation with a trainable convolution network. In particular, the operation of a convolutional layer for segmented speech is similar to the STFT mentioned above, but it uses a set of learnable basis functions. For example,Conv-TasNet [18] directly uses CNN in its encoder part to extract frame-wise features, and SincNet [30] employs the sinc functions, the parameters of which can be learned with CNN to simulate traditional band-pass filters. According to information theory, a signal in the time, frequency, or distortion-free transform domains contains the same information without any loss. However, we cannot confirm that the downstream network processing will not result in information loss or always highlight salient information for SE from any single-domain features. In this way, increasing the redundancy with multiple-domain features at the feature extraction stage might simplify the design of the subsequent network, which is similar to using redundant codes in transmission to detect and correct possible errors. Inevitably, introducing redundancy with multiple-domain features has certain downsides, such as increasing the input size, introducing a potential bias, and complicating, making the fusion of these features less interpretable. In addition to directly using multiple-domain features with their original form (by concatenating them), it is likely that the mutual information among these features, if extracted in advance, can further benefit the entire SE network in terms of both efficiency and efficacy. For example, in [31], both time-domain and frequency-domain features are integrated with a mechanism called bi-projection fusion (BPF), and the resulting feature set produces superior SE performance than Conv-TasNet [18]. BPF is initially developed in image processing, which integrates equirectangular and cubemap panoramic photography to achieve a more accurate depth estimation of panoramic images. In a nutshell, this study is motivated by the following observations:  
1. Features from multiple domains may learn a more effective SE model than single-domain features, according to [31]. 2. The well-known discrete wavelet transform (DWT) may split a signal into many sub-bands without losing information or introducing distortion, and it could be a good choice for producing features or as a pre-processing stage for an SE model. If the used wavelet function has a small support, DWT can be implemented quickly. Furthermore, because of the downsampling process, the total length of the DWT-wise sub-band signals is approximately equal to that of the original signal, and thus DWT does not increase data size when the number of sub-bands is increased. Additionally, our recent research [32; 33; 34] has shown that DWT can be applied in various aspects in SE to promote its performance. 3. The work in [31] employs BPF to fuse two feature sources. We wondered if the BPF mechanism can be extended to fuse more feature sources than just two.