# Signal Separation Based on Knowledge Representation
## 1 Introduction  
Signal separation, as a key signal processing task, aims to untangle complex mixtures. Such problems typically fall into the category of ill-posed problems, thus requiring the extraction of accurate components from mixed signals based on specific assumptions as constraints [1, 2]. It is important in various domains such as speech processing, geophysics, remote sensing, and image processing. The main approach to solving signal separation problems is to utilize the differences in characteristics between target signals, effectively separating them. In this process, the main methods include signal separation methods based on mathematical feature differences, those based on transformation domain feature differences, and artificial intelligence signal separation methods. Signal separation methods based on mathematical prior assumptions are one of the current research focuses. Among them, non-negative matrix factorization (NMF) is a commonly used method based on mathematical features of the target [3, 4]. [PERSON] et al. [5] proposed a new NMF model, which is based on the Kullback-Leible and Itakura-Saito \(\beta\)-divergences and introduces a penalty term to promote small-volume dictionary matrix columns. Under corresponding assumptions and noiseless conditions, it is proven that this model can reliably achieve audio signal separation. [PERSON] et al. [6] found thatsignals recorded by different sensors exhibit certain similarities, especially in their temporal changes. Based on this similarity, assuming that the activation coefficient matrices of their NMF have a similar form and utilizing this similarity for co-factorization, effective signal separation is achieved. NMF assumes that source signals are non-negative, while independent component analysis (ICA) [7] achieves signal separation by finding independent signal components and utilizing the non-Gaussian features in statistical principles. [PERSON] et al. [8] investigated the potential of ICA methods in separating geophysical signals in GRACE data. [PERSON] et al. [9] introduced an implementation using fast independent component analysis (FastICA) for blind image source separation using memristor crossbar arrays. Signal separation methods based on transformation domain features have also received attention [10]. In the field of wavefield separation, [PERSON] et al. [11] introduced the discrete cosine transform (DCT) into seismic wavefield separation, combined with discrete Fourier transform (DFT) and DCT operations, and designed a high-efficiency and precision frequency-space (f-x) domain vertical seismic profiling (VSP) [12] wavefield separation method. This includes methods such as F-K transform [13; 14] and Radon transform [15; 16], among others. Transformation domain-based methods utilize differences in physical features in the transformation domain to achieve separation. While such methods are effective for separating simple signals, energy overlaps in the transformation domain in complex geological models, resulting in an impure separation of wavefield signals. Traditional signal separation methods rely on mathematical prior assumptions, often requiring strict conditions that limit their applicability. Moreover, these methods heavily depend on accurate mathematical models and parameter estimation, leading to poor separation performance when the assumptions are not met. In contrast, transform domain-based signal separation methods require the target signals to exhibit distinct characteristics in the transform domain. However, in complex environments and challenging signal conditions, energy aliasing between the separated signals in the transform domain can result in impure separation outputs. This introduces challenges in feature selection and affects the overall effectiveness of signal separation. Moreover, when there is overlap among the feature domains of target signals, traditional methods often yield poor results. To address this issue, machine learning approaches have been proposed. For instance, [PERSON] et al.