# A Novel Complex-Valued Blind Source Separation and Its Applications in Integrated Reception
## 1 Introduction  
In the complex electromagnetic environment, numerous radar, communication, jamming, and other radiation signals, whether cooperative or non-cooperative, are present. The integrated receiver's wide beam and wide bandwidth coverage often result in overlapping signals in the time-frequency and air domains. Consequently, performing tasks like parameter estimation and information extraction becomes challenging due to the low signal-to-noise ratio (SNR) and high jamming-to-signal ratio (JSR). Under conditions of low SNR, valuable signals may become obscured by strong noise, making it challenging to detect radar signals and increasing the likelihood of bit errors in communication signals. Conversely, if the jamming signal is too strong, jamming can suppress the useful signal, resulting in the failure of or errors in radar signal detection and the inability to demodulate the communication signals. Therefore, the primary concern is to separate these overlapping signals. Blind source separation (BSS) [1, 2] aims to recover or reconstruct the source signal using observed data from the receiver channel, even when the source signal and mixing characteristics are unknown. BSS, as a powerful technology in signal processing, has played a crucial role in various domains, including speech separation [3, 4, 5], wireless communication [6, 7, 8], anti-main lobe jamming [9, 10, 11, 12, 13], and direction of arrival (DOA) estimation [14, 15]. BSS has also provided innovative ideas for ongoing research in integrated-reception processing. In this paper, a complex scenario of integrated reception is considered, as depicted in Figure 1. In this scenario, radar signals, echo signals, jamming signals, and communication signals overlap in the time-frequency domain. After separating the mixed signals, the mixing matrix is used to estimate the DOA. The echo signals undergo pulsed Doppler processing, the communication signal is demodulated, reconnaissance signals are sorted, and the jamming signal is analyzed. This method enables the execution of tasks such as location, detection, reconnaissance, communication, and anti-jamming. When using arrays to receive far-field narrowband signals, which are part of the time delay mixing mathematical model [16], it is necessary to employ complex-valued blind source separation (CVBSS). However, traditional CVBSS algorithms suffer from low separation accuracy, slow convergence, poor robustness, and high sensitivity to noise and jamming. The well-known second-order blind identification (Sobi) [17; 18] is a classical and effective algorithm to solve BSS problems using second-order statistics. Although it has low computational complexity, it exhibits poor separation performance. In [19], the joint approximate diagonalization of eigenmatrices (JADE) algorithm was proposed. This algorithm tackles the objective function maximization problem by jointly diagonalizing the eigenmatrices of a set of fourth-order cumulant matrices, thereby significantly improving the separation performance. The computational complexity is high due to the fourth-order cumulant matrix. To address this, c-FastICA [20] extends the FastICA algorithm from the real domain to the complex-valued domain. However, c-FastICA is limited to circular signals. To handle non-circular signals, [PERSON] et al. developed the nc-FastICA algorithm [21]. Another approach is the RobustICA algorithm proposed in the literature [22; 23], which combines an exact linear search with a cost function called cliquishness. RobustICA addresses the convergence issue of c-FastICA and nc-FastICA, which may lead to a saddle point. However, RobustICA uses cliquishness as the objective function and is more sensitive to noise and outliers. Another algorithm, FAJD [24], is a non-orthogonal joint diagonalization algorithm that avoids degenerate solutions by introducing a penalty  
Figure 1: A type of integrated reception system. term to the non-diagonal error criterion equation. However, the penalty term increases the computational complexity of the algorithm.