# A Novel Complex-Valued Blind Source Separation and Its Applications in Integrated Reception
## 1 Introduction
The algorithms mentioned above all utilize the zero time delay autocorrelation matrix of the observed signal for whitening. However, it is important to note that the zero time delay autocorrelation matrix does not eliminate the impact of additive noise, leading to lower separation accuracy in low SNR and high JSR scenarios. Additionally, most of these algorithms rely on the Newton optimizer, which is highly sensitive to the initial value and prone to becoming stuck in saddle points. In recent years, swarm intelligence-based BSS objective functions such as kurtosis and negative entropy have gained attention [25; 26; 27; 28; 29]. However, these approaches primarily focus on real-valued BSS through instantaneous mixing models, and they rarely consider swarm intelligence as a joint diagonalization optimization algorithm. To address these concerns, this paper introduces a time delay mixing model and proposes a method to correct amplitude and phase errors, ensuring the accurate demodulation of the communication signal. Secondly, the whitening matrix is constructed using multiple time delay correlation matrices of the observed signals, which enhances its anti-noise ability due to the correlation matrix of Gaussian noise being 0. Additionally, the separation matrix is constructed using multiple complex-valued Givens matrices. These matrices not only satisfy the orthogonality constraint of the separation matrix but also effectively reduce the number of parameters to be calculated, simplifying the separation complexity while ensuring accuracy. To further enhance the performance, a nonlinear convergence factor and chaotic perturbation are incorporated into the standard grey wolf optimizer (GWO), resulting in the nonlinear chaotic grey wolf optimizer (NCGWO). Simulation results demonstrate that the proposed algorithm exhibits higher separation accuracy and requires fewer iterations, particularly under low SNR and high JSR conditions. These findings lay a solid foundation for applications such as anti-main lobe jamming, the reduction of the bit error rate (BER), and the joint estimation of the DOA, distance, and velocity. The main contributions or novelties of this paper are as follows. 1. This paper first proposes an integrated receiving system that is based on signal separation. The system utilizes a time delay mixed model, allowing it to receive radar, communication, and jamming signals that suffer from time-frequency aliasing. By applying a blind source separation algorithm, the mixed signals can be effectively separated. This enables the system to perform various tasks such as target position and velocity estimation, the extraction of communication information, anti-interference measures, and direction of arrival (DOA) estimation. 2. A robust whitening method is employed instead of the traditional method. This method utilizes multiple time delay covariance matrices of observed signals to construct a whitening matrix, effectively suppressing the influence of noise. Additionally, we introduce a new method to accurately calculate the coefficients of the delay covariance matrix. 3. The F-norm of the joint diagonalization is used as the objective function. This replaces kurtosis and negative entropy, which are traditionally used in swarm intelligence in solving the blind source separation. Unlike kurtosis and negative entropy, the F-norm does not require multiple averaging operations, making it more efficient when dealing with large amounts of data. The F-norm solely relies on matrix operations, which are known for their fast computation speed. 4. This paper proposes a nonlinear chaotic grey wolf optimizer to solve the objective function. In order to optimize the performance, we apply a restriction on the unitary matrix using multiple complex Givens matrices. This restriction not only reduces the number of parameters in the solution but also accelerates the overall solution process.