Improved MSK Carrier Estimation Method Based on Generalized Cyclic Correntropy

‍ ‍For the minimum shift keying （MSK） signals commonly used in submarine VLF communications， the atmospheric noise generated by lightning strikes during transmission has a wide spectrum in which the high-frequency component decays rapidly with distance. However， the very low-frequency component decays slowly within a spherical waveguide， significantly affecting the transmission of signals far from the field source. In particular， when lightning strikes cause strong impulsive noise， the original estimation algorithm based on the generalized cyclic correlation entropy degrades the suppression performance of such strong impulsive noise， leading to the degradation of the carrier parameter estimation ability or even the failure of the problem. In this study， an improved carrier estimation algorithm based on generalized cyclic correlation entropy is proposed by introducing a non-linear function， the sigmoid generalized cyclic correlation entropy or generalized cyclic correlation entropy， （SGCCE） method. First， the generalized cyclic correlation entropy algorithm achieves the characteristics of carrier estimation under the background of impulse noise， and the reasons for the degradation or even failure of the algorithm are elucidated. Then， based on the characteristic that the sigmoid function can effectively suppress the strong impulse noise and maintain the MSK signal characteristics unchanged， the generalized Gaussian kernel function in the generalized cyclic correlation entropy algorithm is improved using the sigmoid function， and then the SGCCE function of the MSK signal is derived by mathematical modeling. Further， the SGCCE function of the MSK signal is introduced， followed by the generalized cyclic correntropy spectrum of the MSK signal （sigmoid generalized cyclic correntropy spectrum， SGCCES）. Then， the generalized cyclic correntropy spectrum （SGCCES） of the MSK signal is introduced， the relationship between the cyclic frequency and the carrier frequency in the spectral cross-section is verified based on the SGCCES， and the position of the largest spectral peak in the positive semi-axis is retrieved to achieve the carrier frequency estimation. The proposed algorithm uses a bounded nonlinear function to suppress the amplitude of strong impulse noise in the original signal to improve the carrier estimation performance of the original algorithm， and the effect of the algorithm on the performance of carrier frequency estimation in the strong impulse environment is verified via computer simulation. Through comparison with different spectral functions， the simulation results show that the proposed improved carrier estimation algorithm has higher estimation accuracy than the existing algorithm when the alpha-stable distribution noise feature index is small and the generalized signal-to-noise ratio of the signal is low. Moreover， the proposed algorithm can maintain good estimation performance and realize accurate estimation of the carrier frequency. Based on our research results， further studies can be conducted to improve the proposed algorithm for detecting and estimating MSK signals when communicating in extremely low-frequency bands to improve its application in the field of non-cooperative communication.