Research On Adaptive Filtering Algorithms Based On Log Hyperbolic Cosine Cost Function

With the development of the times and the advancement of science and technology,adaptive filtering algorithms have been widely used in the identification of unknown systems,and have become the focus and hotspot of current research.At present,although scholars at home and abroad have proposed many adaptive filtering algorithms,they are all aimed at traditional channels.In practical applications,many unknown channels have a priori sparsity or linear constraint characteristics.The traditional adaptive filtering algorithm is no longer applicable.In addition,in many practical scenarios,environmental noise is not Gaussian distributed noise,but impulse noise with non-Gaussian characteristics.Traditional adaptive filtering algorithms do not perform well in impulse noise.Therefore,we introduce the logarithmic hyperbolic cosine function is used as a cost function to improve the robustness of the algorithm to impulse noise.In response to the above problems,the main work of this article is as follows:（1）For unknown systems with sparse characteristics,such as underwater acoustic communication system,high-definition digital TV transmission system,etc.,based on the minimum logarithmic hyperbolic cosine algorithm（LL）,the Lawson norm constraint is introduced to make full use of the priori sparse characteristic of the unknown system.By adopting the Lawson norm of the system estimation coefficient vector and the logarithmic hyperbolic cosine function of the error to construct a new cost function,and using the gradient descent method to obtain the iterative update equation of the unknown system coefficient vector,a Lawson norm adaptive filtering algorithm（Lawson-lncosh）based hyperbolic cosine function is proposed.Since the Lawson norm can approximately replacel₁ andl₀ norm,and when the parameter p in the Lawson norm is equal to 1 or 0,the norm can be approximated to thel₁ andl₀ norm respectively.Then,using the method of comparative analysis of simulation experiments,the sparse system is estimated under the impulse noise environment to verify the effectiveness of the algorithm.The computer simulation results show that in the case of Gaussian signal input and strong correlation signal input,compared with other existing algorithms,the designed Lawson-lncosh algorithm has better identification ability for sparse system and better robustness to non-Gaussian impulse noise.（2）An unknown system with constrained characteristics means that the parameter vector of the system is subject to a linear constraint.The constrained adaptive filtering algorithm proposed for the constrained system has been widely used in linear system identification,array signal processing,etc.Due to the constraint adaptive filtering algorithm has the advantage of avoiding the accumulation of errors,and has received widespread attention in the past few years.By taking the logarithmic hyperbolic cosine function of the error as the cost function and the constraint characteristic as the constraint term,the Lagrangian multiplier method is adopted to obtain the iterative update equation of the algorithm,and the constrained adaptive filtering algorithm based on the logarithmic hyperbolic cosine function（CLL）is proposed.According to different background noises,the theoretical analysis of the steady-state mean square deviation of the proposed CLL algorithm is carried out,and then a simulation experiment is designed for comparative analysis,which verifies the superiority of the proposed algorithm in the identification of constraint systems and the validity of the theory analysis of the steady-state mean square deviation.（3）Based on the proposed constrained adaptive filtering algorithm based on the logarithmic hyperbolic cosine function,the algorithm has been expanded and developed,that is,on the basis of the cost function of the algorithm,an iterative recursive algorithm has been developed.In the form,a recursive constraint（RCLL）adaptive filtering algorithm based on logarithmic hyperbolic cosine function is proposed.The proposed algorithm only uses the input signal and error signal of the system to complete the iterative update of the adaptive filter weight vector in a recursive manner.Therefore,there is no step size parameter in the iterative equation of the algorithm,and its convergence performance is not affected by the step size.Compared with the above-mentioned CLL algorithm,the RCLL algorithm can obtain better convergence performance and smaller steady-state mean square error.At the same time,the theoretical analysis of the steady-state mean square error of the algorithm was carried out,and a mathematical expression of the theoretical value of the steady-state mean square error was derived,and simulation verification and comparison were carried out under different noise conditions.Experiment results confirm the validity of the theoretical analysis and better convergence performance than other traditional algorithms.