Sparsity Apprised Logarithmic Hyperbolic Tan Adaptive Filters for Nonlinear System Identification and Acoustic Feedback Cancellation

Recently, various robust algorithms based on hyperbolic cosine and sine functions, such as hyperbolic cosine (HCAF), exponential hyperbolic cosine, joint logarithmic hyperbolic cosine adaptive filter, etc., have been predominantly employed for different aspects of adaptive filtering, including nonlinear-system-identification. Further, in this manuscript, an attempt is made to elevate the performance of nonlinear system identification in the wake of impulsive noise interference along with consideration of a sparse environment. Henceforth, in lieu of this, the present paper introduces a new sparsity-apprised logarithmic hyperbolic tan adaptive filter (SA-LHTAF) to handle impulsive noise while dealing with sparse systems. It utilizes a $l_{1}$ norm-related sparsity penalty factor in the robust cost function constructed with a logarithmic hyperbolic tangent function. Further, an improved SA-LHTAF (ISA-LHTAF) is introduced for varying sparsity or moderately sparse systems employing the log sum penalty factor in the proposed technique. The weight update for the proposed technique has been derived from the modified cost function. In addition, the conditions for the upper bound on the convergence factor have been derived. The efficacy of the developed robust techniques is demonstrated for identifying nonlinear systems along with feedback paths of behind-the-ear (BTE) hearing aid. In addition, the proposed techniques are evaluated for training an acoustic feedback canceller for hearing aids.