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Under α stable distribution impulse noise environment, the least mean p-power algorithm (LMP) cannot handle the instability of the algorithm well, due to the large amplitude of the input signal and the large number of useless small weight coefficients in the sparse channel, which delay the convergence speed of the algorithm. In this paper, we develop a new adaptive filtering algorithm, named the proportional normalization least mean p-power (PNLMP) adaptive filtering algorithm. We first introduce a step size control matrix to improve the overall convergence speed and convergence accuracy of the algorithm. Next, to reduce the influence of a large impulse response on the LMP algorithm, a high-order tongue-line function about error is introduced in the normalization processing. Finally, we extend the tongue-line function to ensure that the cost function can also switch freely between V-shaped and M-shaped. This new algorithm overcomes the problem that the traditional LMP algorithm is only applicable to a specific environment and solves the problem of slow convergence speed and low convergence accuracy. Under α stable distribution impulse noise environment, the simulations show that the PNLMP algorithm can improve the convergence speed and stronger system tracking capability compared with existing algorithms, overcoming the problems of slow overall convergence and low convergence accuracy in the traditional algorithm. [ABSTRACT FROM AUTHOR]