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An improved affine projection sign algorithm (APSA) was developed herein using a <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi mathvariant="script">L</mi><mi>p</mi></msub></semantics></math></inline-formula>-norm-like constraint to increase the convergence rate in sparse systems. The proposed APSA is robust against impulsive noise because APSA-type algorithms are generally based on the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi mathvariant="script">L</mi><mn>1</mn></msub></semantics></math></inline-formula>-norm minimization of error signals. Moreover, the proposed algorithm can enhance the filter performance in terms of the convergence rate due to the implementation of the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi mathvariant="script">L</mi><mi>p</mi></msub></semantics></math></inline-formula>-norm-like constraint in sparse systems. Since a novel cost function of the proposed APSA was designed for maintaining the similar form of the original APSA, these have symmetric properties. According to the simulation results, the proposed APSA effectively enhances the filter performance in terms of the convergence rate of sparse system identification in the presence of impulsive noises compared to that achieved using the existing APSA-type algorithms.