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Summary: Greedy algorithms in the compressive sensing theory have been formed the essential method for pruning power amplifier (PA) behavioral models and digital predistorters (DPDs). However, the inherent batch mode of these algorithms limits their application in adaptive digital predistortion framework. In this paper, a powerful subspace pursuit greedy scheme combined with stochastic gradient descent adaptive algorithm is proposed to design a class of adaptive sparse DPDs. According to the given sparsity level, the proposed approach can obtain the sparse terms of DPDs and extract the corresponding coefficients adaptively. Performance improvement of the proposed method is validated by simulation results on the adaptive DPD excited by 15‐MHz 3‐carrier Long‐Term Evolution signals and 50‐MHz 16 amplitude/phase‐shift keying signals. Meanwhile, measurement results on a Doherty PA excited by 30‐MHz 3‐carrier Long‐Term Evolution signals are also performed to verify the advantage of the proposed approach. Simulation and experimental results show that proposed algorithm can efficiently construct the adaptive sparse DPD models with only a small number of parameters; both nonlinear distortions and memory effects in the PA can be almost completely removed. A comparison with the nonsparsity aware DPD techniques and batch mode compressive sensing pruning techniques has been demonstrated that the proposed method exhibit faster convergence, improving tracking capabilities and reduced computational complexity. [ABSTRACT FROM AUTHOR]