In this paper, we propose a new sparse framework for the design of the behavioral model and digital predistorter of a broadband power amplifier (PA). We start by formulating the Volterra kernel to multidimensional memory polynomial by considering the high‐order dynamic truncation of the Volterra model. Then we show how an estimate of the most significant coefficients may be obtained using a matching pursuit (MPT) algorithm by exploiting the sparsity of the model. After the indices of the nonzero coefficients are roughly estimated, the block exact Householder inverse QR‐decomposition‐based recursive least squares (HIQRD‐RLS) algorithm is utilized to estimate the sparse model complex coefficients. For broadband nonlinear PAs, the proposed approach is demonstrated to achieve the best performance among the well‐known traditional approaches in terms of in‐band and out‐of‐band specifications. The proposed approach is also validated by evaluating the digital predistortion (DPD) performance on a Class‐AB PA in terms of adjacent channel power ratio (ACPR). © 2014 Institute of Electrical Engineers of Japan. Published by John Wiley & Sons, Inc.