Sparse system identification is often encountered in applications such as network and acoustic echo cancellation. This work applies the sparsity promoting method to the affine projection algorithm (APA) to develop a sparsity-promoting APA (SAPA). To reduce its computational complexity, the gain matrix of SAPA is periodically updated, which leads to a periodically updated gain matrix based SAPA (PSAPA). In addition, the steady-state and tracking performance of PSAPA is analyzed using an improved weighted energy conservation method, which takes account of the correlation between the adaptive filter weight vector and the noise vector. The results of performance analysis are also suited to other proportionate APAs (PAPAs). Simulation results verify the advantages of the proposed algorithms and show that the theoretical expressions on the steady-state and tracking performance can predict the stochastic behaviors well.