Basis pursuit has been shown to be an effective method of solving inverse problems with a small amount of data when the system to be determined has a sparse representation. Adaptive filters fall under this general category of problems. Here, we use the echo cancellation context to introduce a method of solving the basis pursuit problem with an iterative method based on the proportionate normalized affine projection algorithm (PAPA). Earlier, it has been shown that PAPA can be derived from a basis pursuit perspective. Here we refine the assumptions made in those derivations and show that an iterative form of PAPA yields the same results as basis pursuit without resorting to the simplex method. The resulting algorithm has extremely fast convergence for adaptive filters with very sparse impulse responses. Simulations using the new iterative approach are also presented.