Conventional constant false alarm rate (CFAR) radar target detection algorithms, such as the cell-averaging (CA) CFAR and the order statistic (OS) CFAR, performs poorly with dense targets. In this paper, we develop a sparse signal processing-based CFAR algorithm for dense target detection. Under the ℓ1-norm constraint, we develop the relationship between the coefficient and the false alarm rate and verify the false alarm rate via numerical simulations. Numerical results indicate that the sparse signal processing-based method performs better in dense targets scenarios than CA-CFAR and OS-CFAR. The computation cost is evaluated both theoretically and numerically. We show that the computation time of MM-CFAR and CA-CFAR algorithms increases nearly linearly with the sample size.