Sparse signal recovery problems are common in parameter estimation, image processing, pattern recognition, and so on. The problem of recovering a sparse signal representation from a signal dictionary might be classified as a linear constraint ${\mathrm{\ell }}_{\mathrm{0}}$-quasinorm minimization problem, which is thought to be a Non-deterministic Polynomial-time (NP)-hard problem. Although several approximation methods have been developed to solve this problem via convex relaxation, researchers find the nonconvex methods to be more efficient in solving sparse recovery problems than convex methods. In this paper a nonconvex Exponential Metric Approximation (EMA) method is proposed to solve the sparse signal recovery problem. Our proposed EMA method aims to minimize a nonconvex negative exponential metric function to attain the sparse approximation and, with proper transformation, solve the problem via Difference Convex (DC) programming. Numerical simulations show that exponential metric function approximation yields better sparse recovery performance than other methods, and our proposed EMA-DC method is an efficient way to recover the sparse signals that are buried in noise.