This paper addresses the challenging problem of identifying sparse signals in the presence of noise, a common issue in electrical engineering, applied mathematics, and statistics. The authors introduce a method called convex relaxation, which aims to recover the ideal sparse signal by solving a convex program. This approach is powerful because it can be solved efficiently using standard scientific software. The paper provides general conditions under which convex relaxation succeeds and demonstrates its effectiveness through various applications, including channel coding, linear regression, and numerical analysis. The key contributions include theoretical guarantees for the performance of convex relaxation and practical examples showing its ability to identify sparse signals in different scenarios. The analysis relies on geometric properties of the dictionary, such as coherence, and the correlation condition, which ensures that the global minimizer of the objective function is supported within the desired index set. The paper also discusses the practical implications of these results and their potential impact on real-world signal processing problems.This paper addresses the challenging problem of identifying sparse signals in the presence of noise, a common issue in electrical engineering, applied mathematics, and statistics. The authors introduce a method called convex relaxation, which aims to recover the ideal sparse signal by solving a convex program. This approach is powerful because it can be solved efficiently using standard scientific software. The paper provides general conditions under which convex relaxation succeeds and demonstrates its effectiveness through various applications, including channel coding, linear regression, and numerical analysis. The key contributions include theoretical guarantees for the performance of convex relaxation and practical examples showing its ability to identify sparse signals in different scenarios. The analysis relies on geometric properties of the dictionary, such as coherence, and the correlation condition, which ensures that the global minimizer of the objective function is supported within the desired index set. The paper also discusses the practical implications of these results and their potential impact on real-world signal processing problems.