This paper presents a method for reconstructing a signal from a small number of random measurements. The key idea is that if a signal is sparse or compressible, then it can be accurately recovered from a small number of random measurements by solving a simple linear program. The paper shows that for signals with coefficients that decay rapidly (e.g., following a power law), it is possible to reconstruct the signal to within a high degree of accuracy from a small number of random measurements. The reconstruction is achieved by minimizing the ℓ₁ norm of the signal subject to the constraint that the measurements match the original signal. The paper also shows that this result is optimal in the sense that it is generally impossible to obtain a higher accuracy from any set of K measurements. The methodology extends to various other random measurement ensembles, including randomly sampled Fourier coefficients. The paper also discusses the implications of these results for signal recovery and provides a detailed analysis of the conditions under which the reconstruction is possible. The results are supported by theoretical analysis and numerical experiments.This paper presents a method for reconstructing a signal from a small number of random measurements. The key idea is that if a signal is sparse or compressible, then it can be accurately recovered from a small number of random measurements by solving a simple linear program. The paper shows that for signals with coefficients that decay rapidly (e.g., following a power law), it is possible to reconstruct the signal to within a high degree of accuracy from a small number of random measurements. The reconstruction is achieved by minimizing the ℓ₁ norm of the signal subject to the constraint that the measurements match the original signal. The paper also shows that this result is optimal in the sense that it is generally impossible to obtain a higher accuracy from any set of K measurements. The methodology extends to various other random measurement ensembles, including randomly sampled Fourier coefficients. The paper also discusses the implications of these results for signal recovery and provides a detailed analysis of the conditions under which the reconstruction is possible. The results are supported by theoretical analysis and numerical experiments.