The paper introduces a new iterative recovery algorithm called CoSaMP (Compressive Sampling Matching Pursuit) for approximating compressible signals from noisy samples. CoSaMP is designed to be efficient and robust, offering rigorous bounds on computational cost and storage. The algorithm is based on orthogonal matching pursuit (OMP) but incorporates additional ideas to accelerate convergence and provide strong error guarantees. The main result, Theorem A, guarantees that CoSaMP produces a $2s$-sparse approximation with an error comparable to the scaled $\ell_1$ error of the best $s$-sparse approximation, provided the sampling matrix has a restricted isometry constant $\delta_{2s} \leq c$. The running time is linear in the signal length and proportional to the reconstruction signal-to-noise ratio (R-SNR). The algorithm is particularly efficient for compressible signals, with a runtime complexity of $O(N \log N \cdot |\mathrm{R-SNR}|)$. The paper also discusses the implementation details, resource requirements, and theoretical analysis, including the proof of Theorem A.The paper introduces a new iterative recovery algorithm called CoSaMP (Compressive Sampling Matching Pursuit) for approximating compressible signals from noisy samples. CoSaMP is designed to be efficient and robust, offering rigorous bounds on computational cost and storage. The algorithm is based on orthogonal matching pursuit (OMP) but incorporates additional ideas to accelerate convergence and provide strong error guarantees. The main result, Theorem A, guarantees that CoSaMP produces a $2s$-sparse approximation with an error comparable to the scaled $\ell_1$ error of the best $s$-sparse approximation, provided the sampling matrix has a restricted isometry constant $\delta_{2s} \leq c$. The running time is linear in the signal length and proportional to the reconstruction signal-to-noise ratio (R-SNR). The algorithm is particularly efficient for compressible signals, with a runtime complexity of $O(N \log N \cdot |\mathrm{R-SNR}|)$. The paper also discusses the implementation details, resource requirements, and theoretical analysis, including the proof of Theorem A.