This paper presents a robust uncertainty principle that allows the exact reconstruction of a discrete-time signal from highly incomplete frequency information. The key result is that if a signal f is sparse (i.e., has a small number of non-zero entries), then it can be exactly reconstructed from a small subset of its Fourier coefficients using ℓ₁ minimization. The paper shows that with high probability, the solution to the ℓ₁ minimization problem matches the original signal. This result is extended to higher dimensions and other setups, such as reconstructing piecewise constant images from incomplete frequency samples. The methodology relies on probabilistic arguments and duality theory, and it demonstrates that the ℓ₁ minimization problem can recover signals with a number of non-zero entries much smaller than the number of samples. The paper also discusses the relationship between this result and classical uncertainty principles, and it provides numerical experiments that support the theoretical findings. The main theorem states that for a signal f supported on a set T with size proportional to the logarithm of the signal length, the ℓ₁ minimization problem recovers f exactly with high probability. The paper also addresses the issue of uniqueness and provides conditions under which the reconstruction is unique. The results have important implications for signal processing, image reconstruction, and other applications where sparse signals are involved.This paper presents a robust uncertainty principle that allows the exact reconstruction of a discrete-time signal from highly incomplete frequency information. The key result is that if a signal f is sparse (i.e., has a small number of non-zero entries), then it can be exactly reconstructed from a small subset of its Fourier coefficients using ℓ₁ minimization. The paper shows that with high probability, the solution to the ℓ₁ minimization problem matches the original signal. This result is extended to higher dimensions and other setups, such as reconstructing piecewise constant images from incomplete frequency samples. The methodology relies on probabilistic arguments and duality theory, and it demonstrates that the ℓ₁ minimization problem can recover signals with a number of non-zero entries much smaller than the number of samples. The paper also discusses the relationship between this result and classical uncertainty principles, and it provides numerical experiments that support the theoretical findings. The main theorem states that for a signal f supported on a set T with size proportional to the logarithm of the signal length, the ℓ₁ minimization problem recovers f exactly with high probability. The paper also addresses the issue of uniqueness and provides conditions under which the reconstruction is unique. The results have important implications for signal processing, image reconstruction, and other applications where sparse signals are involved.