The Electric Network Frequency (ENF) serves as a simple means to verify the authenticity of audio recordings. ENF variations contain crucial information, acting as a distinctive "fingerprint" when electronic devices are connected or located near power mains. A novel framework for ENF estimation is proposed. This approach alternates between the Least Absolute Deviation (LAD) regression for determining regression weights and objective function minimization with respect to frequency, adapting them within the context of the ℓ 1 norm or the sum of ℓ 1 norms of the approximation error. This framework is a direct consequence of Laplacian distributed noise. Goodness-of-fit tests are reported, indicating that the Laplacian noise hypothesis is more appropriate than the hypothesis of Gaussian noise in the benchmark ENF-WHU dataset. Extensive evaluation using audio recordings from the aforementioned dataset demonstrates the exceptional performance of the proposed framework outperforming state-of-the-art ENF estimation schemes. These findings provide compelling evidence for the efficacy of the proposed ENF estimation schemes as reliable prerequisites for detecting audio forgeries.