We consider the problem of sinusoidal parameter estimation from signed measurement samples. We formulate the sinusoidal parameter estimation problem as a sparse signal recovery problem and apply the alternating direction method of multipliers (ADMM) to efficiently solve it. The ℓ₁-norm and log-norm approximations are used to replace the ℓ₀-norm. We then judiciously design the detailed update steps of ADMM for the two approximations, so that most or all update steps involve computationally efficient closed-form solutions. Numerical examples are provided to demonstrate the effectiveness of our approaches.