Linear prediction is a commonly used approach to preprocess microphone signals for estimation of time delays from an acoustic source to different microphones. A typical linear prediction algorithm employs the least squares criterion to solve an optimization problem. This criterion, however, is not optimal for non-Gaussian distributed speech signals. In this paper, we propose a sparse preprocessing algorithm to time delay estimation (TDE) from the perspective of linear prediction. Unlike the traditional ℓ₂-norm, we exploit ℓ₁-norm to construct an optimization function not only for the prediction residual vector but for the prediction coefficient vector. We propose to use an augmented Lagrangian alternating direction method of multipliers (ADMM) to solve the ℓ₁/ℓ₁-norm based convex optimization problem. The effectiveness of the proposed algorithm is demonstrated by numerical experiments.