This chapter describes a method for one-dimensional signal denoising that simultaneously utilizes both sparse optimization principles and conventional linear time-invariant (LTI) filtering. The method, called ‘sparsity-assisted signal smoothing’ (SASS), is based on modeling a signal as the sum of a low-pass component and a piecewise smooth component. The problem is formulated as a sparse-regularized linear inverse problem. We provide simple direct methods to set the regularization and non-convexity parameters, the latter if a non-convex penalty is utilized. We derive an iterative optimization algorithm that harnesses the computational efficiency of fast solvers for banded systems. The SASS approach performs a type of wavelet denoising, but does so through sparse optimization rather than through wavelet transforms. The approach is relatively free of the pseudo-Gibbs phenomenon that tends to arise in wavelet denoising.