Simple and explicit formulae, in a linear fractional transformation (LFT) form, for all suboptimal and optimal solutions to the discrete-time version of the one-block Nehari problem, are derived under no restrictive conditions imposed on the initial data. Using the so-called ‘signature condition’—a generalized Popov-theory type argument—general solvability conditions in terms of a certain type of discrete-time Kalman-Szego-Popov-Yakubovich system are obtained. For the class of optimal approximations, derived via singular perturbation techniques, the LFT coefficients are all of McMillan degree n - r, where n and r are the McMillan degree and the multiplicity of the largest Hankel singular value of the system to be approximated, respectively. The results also lead directly to a simple numerical algorithm.