The subsampling of a linear periodically time-varying system results in a collection of linear time-invariant systems with common poles. This key fact, known as "lifting", is used in a two-step realization method. The first step is the realization of the time-invariant dynamics (the lifted system). Computationally, this step is a rank-revealing factorization of a block-Hankel matrix. The second step derives a state space representation of the periodic time-varying system. It is shown that no extra computations are required in the second step. The computational complexity of the overall method is therefore equal to the complexity for the realization of the lifted system. A modification of the realization method is proposed, which makes the complexity independent of the parameter variation period. Replacing the rank-revealing factorization in the realization algorithm by structured low-rank approximation yields a maximum likelihood identification method. Existing methods for structured low-rank approximation are used to identify efficiently a linear periodically time-varying system. These methods can deal with missing data. (C) 2014 Elsevier Ltd. All rights reserved.