In this paper we present a new subspace algorithm for the identification of multi-input multi-output linear discrete time systems from measured power spectrum data. We show how the state space system matrices can be determined by taking the inverse discrete Fourier transform of the given data and applying the result to a new realization algorithm. Special attention is paid to ensure the positive realness of the identified power spectrum. The computational speed is improved by applying a Lanczos algorithm. The algorithm is illustrated with two practical examples.