In the general case of non-uniformly spaced frequency-domain data and/or arbitrarily Coloured disturbing noise, the frequency-domain subspace identification algorithms described in McKelvey, Akcay, and Ljung (IEEE Trans. Automatic Control 41(7) (1996) 960) and Van Overschee and De Moor (Signal Processing 52(2) (1996) 179) are consistent only if the covariance matrix of the disturbing noise is known. This paper studies the asymptotic properties (strong convergence, convergence rate, asymptotic normality, strong consistency and loss in efficiency) of these algorithms when the true noise covariance matrix is replaced by the sample noise covariance matrix obtained from a small number of independent repeated experiments. As an additional result the strong convergence (in case of model errors), the convergence rate and the asymptotic normality of the subspace algorithms with known noise covariance matrix follows.