We discuss the generation, using the "Zeldovich approximation'', of initial conditions for cosmological N-body simulations. We give an exact analytical formula for the power spectrum (PS) of such initial conditions, including the contribution coming from the "pre-initial'' (unperturbed) distribution. Using one dimensional simulations we study both the ensemble properties of the initial conditions with respect to the theoretical ones, and finite-size effects in this representation. While the PS typically represents well the input (theoretical) PS at wavenumbers below that characteristic of the discreteness, we find that the range of scale, if any, over which the theoretical real space correlation properties (e.g. the two point correlation function and real space mass variance) are well represented by the those of the discretised system, is strongly model dependent.