Following the procedure proposed recently by Martin and Brandenberger we investigate the spectrum of the cosmological perturbations in the case when the ``trans-Plackian'' dispersion relations are motivated by the quantum $\kappa$-Poincar\'e algebra. We find that depending on the choice of initial conditions of the perturbations, the spectrum either differs from the flat one for for instantaneous Minkowski vacuum or, in the case of initial conditions minimizing energy density, leads to the observed scale-invariant Harrison-Zel'dovich spectrum in the Friedmann epoch.