Fast Fourier transform based estimators are formulated for measuring momentum power spectra,including the auto power spectra of the momentum,the momentum divergence,and the cross spectrum of density fluctuation and momentum divergence.Algorithms using the third order Bettle-Lemarie scaling function to assign discrete objects to regular grids for fast Fourier transform are proposed to clean alias effects.Numerical experiments prove that the implementation can achieve sub-percent precision till close to the Nyquist frequency.The impact of removing bulk flow on the estimation of momentum power spectra is derived theoretically and verified numerically.Subtracting bulk flow has little effects at large scales but might induce meaningful differences in nonlinear regime,and probably it is not necessary to subtract bulk flow for samples which peculiar velocities are exact or sufficiently accurate.Momentum power spectra of dark matter samples from N-body simulation are measured and discussed.As expected,the prediction of the one loop Eulerian perturbation theory agrees with simulation only slightly better than the linear theory at z = 0,but can be applied to higher redshift with improved accuracy.Measurements of simulation data and the one loop Eulerian theory both reveal that the momentum field contains strong rotational part,and there is a large stochastic component in the divergence of momentum which is not correlated with the density field.The three kinds of momentum power spectra have their own characteristics.