  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-Lemari\'{e} 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. Impact of removing bulk flow on estimation of momentum power spectra
are 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, 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.
