  Cross-correlation techniques have been used since 1974 for measuring velocity
shifts and velocity dispersions from stellar and nebular spectra and, since
1979, the analysis based on the Fourier Method has been applied. However, we
are currently obtaining data with spectral resolutions higher than those for
which this technique was developed, hence some revision seems timely. The
principal aim of this work is adapting Tonry and Davis' method and implementing
it for the treatment of very high spectral resolution data. We have applied
this technique to two different sets of spectroscopic data of moderate and high
resolutions obtained with the MUSE and MEGARA spectrographs respectively. Using
stellar spectra obtained with these two instruments (i) we have optimised the
input parameters; (ii) we have analysed the method assumptions; and (iii) we
have compared the results for the two sets of data. For MEGARA data, we have
found that the cross-correlation function lost its Gaussian behavior at higher
resolutions. Thus, we have developed an equivalent mathematical method that can
be used for this kind of data. Additionally, the velocity dispersion error
analysis suggests that the greatest error introduced in this method is due to
the subtraction or masking of the nebular lines. For the application of
cross-correlation techniques to high spectral resolution data, we propose to
calculate the galaxy-galaxy and star-galaxy correlations, with widths
$\mu_{gg}$ and $\mu_{gt}$ respectively. Then, the width of the broadening
function can be calculated as $\sigma = \sqrt{\mu_{gg}^2 - \mu_{gt}^2}$.
