  We determine the frequency dependence of four independent CdWO$_4$ Cartesian
dielectric function tensor elements by generalized spectroscopic ellipsometry
within mid-infrared and far-infrared spectral regions. Single crystal surfaces
cut under different angles from a bulk crystal, (010) and (001), are
investigated. From the spectral dependencies of the dielectric function tensor
and its inverse we determine all long wavelength active transverse and
longitudinal optic phonon modes with $A_u$ and $B_u$ symmetry as well as their
eigenvectors within the monoclinic lattice. We thereby demonstrate that such
information can be obtained completely without physical model line shape
analysis in materials with monoclinic symmetry. We then augment the effect of
lattice anharmonicity onto our recently described dielectric function tensor
model approach for materials with monoclinic and triclinic crystal symmetries
[Phys. Rev. B, 125209 (2016)], and we obtain excellent match between all
measured and modeled dielectric function tensor elements. All phonon mode
frequency and broadening parameters are determined in our model approach. We
also perform density functional theory phonon mode calculations, and we compare
our results obtained from theory, from direct dielectric function tensor
analysis, and from model lineshape analysis, and we find excellent agreement
between all approaches. We also discuss and present static and above
reststrahlen spectral range dielectric constants. Our data for CdWO$_4$ are in
excellent agreement with a recently proposed generalization of the
Lyddane-Sachs-Teller relation for materials with low crystal symmetry [Phys.
Rev. Lett. 117, 215502 (2016)].
