Sub-Doppler, saturation dip, spectra of lines in the $v_1 + v_3$, $v_1 + 2v_4$ and $v_3 + 2v_4$ bands of $^{14}$NH$_3$ have been measured by frequency comb-referenced diode laser absorption spectroscopy. The observed spectral line widths are dominated by transit time broadening, but show resolved or partially-resolved hyperfine splittings that are primarily determined by the $^{14}$N quadrupole coupling. Modeling of the observed line shapes based on the known hyperfine level structure of the ground state of the molecule shows that, in nearly all cases, the excited state level has hyperfine splittings similar to the same rotational level in the ground state. The data provide accurate frequencies for the line positions and easily separate lines overlapped in Doppler-limited spectra. The observed hyperfine splittings can be used to make and confirm rotational assignments and ground state combination differences obtained from the measured frequencies are comparable in accuracy to those obtained from conventional microwave spectroscopy. One upper state level shows very clear differences from the expected splittings. Examination of the known vibration-rotation level structure shows there is a near degeneracy between this level in $v_1+v_3$ and a rotational level in the $v_1 + 2v_4$ manifold which is of the appropriate symmetry to be mixed by magnetic hyperfine terms that couple ortho- and para- forms of the molecule.
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