This paper reports on a detailed performance characterization of a recently developed optical single-sideband (OSSB) laser system based on an IQ modulator and second-harmonic generation for rubidium atom interferometry experiments. The measured performance is used to evaluate the noise contributions of this OSSB laser system when it is applied to drive stimulated Raman transitions in $^{87}$Rb for precision measurements of gravitational acceleration. The laser system suppresses unwanted sideband components, but additional phase shift compensation needs to be applied when performing frequency chirps with such an OSSB laser system. The total phase noise contribution of the OSSB laser system in the current experiment is 72 mrad for a single atom-interferometry sequence with interrogation times of $T=120$ ms, which corresponds to a relative precision of 32 n$g$ per shot. The dominant noise sources are found in the relative intensity fluctuations between sideband and carrier components and the phase noise of the microwave source.
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