We demonstrate that using appropriate signal-processing techniques allows us to greatly improve the signal-to-noise ratio and accuracy of frequency comb interference spectroscopy measurements. We show that the phase noise from the continuous wave laser used as local oscillator is common to all beat notes and can be removed, enabling longer coherent integration time. An on-line calibration of the spectrum normalizes the frequency response of the electronics. The signal power-to-noise ratio of the spectra thus obtained is a factor of 16,000 (42 dB) higher than in previously demonstrated results, and the quality of the spectra is much higher.