We demonstrate continuous infrared spectra from 20 nm sample spots, by combining dispersive Fourier-transform infrared spectroscopy (FTIR) with scattering near-field microscopy (s-SNOM). With the "apertureless" tip of a standard AFM cantilever in one arm of a Michelson interferometer the spectra arise simultaneously in amplitude and phase. The effect of near-field phonon resonance of SiC is used to verify background-free s-SNOM operation, and to determine the absolute scattering efficiency, at 6 cm(-1) spectral resolution. We further report first evidence of free-induction decay from a scatterer composed of parts coupled by near-fields. This is possible only with broadband illumination. It offers a new, unique tool to discriminate against background scattering artifacts.