  Nonlinear interferometers allow spectroscopy in the mid-infrared range by
detecting correlated visible light, for which non-cooled detectors with higher
specific detectivity and lower dark count rates are available. We present a new
approach for the registration of spectral information, which combines a
nonlinear interferometer using non-degenerate spontaneous parametric
down-conversion (SPDC) with a Fourier-transform spectroscopy concept. In order
to increase the spectral coverage, we use broadband non-collinear SPDC in
periodically poled LiNbO$_3$. Without the need for spectrally selective
detection, continuous spectra with a spectral bandwidth of more than
100$\,$cm$^{-1}$ are achieved. We demonstrate transmission spectra of a
polypropylene sample measured with ~ 6$\,$cm$^{-1}$ resolution in the spectral
range between 3.2$\,\mu$m to 3.9$\,\mu$m.
