  The ASPECT-BET (An sdd-SPECTrometer for BETa decay studies) project aims to
develop a novel technique for the precise measurement of forbidden $\beta$
spectra in the 10 keV - 1 MeV range. This technique uses a Silicon Drift
Detector (SDD) as the main spectrometer, surrounded, if necessary, by a veto
system to reject events with only partial energy deposition in the SDD.
Accurate knowledge of the spectrometer's response to electrons is essential to
reconstruct the theoretical shape of the $\beta$ spectrum. To compute this
response, GEANT4 simulations optimized for low-energy electron interactions are
used. In this article, we present the performance of these simulations in
reconstructing the electron spectra, measured with SDDs, of a $^{109}$Cd
monochromatic source, both in vacuum and in air. The allowed $\beta$ spectrum
of a $^{14}$C source is also measured and analyzed, and it is shown that the
experimental shape factor commonly used in the literature to reconstruct the
measured spectrum is not necessary to explain the spectrum.
