  Echo-enabled harmonic generation free-electron lasers (EEHG FELs) are
promising candidates to produce fully coherent soft x-ray pulses by virtue of
efficient high harmonic frequency up-conversion from UV lasers. The ultimate
spectral limit of EEHG, however, remains unclear, because of the broadening and
distortions induced in the output spectrum by residual broadband energy
modulations in the electron beam. We present a mathematical description of the
impact of incoherent (broadband) energy modulations on the bunching spectrum
produced by the microbunching instability through both the accelerator and the
EEHG line. The model is in agreement with a systematic experimental
characterization of the FERMI EEHG FEL in the photon energy range $130-210$ eV.
We find that amplification of electron beam energy distortions primarily in the
EEHG dispersive sections explains an observed reduction of the FEL spectral
brightness that is proportional to the EEHG harmonic number. Local maxima of
the FEL spectral brightness and of the spectral stability are found for a
suitable balance of the dispersive sections' strength and the first seed laser
pulse energy. Such characterization provides a benchmark for user experiments
and future EEHG implementations designed to reach shorter wavelengths.
