Summary: Wavelength conversion using the cross-gain modulation (XGM) of amplified spontaneous emission (ASE) in a traveling-wave type semiconductor optical amplifier (TW-SOA) is theoretically studied. Taking into account the spatial and temporal variations of carrier density along the SOA length, output signal and converted ASE waveforms are analyzed. We also reveal the dependency of the signal and converted ASE waveforms on input signal power and repetition frequency, and confirm that numerical analyses well agree with the experimental results. Finally we qualitatively clarify the way to improve frequency response by simulating eye-diagrams for long SOAs and assist light pumping for the first time.