Frequency-shifting loops (FSL) are ring resonators containing both an amplifier and a frequency-shifter, usually an acousto-optic modulator (AOFS) [1]. They are promising solutions to generate pulses with tunable and high repetition rates or for arbitrary RF waveform generation. Here we investigate an all-fibered frequency-shifting loop that includes an electro-optic amplitude modulator (EOM). At variance with preceding research, e.g. [1-3], the EOM creates at each round-trip two side-bands that recirculate inside the loop. We demonstrate an original double-pulse regime when the loop length is a multiple of the RF modulation wavelength applied to the modulator. By changing the bias voltage applied to the EOM, the interval between the pulses can be continuously adjusted. The system is modeled by a linear interference model that takes the amplitude modulation function and loop delay into account.