We investigate theoretically and experimentally an all-fibered frequency-shifting loop which includes an electro-optic amplitude modulator (EOM) and an optical amplifier, and is seeded by a continuous-wave laser. At variance with frequency-shifted feedback lasers, or Talbot lasers, that contain an acousto-optic frequency shifter, the EOM creates at each round-trip two side-bands that recirculate inside the loop. Benefiting from the high modulation frequency of the EOM, a wide optical frequency comb up to 40 GHz is generated. We demonstrate an original double-pulse regime when the loop length is a multiple of the RF modulation wavelength applied to the modulator. The inter-pulse interval is governed by both the bias voltage and modulation depth of the EOM. Besides, some typical waveforms such as saw-tooth and rectangle are experimentally obtained by properly setting operating frequency, bias voltage and the RF power. The system is modeled by a linear interference model that takes the amplitude modulation function and loop delay into account. The model explains the formation of pulse doublets and reproduces well all the experimental waveforms. Furthermore, the un-seeded loop driven above threshold also generates mode-locked picosecond pulse doublets with a continuously adjustable delay up to the modulation period.