Frequency-shifting loops (FSLs) are analyzed theoretically in cases where the intracavity modulator induces two sidebands at each round trip, a situation that can be commonly obtained with electro-optic intensity or phase modulators. Using a simple model, we discuss the ability of such loops to perform frequency-to-time mapping, in the integer Talbot condition, or pulse repetition rate enhancement, in the fractional Talbot condition. The results are compared to the established acousto-optic FSL with pure frequency shift. We show that, in spite of a more complicated situation resulting from the dual sideband modulation, pulse repetition rate amplification can be obtained with an amplitude modulator, and frequency-to-time mapping can be obtained with a phase modulator. This opens new routes to high-frequency manipulation of microwave-optical signals with high-bandwidth (multi-gigahertz) modulators.