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A novel broadband photonic RF phase shifter is proposed that employs single-sideband modulation of the broadband RF signal onto an optical carrier. The phase of the carrier is shifted relative to the modulated signal using an optical allpass filter. Ring-resonator-based optical allpass filters are tuned across the carrier frequency to yield an RF phase shift of over 360 degrees. Results from a proof-of-concept experimental demonstration are presented for modulation frequencies in the range of 2-4 GHz.