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A novel all-optical design for implementing THz-bandwidth real-time Hilbert transformers is proposed and numerically demonstrated. We show that an all-optical Hilbert transformer can be implemented using a uniform-period long-period fiber grating (LPG) with a properly designed amplitude-only grating apodization profile incorporating a single/multiple π-phase-shift(s) along the grating length. The designed LPG for implementation of Hilbert transformer operates in the cross-coupling mode, which can be practically implemented based on either a fiber-optic approach or integrated-waveguide technology. All-optical Hilbert transformers capable of processing arbitrary optical signals with bandwidths well in the THz range can be implemented using feasible LPG designs.