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We propose the concept of ultra-broadband terahertz communication, based on directed non-line-of-sight (NLOS) transmissions. Potential applications of such a system supporting multi-gigabit data rates are given, and put into the context of currently emerging WLANs/WPANs. The technology and propagation constraints serve as boundary conditions for the determination of the required antenna gain to support ultra-broadband communication. Resulting high-gain antenna requirements will necessitate highly directed transmissions. We propose the use of omni-directional dielectric mirrors to support directed NLOS paths. Their performance is investigated with ray-tracing simulations of a terahertz propagation channel in a dynamic office environment, which is calibrated with measured building-material and mirror parameters. We demonstrate that a directed NLOS path scheme will make a terahertz communication system robust to shadowing. Furthermore, we show that dielectric mirrors covering only parts of the walls will significantly enhance the signal coverage in a typical indoor scenario.