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The heterodyne mixing and direct detection performance of a superconducting NbN hot-electron bolometer (HEB) integrated with a log-spiral antenna have been thoroughly characterized. The corrected receiver noise temperature and IF gain bandwidth are approximately 800 K at 0.5 THz, 750 K at 0.85 THz and 3 GHz at its optimum bias point. In addition, both the receiver noise temperature and IF gain bandwidth were found insensitive to the bath temperature, while the bias point was fixed, in good agreement with those simulated with the hot-spot model taking account of the HEB's current-dependent resistive transition. Furthermore, the HEB's frequency response was measured by a Fourier transform spectrometer at different bias points and bath temperatures. The estimated noise equivalent power (NEP) was close to 10-13 W/Hz0.5 around the HEB's transition temperature.