Terahertz spectrometers with a wide instantaneous frequency coverage for passive remote sensing are enormously attractive for many terahertz applications, such as astronomy, atmospheric science and security. Here we demonstrate a wide-band terahertz spectrometer based on a single superconducting chip. The chip consists of an antenna coupled to a transmission line filterbank, with a microwave kinetic inductance detector behind each filter. Using frequency division multiplexing, all detectors are read-out simultaneously creating a wide-band spectrometer with an instantaneous bandwidth of 45 GHz centered around 350 GHz. The spectrometer has a spectral resolution of $F/Delta F=380$ and reaches photon-noise limited sensitivity. We discuss the chip design and fabrication, as well as the system integration and testing. We confirm full system operation by the detection of an emission line spectrum of methanol gas. The proposed concept allows for spectroscopic radiation detection over large bandwidths and resolutions up to $F/Delta Fsim1000$, all using a chip area of a few $mathrm{cm^2}$. This will allow the construction of medium resolution imaging spectrometers with unprecedented speed and sensitivity.
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