An HTS Josephson spectrometer has been designed, fabricated and experimentally studied. The spectrometer circuit consists of a YBCO bicrystal Josephson junction integrated with a double-slot or log-periodic antenna, connected in parallel with a gold low-inductance shunt. The selective detector response and RF response at intermediate frequency fIF = 1.4 GHz were measured in the signal frequency range fs = 400-1250 GHz and the linewidth of Josephson oscillations (LJO) was determined by using three different methods. (1) The Hilbert spectroscopy approach was used in the processing of the selective detector response. (2) In case of fIF LJO corresponds to the frequency down-conversion in a self-pumped Josephson mixer mode. For the processing of the RF response in the high fIF limit we suggest a novel method that allows extraction of the signal spectrum by means of simple operations: shifting, summing and subtracting. The advantage of the RF response method is in its simplicity and higher sensitivity that allows it to be proposed for spectroscopy applications.