  We report on the noise properties of ion-irradiated YBa$_2$Cu$_3$O$_7$
Josephson junctions. This work aims at investigating the linewidth of the
Josephson oscillation with a detector response experiment at $\simeq$132 GHz.
Experimental results are compared with a simple analytical model based on the
Likharev-Semenov equation and the de Gennes dirty limit approximation. We show
that the main source of low-frequency fluctuations in these junctions is the
broadband Johnson noise and that the excess ($\frac{1}{f}$) noise contribution
does not prevail in the temperature range of interest, as reported in some
other types of high-T$_c$ superconducting Josephson junctions. Finally, we
discuss the interest of ion-irradiated junctions to implement frequency-tunable
oscillators consisting of synchronized arrays of Josephson junctions.
