  Radio observations being insensitive to the dust-obscuration, have been
exploited to unveil the population of Active Galactic Nuclei (AGN) residing in
galaxies with large dust content. In this paper, we investigate the radio
characteristics of 321 dust-obscured galaxies (DOGs; S$_{\rm 24
{\mu}m}$/S$_{\rm r band}$ $\geq$1000) by using mainly deep band-3 (250$-$550
MHz) observations from the upgraded Giant Metrewave Radio Telescope (uGMRT) and
1.5 GHz Jansky Very Large Array (JVLA) observations. We find that, for our
sample of DOGs, deep (median noise-rms $=$ 30 $\mu$Jy beam$^{-1}$) 400 MHz
band-3 uGMRT observations yield the highest detection rate (28 per cent) among
those obtained with the JVLA, and the LOw Frequency ARray (LOFAR) radio
observations and the ${\it XMM-N}$ X-ray observations. The radio
characteristics of our sample sources, ${\it i.e.,}$ linear extent ($<$40 kpc
at $z$ $<$1.2), bimodal spectral index (${\alpha}_{\rm 400 MHz}^{\rm 1.5 GHz}$)
distribution and the radio luminosities (L$_{\rm 1.5 GHz}$ $>$5.0 $\times$
10$^{23}$ W Hz$^{-1}$), suggest them to be mainly consist of
Compact$-$Steep$-$Spectrum (CSS) or Peaked-Spectrum (PS) sources representing
an early phase of the AGN-jet activity in dust-obscured environments. With
stacking we find the existence of faint radio emission (S$_{\rm 400 MHz}$ =
72.9 $\mu$Jy beam$^{-1}$ and S$_{\rm 1.5 GHz}$ = 29 $\mu$Jy beam$^{-1}$ with
signal-to-noise ratio $\sim$ 20) in otherwise radio-undetected DOGs. Our study
revealing the faint emission at a few tens of $\mu$Jy level in high$-z$ DOGs
can be used as a test-bed for the deeper radio continuum surveys planned with
the Square-Kilometer Array (SKA) and its pathfinders.
