  The possible formation of Direct Collapse Black Holes (DCBHs) in the first
metal-free atomic cooling halos at high redshifts ($z > 10$) is nowadays object
of intense study and several methods to prove their existence are currently
under development. The abrupt collapse of a massive ($\sim 10^4 - 10^5 \,
\mathrm{M_{\odot}}$) and rotating object is a powerful source of gravitational
waves emission. In this work, we employ modern waveforms and the improved
knowledge on the DCBHs formation rate to estimate the gravitational signal
emitted by these sources at cosmological distances. Their formation rate is
very high ($\sim 10^4 \, \mathrm{yr^{-1}}$ up to $z\sim20$), but due to a short
duration of the collapse event ($\sim 2-30\, \mathrm{s}$, depending on the DCBH
mass) the integrated signal from these sources is characterized by a very low
duty-cycle (${\cal D}\sim 10^{-3}$), i.e. a shot-noise signal. Our results show
that the estimated signal lies above the foreseen sensitivity of the
Ultimate-DECIGO observatory in the frequency range $(0.8-300) \, \mathrm{mHz}$,
with a peak amplitude $\Omega_{gw} = 1.1 \times 10^{-54}$ at $\nu_{max} = 0.9
\, \mathrm{mHz}$ and a peak Signal-to-Noise Ratio $\mathrm{SNR}\sim 22$ at $\nu
= 20 \, \mathrm{mHz}$. This amplitude is lower than the Galactic confusion
noise, generated by binary systems of compact objects in the same frequency
band. For this reason, advanced techniques will be required to separate this
signal from background and foreground noise components. As a proof-of-concept,
we conclude by proposing a simple method, based on the auto-correlation
function, to recognize the presence of a ${\cal D} \ll 1$ signal buried into
the continuous noise. The aim of this work is to test the existence of a large
population of high-z DCBHs, by observing the gravitational waves emitted during
their infancy.
