Protonated molecular species have been proven to be abundant in the interstellar gas. This class of molecules is also pivotal for the determination of important physical parameters for the ISM evolution (e.g. gas ionisation fraction) or as tracers of non-polar, hence not directly observable, species. The identification of these molecular species through radioastronomical observations is directly linked to a precise laboratory spectral characterisation. The goal of the present work is to extend the laboratory measurements of the pure rotational spectrum of the ground electronic state of protonated carbonyl sulfide (HSCO$^+$) and its deuterium substituted isotopomer (DSCO$^+$). At the same time, we show how implementing different laboratory techniques allows the determination of different spectroscopical properties of asymmetric-top protonated species. Three different high-resolution experiments were involved to detected for the first time the $b-$type rotational spectrum of HSCO$^+$, and to extend, well into the sub-millimeter region, the $a-$type spectrum of the same molecular species and DSCO$^+$. The electronic ground-state of both ions have been investigated in the 273-405 GHz frequency range, allowing the detection of 60 and 50 new rotational transitions for HSCO$^+$ and DSCO$^+$, respectively. The combination of our new measurements with the three rotational transitions previously observed in the microwave region permits the rest frequencies of the astronomically most relevant transitions to be predicted to better than 100 kHz for both HSCO$^+$ and DSCO$^+$ up to 500 GHz, equivalent to better than 60 m/s in terms of equivalent radial velocity. The present work illustrates the importance of using different laboratory techniques to spectroscopically characterise a protonated species at high frequency, and how a similar approach can be adopted when dealing with reactive species.
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