Unsupervised target generation for hyperspectral imagery (HSI) have generated great interest in the hyperspectral community. However, most of the current unsupervised target generation algorithms have to process large HSI data, which is acquired using the traditional Nyquist-Shannon sampling theorem, resulting in data with high band-to-band correlation. As a consequence, these algorithms end up processing redundant information, raising the demand for large memory storage, processing time, and transmission bandwidth. In the past, some efforts have been dedicated to dealing with the redundant information via data reduction (DR) or data compression post-acquisition. However, to the best of our knowledge, this challenge has been addressed outside the context of Compressive Sensing (CS). This paper applies CS data acquisition process at the sensor level so that the redundant information is removed at the early stage of the data processing chain. The main advantage of our approach is that it employs a random sensing process, and the concept of universality, to randomly sense the HSI bands and produce data containing the bare minimum information. We take advantage of CS Restricted Isometric Properties (RIP), Restricted Conformal Properties (RCP), and newly derived orthogonal sub-space projection (OSP) properties to perform automatic target generation process (ATGP) in the compressively sensed band domain (CSBD), instead of in the original data space (ODS), where the HSI data contains full spectral bands. Our experimental results show that, by working in the CSBD, we avoid processing redundant data and still maintain performance results that are comparable with the performance results obtained in the ODS.