In compressive sensing (CS)-based spectrum sensing literature, most studies consider accurate reconstruction of the primary user signal rather than detection of the signal. Furthermore, possible absence of the signal is not taken into account while evaluating the spectrum sensing performance. In this study, Bayesian CS is studied in detail for primary user detection. In addition to assessing the signal reconstruction performance and comparing it with the conventional basis pursuit approach and the corresponding lower bounds, signal detection performance is also considered both analytically and through simulation studies. In the absence of a primary user signal, the trade-off between probabilities of detection and false alarm is studied as it is equally important to determine the performance of a CS approach when there is no active primary user. To reduce the computation time and yet achieve a similar detection performance, finally the effect of number of iterations is studied for various systems parameters including signal-to-noise-ratio, compression ratio, mean value of accumulated energy and threshold values. The presented framework in this study is important in the overall implementation of CS-based approaches for primary user detection in practical realisations such as LTE downlink OFDMA as it considers both signal reconstruction and detection.