In this paper we propose a framework for using duplex Doppler ultrasound systems. These type of systems need to interleave the acquisition and display of a B-mode image and of the pulsed Doppler spectrogram. In a recent study [1] we have shown that compressed sensing-based reconstruction of Doppler signal allowed reducing the number of Doppler emissions and yielded better results than traditional interpolation and at least equivalent or even better depending on the configuration than the study estimating the signal from sparse data sets given in [2]. We propose here to improve over this study by using a novel framework for randomly interleaving Doppler and US emissions. The proposed method reconstructs the Doppler signal segment by segment using a block sparse Bayesian learning (BSBL) algorithm based CS reconstruction. The interest of using such framework in the context of duplex Doppler is linked to the unique ability of BSBL to exploit block-correlated signals and to recover nonsparse signals. The performance of the technique is evaluated from simulated data as well as experimental in vivo data and compared to the recent results in [1].