Telehealth and wearable equipment can deliver personal healthcare and necessary treatment remotely. One major challenge is transmitting large amount of biosignals through wireless networks. The limited battery life calls for low-power data compressors. Compressive Sensing (CS) has proved to be a low-power compressor. In this study, we apply CS on the compression of multichannel biosignals. We firstly develop an efficient CS algorithm from the Block Sparse Bayesian Learning (BSBL) framework. It is based on a combination of the block sparse model and multiple measurement vector model. Experiments on real-life Fetal ECGs showed that the proposed algorithm has high fidelity and efficiency. Implemented in hardware, the proposed algorithm was compared to a Discrete Wavelet Transform (DWT) based algorithm, verifying the proposed one has low power consumption and occupies less computational resources.