Cloud radio access network (C-RAN) has recently attracted much attention as a promising architecture for future mobile networks to sustain the exponential growth of data rate. In C-RAN, one data processing center or baseband unit (BBU) communicates with users via distributed remote radio heads (RRHs), which are connected to the BBU via high capacity, low latency fronthaul links. In this paper, we study the compression on fron- thaul uplinks and propose a joint decompression algorithm at the BBU. The central premise behind the proposed algorithm is to ex- ploit the correlation between RRHs. Our contribution is threefold. First, we propose a joint decompression and detection (JDD) algorithm which jointly performs decompressing and detecting. The JDD algorithm takes into consideration both the fading and compression effect in a single decoding step. Second, block error rate (BLER) of the proposed algorithm is analyzed in closed-form by using pair-wise error probability analysis. Third, based on the analyzed BLER, we propose adaptive compression schemes subject to quality of service (QoS) constraints to minimize the fronthaul transmission rate while satisfying the pre-defined target QoS. As a dual problem, we also propose a scheme to minimize the signal distortion subject to fronthaul rate constraint. Numerical re- sults demonstrate that the proposed adaptive compression schemes can achieve a compression ratio of 300% in experimental setups.