Cloud Radio Access Network (C-RAN) is a promising candidate for future mobile networks to sustain the exponentially increasing demand for data rate. The centralized architecture enables C-RAN to exploit multi-cell cooperation and interference management effectively. In C-RAN, one baseband unit (BBU) communicates with users through distributed Remote Radio Heads (RRHs) which are connected to the BBU via high capacity, low latency fronthaul links and perform ''soft» relaying. However, the architecture of C-RAN imposes a shortage of fronthaul bandwidth because raw In-phase/Quadrature-phase (I/Q) samples are exchanged between the RRHs and the BBU. In this paper, we leverage on advanced signal processing to improve the compression efficiency in fronthaul uplinks. Specifically, we propose a joint decoding algorithm at the BBU that exploits the correlation among the RRHs and jointly performs decompressing and decoding. An upper bound of the Block Error Rate (BLER) of the proposed algorithm is derived using pair-wise error probability analysis. Based on the BLER upper bound, we propose an adaptive compression scheme which minimizes the fronthaul transmission rate while satisfying a target quality of service constrain on the BLER. Our proposed adaptive compressor originates from practical scenarios in which most applications tolerate certain non-zero BLER thresholds.