Ambient backscatter communication (AmBC) leverages the existing ambient radio frequency (RF) environment to implement communication with battery-free devices. One critical challenge of AmBC systems is signal recovery because the transmitted information bits are embedded in the ambient RF signals and these are unknown and uncontrollable. To address this problem, most existing approaches use averaging-based energy detectors and consequently the data rate is low and there is an error floor. Here we propose a new detection strategy based on the ratio between signals received from a multiple-antenna Reader. The advantage of using the ratio is that ambient RF signals are removed directly from the embedded signals without averaging and hence it can increase data rates and avoid the error floor. Different from original ratio detectors that use the magnitude ratio of the signals between two Reader antennas, in our proposed approach, we utilize the complex ratio so that phase information is preserved and propose an accurate linear channel model approximation. This allows the application of existing linear detection techniques from which we can obtain a minimum distance detector and closed-form expressions for bit error rate (BER). In addition, averaging, coding and interleaving can also be included to further enhance the BER. The results are also general, allowing any number of Reader antennas to be utilized in the approach. Numerical results demonstrate that the proposed approach performs better than approaches based on energy detection and original ratio detectors.