Fading is usually characterized in terms of the resulting strength of the desired signal. From this viewpoint, the key metric of channel quality is the signal-to-interference-and-noise ratio which is calculated from the magnitudes of the fading coefficients. Here, we take an alternate view of fading as a (noisy) equation of the transmitted signals, desired or not. Using structured codes, receivers can decode these equations reliably, often at higher rates than the messages individually. The desired messages can be recovered by collecting and solving an appropriate set of equations. In wireless networks consisting of several sources, relays, and destinations, this approach can increase end-to-end throughputs. For simpler topologies such as multiple-access and point-to-point MIMO, we show that this perspective naturally leads to a new receiver architecture that offers performance close to that of a joint decoder while maintaining complexity close to that of a decorrelator.