Systems, methods, and apparatuses for analyzing and synthesizing wireless communication signals are provided. A receiver might transform a received signal into a basis in which the transformed signal is sparse, which can reduce the complexity of joint detection by facilitating message passing algorithm (MPA) decoding. A transmitter might employ dense codewords in a first basis, which may facilitate certain signal-processing operations and may provide a transmission with a low peak-to-average-power ratio. The codewords can be designed to be sparse when transformed to a second basis. The codewords may be configured for non-orthogonal multiple access.