Wireless Physical Layer Network Coding (WNC) techniques exploit the inherent nature of wireless channels to improve the performance of wireless communication systems. Unlike conventional Network Coding, in WNC the transmitted signals constructively interact directly in the constellation space, thus inducing specific requirements on the source constellation design. Suitable multi-source constellations should enable direct decoding of WNC functions of user data (from the observed superimposed constellations at receiving nodes) and simultaneously they should allow delivery of partial information to nodes with worse channel conditions (exploiting the natural broadcast property of wireless channels). Source constellations possessing both the aforementioned attributes simultaneously can be desirable in a WNC-based system and thus proper constellation design can become a relatively challenging task. In this paper we focus on this problem and we introduce a systematic constellation design algorithm for a 5-node Wireless Butterfly Network (WBN) with WNC processing, where the basic principles of multi-source constellation design for WNC systems can be demonstrated. We show that the proposed constellations outperform the conventional approaches over the whole range of SNR conditions in the system.
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