Physical Layer Network Coding for Two-Way Relaying with QAM

Abstract

The design of modulation schemes for the physical layer network-coded two way\nrelaying scenario was studied in [1], [3], [4] and [5]. In [7] it was shown\nthat every network coding map that satisfies the exclusive law is representable\nby a Latin Square and conversely, and this relationship can be used to get the\nnetwork coding maps satisfying the exclusive law. But, only the scenario in\nwhich the end nodes use $M$-PSK signal sets is addressed in [7] and [8]. In\nthis paper, we address the case in which the end nodes use $M$-QAM signal sets.\nIn a fading scenario, for certain channel conditions $\\gamma e^{j \\theta}$,\ntermed singular fade states, the MA phase performance is greatly reduced. By\nformulating a procedure for finding the exact number of singular fade states\nfor QAM, we show that square QAM signal sets give lesser number of singular\nfade states compared to PSK signal sets. This results in superior performance\nof $M$-QAM over $M$-PSK. It is shown that the criterion for partitioning the\ncomplex plane, for the purpose of using a particular network code for a\nparticular fade state, is different from that used for $M$-PSK. Using a\nmodified criterion, we describe a procedure to analytically partition the\ncomplex plane representing the channel condition. We show that when $M$-QAM ($M\n>4$) signal set is used, the conventional XOR network mapping fails to remove\nthe ill effects of $\\gamma e^{j \\theta}=1$, which is a singular fade state for\nall signal sets of arbitrary size. We show that a doubly block circulant Latin\nSquare removes this singular fade state for $M$-QAM.\n