#### User-Antenna Selection for Physical-Layer Network Coding based on Euclidean Distance

##### Vaibhav Kumar, Barry Cardiff, Mark F. Flanagan

In this paper, we present the error performance analysis of a multiple-input multiple-output (MIMO) physical-layer network coding (PNC) system with two different user-antenna selection (AS) schemes in asymmetric channel conditions. For the first antenna selection scheme (AS1), where the user-antenna is selected in order to maximize the overall channel gain between the user and the relay, we give an explicit analytical proof that for binary modulations, the system achieves full diversity order of $min(N_A , N_B ) \times N_R$ in the multiple-access (MA) phase, where $N_A$, $N_B$ and $N_R$ denote the number of antennas at user $A$, user $B$ and relay $R$ respectively. We present a detailed investigation of the diversity order for the MIMO-PNC system with AS1 in the MA phase for any modulation order. A tight closed-form upper bound on the average SER is also derived for the special case when $N_R = 1$, which is valid for any modulation order. We show that in this case the system fails to achieve transmit diversity in the MA phase, as the system diversity order drops to $1$ irrespective of the number of transmit antennas at the user nodes. Additionally, we propose a Euclidean distance (ED) based user-antenna selection scheme (AS2) which outperforms the first scheme in terms of error performance. Moreover, by deriving upper and lower bounds on the diversity order for the MIMO-PNC system with AS2, we show that this system enjoys both transmit and receive diversity, achieving full diversity order of $\min(N_A, N_B) \times N_R$ in the MA phase for any modulation order. Monte Carlo simulations are provided which confirm the correctness of the derived analytical results.

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