Myopic Multi-Hop Transmission Strategies in Layered Wireless Networks Maksym A. Girnyk and Lars K. Rasmussen School of Electrical Engineering and ACCESS Linnaeus Center KTH - Royal Institute of Technology, Stockholm, Sweden email: {mgyr, lkra}@kth.se Abstract—A layered wireless network is considered, where information is transmitted in a multi-hop MIMO-like fashion from the source node layer through several intermediate layers of nodes before reaching the sink node layer. Analog network coding is investigated as the network transmission strategy, exploiting the inherent superposition features of the wireless channel. A general myopic transmission protocol is proposed, allowing overhearing across several layers. Special cases are investigated in terms of diversity and bit error rate when using different types of linear detectors. Conditions for successful decoding of all source node messages at each sink node are derived. I. I NTRODUCTION It has been shown that random coding over networks outperforms the best routing strategies in terms of achievable simultaneous rate for all users [1]. Intermediate nodes in the network are allowed to mix the incoming packets via random linear combinations over a finite field. To allow decoding at sink nodes, each intermediate node appends the random weighting factors to the transmitted packet as an overhead. This approach can be adopted to wireless networks, since in the wireless setting, simultaneously transmitted signals are inherently mixed through the random superposition features of the wireless multiple-access channel. The weighting coef- ficients are in this case the random complex channel gains. Intermediate nodes receive a superposition of signals, and subsequently conduct amplify-and-forward to upstream nodes. Assuming that channel state information (CSI) is available at receiving nodes, channel gains are also forwarded as overhead. Sink nodes are therefore able to decode the transmitted mes- sages if a sufficient number of independent linear combinations is received, leading to a solvable linear equation system. This strategy is termed analog network coding (ANC) [2], and has been shown to achieve significantly higher throughput than either routing or digital network coding for wireless channels. A similar amplify-and-forward-based transmission strat- egy was investigated in [3] for wireless networks with a regular structure. It was assumed that source transmissions traverse several layers of intermediate nodes in a multiple- input multiple-output (MIMO) fashion before reaching the The research leading to these results has received funding from the Euro- pean Research Council under the European Community’s Seventh Framework Programme (FP7/2007-2013) / ERC grant agreement n ◦ 228044. The work has further been supported in parts by the ARC Grant DP0986089, and the VR grant 621-2009-4666. destination. This strategy leads to a layered network structure as shown in Fig. 1. The ANC multi-hop transmission strat- egy was shown to be degree-of-freedom optimal for layered networks at high signal-to-noise ratio (SNR) and a trade-off between network size, rate and diversity was studied. High- SNR conditions for ANC in such networks are derived in [4]. At every hop in a layered structure the ANC-based trans- mission strategy behaves as a MIMO channel [5], where corresponding sets of cooperating single-antenna nodes act as multiple transmit/receive antennas, respectively. Similar multi- hop channels were studied in [6]. Here the resulting channel between a pair of multi-antenna source and destination is formed by a number of serially concatenated Rayleigh-fading MIMO channels. Several relaying strategies were analyzed for this scenario from a diversity-multiplexing point-of-view, and approximately universal coding schemes were proposed for achieving the optimal diversity-multiplexing trade-off. In [7] the diversity order of ANC-based multi-hop trans- mission was studied with various types of connections among nodes in a single-antenna multi-hop network. Later, in [8] this approach was further generalized to so-called omniscient and myopic coding strategies. The former assumes cooperation among all the nodes in the network in order to deliver the message to the destination. The latter assumes that cooperation can only be done within several neighboring nodes upstream. It was shown that commonly discarded overheard information can be utilized and lead to increased achievable rates and link reliability at the expense of implementation complexity. In this paper we consider a general ℓ multi-hop myopic transmission strategy for wireless layered networks with over- hearing. We develop a framework for analysis using so- called transmission diagrams. Using the framework, for the case of no overhearing we obtain a closed-form expression of the instantaneous bit error rate (BER). This performance metric can be useful, e.g., for opportunistic relaying [9] and provides us with insights for future average BER analysis. For networks with overhearing we introduce the individual node decoding strategy and derive conditions for successful decoding under the assumption of no cooperation among sink nodes. Simulation results are obtained to support analytical conclusions. For notation, we use boldface upper-case letters, A, to denote matrices, boldface lower-case letters, s, to denote