Network Coding for Wirless Relaying and Wireline Networks
Abstract
Network coding has emerged as an attractive alternative to routing because of the through put improvement it provides by reducing the number of channel uses. In a wireless scenario, in addition, further improvement can be obtained through Physical layer Network Coding (PNC), a technique in which nodes are allowed to transmit simultaneously, instead of transmitting in orthogonal slots. In this thesis, the design and analysis of network coding schemes are considered, for wireless twoway relaying, multiuser Multiple Access Relay Channel (MARC) and wireline networks.
In a wireless twoway relay channel with PNC, the simultaneous transmissions of user nodes result in Multiple Access Interference (MAI) at there lay node. The harmful eﬀect of MAI is the presence of signal set dependent deep channel fade conditions, called singular fade states, under which the minimum distance of the eﬀective constellation at the relay become zero. Adaptively changing the network coding map used at the relay according to channel conditions greatly reduces the impact of this MAI. In this work, we obtain these adaptive PNC maps, which are ﬁnite in number ,by completing partially ﬁlled Latin Squares and using graph vertex coloring. Having obtained the network coding maps, the set of all possible channel realizations is quantized into a ﬁnite number of regions, with a speciﬁc network coding map chosen in a particular region and such a quantization is obtained analytically for 2λPSK signal set. The performance of the adaptive PNC scheme for twoway relaying is analyzed and tight high SNR upper bounds are obtained for the average endtoend symbol error probability, in terms of the average error probability of a pointtopoint fading channel. The adaptive PNC scheme is generalized for twoway relaying with multiple antennas at the nodes.
As an alternative to the adaptive PNC scheme for twoway relaying, a Distributed Space Time Coding (DSTC) scheme is proposed, which eﬀectively removes the eﬀect of singular fade states at the transmitting nodes itself without any Channel State Information at the Transmitter (CSIT), and without any need to change the PNC map as a function of channel fade conditions. It is shown that the singular fade states can be viewed equivalently as vector subspaces of C2, which are referred to as the singular fade subspaces. DSTC design criterion to minimize the number of singular fade subspaces and maximize the coding gain is formulated and explicit low decoding complexity DSTC designs are provided.
For the Kuser MARC, in which K source nodes want to transmit messages to a destination node D with the help of are lay node R, a new PNC scheme is proposed. Use of a manytoone PNC map with conventional minimum squared Euclidean distance decoding at D, results in a loss of diversity order due to error propagation from the relay node. To counter this, we propose a novel low complexity decoder which offers the maximum diversity order of two.
Next, we consider wire line networks and explore the connections between linear network coding, linear index coding and discrete polymatroids, which are the multiset analogue of matroids.
We define a discrete polymatroidal network and show that a fractional vector linear solution over a field Fq exists for a network if and only if the network is discrete polymatroidal with respect to a discrete polymatroid representable over Fq.An algorithm to construct networks starting from certain class of discrete polymatroids is provided. Every representation over Fq for the discrete polymatroid, results in a fractional vector linear solution over Fq for the constructed network.
It is shown that a linear solution to an index coding problem exists if and only if there exists a representable discrete polymatroid satisfying certain conditions which are determined by the index coding problem considered. El Rouayheb et. al. showed that the problem of ﬁnding a multilinear representation for a matroid can be reduced to finding a perfect linear index coding solution for an index coding problem obtained from that matroid. Multilinear representation of a matroid can be viewed as a special case of representation of an appropriate discrete polymatroid. We generalize the result of El Rouayheb et. al. by showing that the problem of finding a representation for a discrete polymatroid can be reduced to finding a perfect linear index coding solution for an index coding problem obtained from that discrete polymatroid.
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