Near-Optimal Quantization and Linear Network Coding for Relay Networks

Abstract

We introduce a discrete network corresponding to any Gaussian wireless network that is obtained by simply quantizing the received signals and restricting the transmitted signals to a finite precision. Since signals in the discrete network are obtained from those of a Gaussian network, the Gaussian network can be operated on the quantization-based digital interface defined by the discrete network. We prove that this digital interface is near optimal for Gaussian relay networks and the capacities of the Gaussian and the discrete networks are within a bounded gap of <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">O</i> ( <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">M</i> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> ) bits, where <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">M</i> is the number of nodes. We also prove that any near-optimal coding strategy for the discrete network can be naturally transformed into a near-optimal coding strategy for the Gaussian network merely by quantization. We exploit this property by designing a linear coding strategy for the case of layered discrete relay networks. The linear coding strategy is near optimal and achieves all rates within <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">O</i> ( <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">M</i> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> ) bits of the capacity, independent of channel gains or signal-to-noise ratio. The linear code is therefore a near-optimal strategy for layered Gaussian relay networks and can be used as on the Gaussian network after simply quantizing the signals. The relays in the linear code need not know the channel gains on either the incoming or the outgoing links. The transmit and receive signals at all relays are simply quantized to binary tuples of the same length <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i> , which is all that the nodes need to know. The linear network code is a particularly simple scheme and requires all the relay nodes to collect the received binary tuples into a long binary vector and apply a linear transformation on the long vector. The resulting binary vector is split into smaller binary tuples for transmission by the relays. The quantization requirements of the linear network code are completely defined by the parameter <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i> , which therefore also determines the resolution of the analog-to-digital and digital-to-analog converters that are required for operating the network within a bounded gap of the network's capacity. As evident from the description, the linear network code explicitly connects network coding for wireline networks with codes for Gaussian networks.