Phase Precoded Compute-and-Forward with Partial Feedback

In this work, we propose phase precoding for the compute-and-forward (CoF) protocol. We derive the phase precoded computation rate and show that it is greater than the original computation rate of CoF protocol without precoder. To maximize the phase precoded computation rate, we need to 'jointly' find the optimum phase precoding matrix and the corresponding network equation coefficients. This is a mixed integer programming problem where the optimum precoders should be obtained at the transmitters and the network equation coefficients have to be computed at the relays. To solve this problem, we introduce phase precoded CoF with partial feedback. It is a quantized precoding system where the relay jointly computes both a quasi-optimal precoder from a finite codebook and the corresponding network equations. The index of the obtained phase precoder within the codebook will then be fedback to the transmitters. A "deep hole phase precoder" is presented as an example of such a scheme. We further simulate our scheme with a lattice code carved out of the Gosset lattice and show that significant coding gains can be obtained in terms of equation error performance.Comment: 5 Pages, 4 figures, submitted to ISIT 201

A Linearithmic Time Algorithm for a Shortest Vector Problem in Compute-and-Forward Design

We propose an algorithm with expected complexity of \bigO(n\log n) arithmetic operations to solve a special shortest vector problem arising in computer-and-forward design, where $n$ is the dimension of the channel vector. This algorithm is more efficient than the best known algorithms with proved complexity.Comment: It has been submitted to ISIT 201

Integer-Forcing MIMO Linear Receivers Based on Lattice Reduction

A new architecture called integer-forcing (IF) linear receiver has been recently proposed for multiple-input multiple-output (MIMO) fading channels, wherein an appropriate integer linear combination of the received symbols has to be computed as a part of the decoding process. In this paper, we propose a method based on Hermite-Korkine-Zolotareff (HKZ) and Minkowski lattice basis reduction algorithms to obtain the integer coefficients for the IF receiver. We show that the proposed method provides a lower bound on the ergodic rate, and achieves the full receive diversity. Suitability of complex Lenstra-Lenstra-Lovasz (LLL) lattice reduction algorithm (CLLL) to solve the problem is also investigated. Furthermore, we establish the connection between the proposed IF linear receivers and lattice reduction-aided MIMO detectors (with equivalent complexity), and point out the advantages of the former class of receivers over the latter. For the $2 \times 2$ and $4\times 4$ MIMO channels, we compare the coded-block error rate and bit error rate of the proposed approach with that of other linear receivers. Simulation results show that the proposed approach outperforms the zero-forcing (ZF) receiver, minimum mean square error (MMSE) receiver, and the lattice reduction-aided MIMO detectors.Comment: 9 figures and 11 pages. Modified the title, abstract and some parts of the paper. Major change from v1: Added new results on applicability of the CLLL reductio

Efficient Integer Coefficient Search for Compute-and-Forward

Integer coefficient selection is an important decoding step in the implementation of compute-and-forward (C-F) relaying scheme. Choosing the optimal integer coefficients in C-F has been shown to be a shortest vector problem (SVP) which is known to be NP hard in its general form. Exhaustive search of the integer coefficients is only feasible in complexity for small number of users while approximation algorithms such as Lenstra-Lenstra-Lovasz (LLL) lattice reduction algorithm only find a vector within an exponential factor of the shortest vector. An optimal deterministic algorithm was proposed for C-F by Sahraei and Gastpar specifically for the real valued channel case. In this paper, we adapt their idea to the complex valued channel and propose an efficient search algorithm to find the optimal integer coefficient vectors over the ring of Gaussian integers and the ring of Eisenstein integers. A second algorithm is then proposed that generalises our search algorithm to the Integer-Forcing MIMO C-F receiver. Performance and efficiency of the proposed algorithms are evaluated through simulations and theoretical analysis.Comment: IEEE Transactions on Wireless Communications, to appear.12 pages, 8 figure

Lattice Network Coding in Distributed Massive MIMO Systems

In this thesis, the uplink of distributed massive MIMO where a large number of distributed access point antennas simultaneously serve a relatively smaller number of users is considered. Lattice network coding (LNC), which comprises compute and forward (C&F) and integer forcing (IF), is employed to avoid the potentially enormous backhaul load. Firstly, novel algorithms for coefficient selection in C&F are proposed. For the first time, we propose a low polynomial complexity algorithm to find the optimal solution for the complex valued case. Then we propose a sub-optimal simple linear search algorithm which is conceptually sub-optimal, however numerical results show that the performance degradation is negligible compared to the exhaustive method. The complexity of both algorithms are investigated both theoretically and numerically. The results show that our proposed algorithms achieve better performance-complexity trade-offs compared to the existing algorithms. Both algorithms are suitable for lattices over a wide range of algebraic integer domains. Secondly, the performance of LNC in a realistic distributed massive MIMO model (including fading, pathloss and correlated shadowing) is investigated in this thesis. By utilising the characteristic of pathloss, a low complexity coefficient selection algorithm for LNC is proposed. A greedy algorithm for selecting the global coefficient matrix is proposed. Comprehensive comparisons between LNC and some other promising linear strategies for massive MIMO, such as small cells (SC), maximum ratio combining (MRC), and minimum mean square error (MMSE) are also provided. Numerical results reveal that LNC not only reduces the backhaul load, but also provides uniformly good service to all users in a wide range of applications. Thirdly, the inevitable loss of information due to the quantisation and modulo operation under different backhaul constraints are investigated. An extended C\&F with flexible cardinalities is proposed to adapt to the different backhaul constraints. Numerical results show that by slightly increasing the cardinality, the gap between C\&F to the infinite backhaul case can be significantly reduced