Integer-forcing Research Articles

Full Diversity Unitary Precoded Integer-Forcing

We consider a point-to-point flat-fading MIMO channel with channel state information known both at transmitter and receiver. At the transmitter side, a lattice coding scheme is employed at each antenna to map information symbols to independent lattice codewords drawn from the same codebook. Each lattice codeword is then multiplied by a unitary precoding matrix P and sent through the channel. At the receiver side, an integer-forcing (IF) linear receiver is employed. We denote this scheme as unitary precoded integer-forcing (UPIF). We show that UPIF can achieve full-diversity under a constraint based on the shortest vector of a lattice generated by the precoding matrix P. This constraint and a simpler version of that provide design criteria for two types of full-diversity UPIF. Type I uses a unitary precoder that adapts at each channel realization. Type II uses a unitary precoder, which remains fixed for all channel realizations. We then verify our results by computer simulations in 2×2, and 4×4 MIMO using different QAM constellations. We finally show that the proposed Type II UPIF outperform the MIMO precoding X-codes at high data rates.

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 × 2 and 4× 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.

Physical Layer Network Coding Based on Integer Forcing Precoded Compute and Forward

In this paper, we address the implementation of physical layer network coding (PNC) based on compute and forward (CF) in relay networks. It is known that the maximum achievable rates in CF-based transmission is limited due to the channel approximations at the relay. In this work, we propose the integer forcing precoder (IFP), which bypasses this maximum rate achievability limitation. Our precoder requires channel state information (CSI) at the transmitter, but only that of the channel between the transmitter and the relay, which is a feasible assumption. The overall contributions of this paper are three-fold. Firstly, we propose an implementation of CF using IFP and prove that this implementation achieves higher rates as compared to traditional relaying schemes. Further, the probability of error from the proposed scheme is shown to have up to 2 dB of gain over the existent lattice network coding-based implementation of CF. Secondly, we analyze the two phases of transmission in the CF scheme, thereby characterizing the end-to-end behavior of the CF and not only one-phase behavior, as in previous proposals. Finally, we develop decoders for both the relay and the destination. We use a generalization of Bezout’s theorem to justify the construction of these decoders. Further, we make an analytical derivation of the end-to-end probability of error for cubic lattices using the proposed scheme.

Integer-Forcing Linear Receiver Design with Slowest Descent Method

Compute-and-forward (CPF) strategy is one category of network coding in which a relay will compute and forward a linear combination of source messages according to the observed channel coefficients, based on the algebraic structure of lattice codes. Recently, based on the idea of CPF, integer forcing (IF) linear receiver architecture for MIMO system has been proposed to recover different integer combinations of lattice codewords for further original message detection. In this paper, we consider the problem of IF linear receiver design with respect to the channel conditions. Instead of exhaustive search, we present practical and efficient suboptimal algorithms to design the IF coefficient matrix with full rank such that the total achievable rate is maximized, based on the slowest descent method. Numerical results demonstrate the effectiveness of our proposed algorithms.