Integer-forcing Research Articles

Successive Cancellation Integer Forcing via Practical Binary Codes

A new multiple-input multiple-output (MIMO) receiver scheme for practical binary codes is proposed that provides consistent gains over conventional linear receivers. We first develop a practical successive integer forcing (IF) scheme based on practical binary codes rather than lattice codes. We then present the successive cancellation integer forcing (SC-IF) scheme, which combines and enhances successive IF and minimum mean squared error successive interference cancellation (MMSE-SIC). In this scheme, the receiver first decides whether individual decoding or IF sum decoding is appropriate for each data stream, and then conducts successive IF sum decoding only for selected streams while decoding the remaining streams using MMSE-SIC. The proposed SC-IF methodology mitigates the performance loss caused by mismatched IF filtering in fading channels, while attenuating the noise amplification caused by MMSE filtering. Extensive link-level simulations demonstrate that the proposed successive IF significantly improves the basic IF, and the SC-IF improves both the successive IF and MMSE-SIC, offering uniform improvements over conventional linear receivers for most channel correlation and variation parameters and modulation orders at comparable computational costs. These results illustrate the viability of SC-IF as a fundamental building block for high-performance MIMO receivers in 5G-Advanced and/or subsequent-generation communication systems.

Integer Forcing Interference Management for the MIMO Interference Channel

A new interference management scheme based on integer forcing (IF) receivers is studied for the two-user multiple-input and multiple-output (MIMO) interference channel. The proposed scheme employs a message splitting method that divides each data stream into common and private sub-streams, in which the private stream is recovered by the dedicated receiver only while the common stream is required to be recovered by both receivers. Specifically, to enable IF sum decoding at the receiver side, all streams are encoded using the same lattice code. Additionally, the number of common and private streams of each user is carefully determined by considering the number of antennas at transmitters and receivers, the channel matrices, and the effective signal-to- noise ratio (SNR) at each receiver to maximize the achievable rate. Furthermore, we consider various assumptions of channel state information at the transmitter side (CSIT) and propose low-complexity linear transmit beamforming suitable for each CSIT assumption. The achievable sum rate and rate region are analytically derived and extensively evaluated by simulation for various environments, demonstrating that the proposed interference management scheme strictly outperforms the previous benchmark schemes in a wide range of channel parameters due to the gain from IF sum decoding.

Optimization of Integer-Forcing Precoding for Multi-User MIMO Downlink

Integer-forcing (IF) precoding is an alternative to linear precoding for multi-user (MU) multiple-input-multiple-output (MIMO) channels, with the potential to offer superior performance at a similar complexity. In this letter, a low-complexity suboptimal method is proposed to optimize the parameters of an IF scheme for any number of $K$ users. The proposed method involves solving a relaxation of the problem followed by the application of a lattice reduction algorithm and is shown to have an overall complexity of $\mathcal{O}(K^3)$. Simulation results show that the proposed method achieves a higher sum rate than a heuristic choice of parameters and significantly outperforms conventional linear precoding in all simulated scenarios.

Multimode Integer-Forcing Receivers for Block Fading Channels

Integer-forcing (IF) receivers are a class of linear multiple-input multiple-output (MIMO) receivers to achieve near-optimal performance under quasi-static channels, with almost the same complexity as that of conventional linear receivers. Instead of trying to recover the transmitted messages directly, an IF receiver recovers their integer-linear combinations corresponding to a chosen integer-valued effective channel matrix, and then obtains them. Recently, the IF strategy has been extended to a block fading scenario. In this paper, we propose a novel IF receiver for block fading channels, called a multimode IF (M-IF) receiver. Unlike conventional IF receivers, the proposed M-IF receiver employs several integer-valued effective channel matrices for IF decoding. We also propose an efficient search algorithm for selecting such matrices with the aid of Gaussian elimination. We then show that for any channel realizations, the proposed M-IF receiver has better performance in terms of the effective noise variance than conventional IF receivers. Simulation results demonstrate that it works really well in various environments.

Capacity Analysis of Lattice Reduction Aided Equalizers for Massive MIMO Systems

Massive multi-input-multi-output (MIMO) systems are the future of the communication system. The proper design of the MIMO system needs an appropriate choice of detection algorithms. At the same time, Lattice reduction (LR)-aided equalizers have been well investigated for MIMO systems. Many studies have been carried out over the Korkine–Zolotareff (KZ) and Lenstra–Lenstra–Lovász (LLL) algorithms. This paper presents an analysis of the channel capacity of the massive MIMO system. The mathematical calculations included in this paper correspond to the channel correlation effect on the channel capacity. Besides, the achievable gain over the linear receiver is also highlighted. In this study, all the calculations were further verified through the simulated results. The simulated results show the performance comparison between zero forcing (ZF), minimum mean squared error (MMSE), integer forcing (IF) receivers with log-likelihood ratio (LLR)-ZF, LLR-MMSE, KZ-ZF, and KZ-MMSE. The main objective of this work is to show that, when a lattice reduction algorithm is combined with the convention linear MIMO receiver, it improves the capacity tremendously. The same is proven here, as the KZ-MMSE receiver outperforms its counterparts in a significant margin.

Spatially Modulated Integer-Forcing Transceivers With Practical Binary Codes

A new practical multiple-input multiple-output (MIMO) transceiver technique called spatially modulated integer-forcing (SM-IF) is proposed for hybrid beamforming array systems, which combines generalized spatial modulation (GSM) with integer-forcing (IF) MIMO multiplexing transmission to achieve improved spectral efficiency with low system complexity. In the proposed scheme, the transmitter first activates a subset of antennas using analog beamforming and delivers an SM information stream via the index of the activated antenna group. Then MIMO digital multiplexing is applied to send multiple MIMO information streams via multilevel coding (MLC) with binary channel codes through the activated antennas. At the receiver side, the receiver first estimates the index of the activated antenna group to recover the SM stream and then recovers the MIMO streams by IF sum decoding in conjunction with multi-stage decoding. The proposed scheme provides a unified coding framework to achieve a synergistic gain by integrating off-the-shelf practical binary codes and low complexity IF decoding into hybrid beamforming array systems. By extensive link-level numerical simulations, the proposed scheme is shown to outperform both the conventional plain IF MIMO and plain SM approaches. The results demonstrate that the proposed SM-IF can be a promising transceiver technique implementable in practical hybrid beamforming array systems.

Steepest Gradient-Based Orthogonal Precoder for Integer-Forcing MIMO

In this paper, we develop an orthogonal precoding scheme for integer-forcing (IF) linear receivers using the steepest gradient algorithm. Although this scheme can be viewed as a special case of the unitary precoded integer-forcing (UPIF), it has two major advantages. First, the orthogonal precoding outperforms its unitary counterpart in terms of achievable rate, outage probability, and error rate. We verify this advantage via theoretical and numerical analyses. Second, it exhibits lower complexity as the dimension of orthogonal matrices is half that of unitary matrices in the real-valued domain. For finding “good” orthogonal precoder matrices, we propose an efficient algorithm based on the steepest gradient algorithm that exploits the geometrical properties of orthogonal matrices as a Lie group. The proposed algorithm has low complexity and can be easily applied to an arbitrary MIMO configuration. We also confirm numerically that the proposed orthogonal precoding outperforms UPIF type II in some scenarios and the X-precoder in high-order QAM schemes, e.g., 64- and 256-QAM.

Transmit Beamforming and Integer Matrix Design for MISO SWIPT Systems With Integer Forcing

This letter considers simultaneous wireless information and power transfer (SWIPT) in multiple-input single-output broadcasting systems based on the integer forcing (IF) technique where multiple receivers are equipped with a power splitting circuit and able to harvest energy and decode information at the same time. The optimization problem that incorporates the beamforming matrix and IF coefficients matrix designs is formulated to minimize the transmit power at transmitter, while guaranteeing the achievable rate and harvested energy threshold of each receiver. To tackle the NP-hardness and non-convexity of the optimization problem, we propose an iterative algorithm consisting of two sub-problems. Simulation results provide the comparison of the performance of SWIPT systems with and without IF technique.

Blind Integer-Forcing Interference Alignment for Downlink Cellular Networks

A novel blind interference alignment (IA) scheme in conjunction with interference decoding by virtue of integer forcing is proposed for downlink cellular networks without channel state information at the base station. In the proposed scheme, inter-user interference at each mobile user is aligned in blind manner, and by employing the same lattice code for all users, each user directly decodes integer-linear combinations of interfering codewords from the aligned interference signals in contrast to the previous works that simply null out the aligned interference by zero forcing. Numerical simulations demonstrate that the proposed scheme significantly improves the sum rate compared with the previous blind IA schemes.

Performance analysis for low-complexity detection of MIMO V2V communication systems

In Vehicle-to-Vehicle (V2V) communication, the rapid movement of vehicles brings fast attenuation of signal propagation and serious shift effect of Doppler frequency, leading to serious transmission performance deterioration. Multiple-Input Multiple-Output (MIMO) technique can be well applied in V2V communication by exploiting diversity and multiplexing gain to improve the system reliability and increase the data rate. In an MIMO system, the achieved diversity and multiplexing gain is closely related to its detection method. Integer-Forcing (IF) detection is one of the linear detections proposed in recent years. Compared to the traditional linear and Maximum Likelihood (ML) detection, IF has a good tradeoff between the complexity and system performance. In this paper, considering its good detection performance, IF is applied to the detection of a MIMO V2V communication system and the performance is further analyzed theoretically by employing Alamouti code at the transmitters. The diversity gain proves to be full and is confirmed by the simulation results.

Outage Behavior of Integer Forcing With Random Unitary Pre-Processing

Integer forcing is an equalization scheme for the multiple-input multiple-output communication channel that has been demonstrated to allow operating close to capacity for “most” channels. In this paper, the measure of “bad” channels is quantified by considering a compound channel setting, where the transmitter communicates over a fixed channel but knows only its mutual information. The transmitter encodes the data into independent streams, all taken from the same linear code. The coded streams are transmitted after applying a unitary transformation. At the receiver side, integer-forcing equalization is applied, followed by standard single-stream decoding. Considering pre-processing matrices drawn from a random ensemble, outage corresponds to the event that the target rate exceeds the achievable rate of integer forcing for a given channel matrix. For the case of the circular unitary ensemble, an explicit universal bound on the outage probability for a given target rate is derived that holds for any channel in the compound class. The derived bound depends only on the gap-to-capacity and the number of transmit antennas. The results are also applied to obtain universal bounds on the gap-to-capacity of multiple-antenna closed-loop multicast, achievable via linear pre-processed integer forcing.

Transceiver Design for MIMO VLC Systems With Integer-Forcing Receivers

In this paper, we investigate the transceiver design for multiple-input multiple-output visible light communication (VLC) systems that employ the integer-forcing (IF) lattice decoding technique. To facilitate the joint design of the transmitter precoder, the integer matrix and the receiver equalizer, we first give a necessary and sufficient condition for the integer matrix to be invertible over 1-D lattice. Based on this condition, we then propose a new method for choosing the integer matrix which achieves better performance than the existing lattice reduction method. Moreover, taking into account two typical constraints in VLC, we optimize transmit and receive matrices using the conditional gradient method or the projected gradient method. Finally, the integer matrix and the transmit and receive matrices are jointly optimized in an iterative manner. Simulation results are provided to demonstrate the performance improvement achieved by the proposed framework.

Approximate Sum Rate for Integer-Forcing Receiver

The computation of the achievable sum rate for integer-forcing (IF) linear receivers requires the search for the shortest set of linearly independent lattice vectors (SIVP). Since SIVP is considered an NP-hard problem, we propose two approximations to the IF achievable sum rate. The first approximation is suitable for systems with two antennas at both transmitter and receiver ( $2\times 2$ arrays) and is based on the analysis of the Gauss-Lagrange algorithm. The second approximation considers $n\times n$ array systems and is based on Minkowski’s second theorem. Assuming a block-fading channel with Rayleigh distribution, we have derived the exact probability density function of both approximations. We have used these results to obtain the mean sum-rate, outage probability, and outage rate for uncorrelated fading channels. For the second approximation, the effect of channel correlation on these metrics was also analyzed. An exhaustive series of simulations have been conducted to assess the tightness and robustness of both solutions.

Boosted KZ and LLL Algorithms

There exist two issues among popular lattice reduction (LR) algorithms that should cause our concern. The first one is Korkine-Zolotarev (KZ) and Lenstra-Lenstra-Lovasz (LLL) algorithms may increase the lengths of basis vectors. The other is KZ reduction suffers much worse performance than Minkowski reduction in terms of providing short basis vectors, despite its superior theoretical upper bounds. To address these limitations, we improve the size reduction steps in KZ and LLL to set up two new efficient algorithms, referred to as boosted KZ and LLL, for solving the shortest basis problem (SBP) with exponential and polynomial complexity, respectively. Both of them offer better actual performance than their classic counterparts, and the performance bounds for KZ are also improved. We apply them to designing integer-forcing (IF) linear receivers for multi-input multi-output (MIMO) communications. Our simulations confirm their rate and complexity advantages.

Multilevel Coding Scheme for Integer-Forcing MIMO Receivers With Binary Codes

An integer-forcing (IF) linear multiple-input multiple-output (MIMO) receiver has recently been proposed, which is theoretically shown to achieve near capacity with almost the same complexity as that of conventional linear receivers. The key idea is that the receiver attempts to directly decode integer-linear combinations of codewords. To ensure that this sum-decoding operation is feasible, in previous works, lattice codes over $\mathbb {Z}_{q}$ were employed. Although those codes can attain good theoretical performance, however, its implementation complexity can be considerably high in practice, especially when $q$ is large to support high-order modulations. In this paper, we propose a practical multilevel coding scheme for IF MIMO, in which multilevel encoding composed of binary linear codes $(q = 2)$ in conjunction with the natural mapping is employed on the transmitter side and multistage decoding adapted to the IF operation is employed on the receiver side. The performance of the proposed scheme is extensively evaluated both analytically and numerically, showing that the gain of IF over conventional receivers is indeed achievable in practical settings with almost the same complexity. Our results imply that the proposed IF MIMO can be an attractive solution for the 5G communications due to its ability of supporting high spectral efficiency with low complexity.

A polynomial time algorithm for a variant of shortest vector problem with applications in compute-and-forward design

In wireless communication networks, one of the most significant problems in Compute-and-Forward (CPF) strategy based on Integer-Forcing (IF) technique is to search the optimal integer coefficient vector for a relay to decode the received messages. While this is a variant of shortest vector problem (SVP), which is prohibitively complex in its general form. In this paper, we mainly focus on the optimal integer solution with non-zero entries in multi-source one-relay model. For comparison we extend the number (or dimension) of communication nodes from 2 to any larger integer n based on an existing 2-dimensional search algorithm with non-zero entries. Then we add this non-zero-entries constraint to an existing search algorithm and make some improvements to search the optimal integer vector by solving an optimization problem over only one variable instead of searching all vector components, which has a much lower time complexity. We prove that our new search algorithm is more efficient with polynomial complexity compared with the extended n-dimensional algorithm, meanwhile the same optimal integer solution is guaranteed.

On Integer-Forcing Precoding for the Gaussian MIMO Broadcast Channel

Integer-forcing (IF) precoding, also known as downlink IF, is a promising new approach for communication over multiple-input multiple-output (MIMO) broadcast channels. Inspired by the integer-forcing linear receiver for multiple-access channels, it generalizes linear precoding by inducing an effective channel matrix that is approximately integer, rather than approximately identity. Combined with lattice encoding and a pre-inversion of the channel matrix at the transmitter, the scheme has the potential to outperform any linear precoding scheme, despite enjoying similar low complexity. In this paper, a specific IF precoding scheme, called diagonally-scaled exact IF (DIF), is proposed and shown to achieve maximum spatial multiplexing gain. For the special case of two receivers, in the high SNR regime, an optimal choice of parameters is derived analytically, leading to an almost closed-form expression for the achievable sum rate. In particular, it is shown that the gap to the sum capacity is upper bounded by 0.27 bits for any channel realization. For general SNR, a regularized version of DIF (RDIF) is proposed. Numerical results for two receivers under Rayleigh fading show that RDIF can achieve performance superior to optimal linear precoding and very close to the sum capacity.

Integer-Forcing Source Coding

Integer-Forcing (IF) is a new framework, based on compute-and-forward, for decoding multiple integer linear combinations from the output of a Gaussian multiple-input multiple-output channel. This paper applies the IF approach to arrive at a new low-complexity scheme, IF source coding, for distributed lossy compression of correlated Gaussian sources under a minimum mean squared error distortion measure. All encoders use the same nested lattice codebook. Each encoder quantizes its observation using the fine lattice as a quantizer and reduces the result modulo the coarse lattice, which plays the role of binning. Rather than directly recovering the individual quantized signals, the decoder first recovers a full-rank set of judiciously chosen integer linear combinations of the quantized signals, and then inverts it. In general, the linear combinations have smaller average powers than the original signals. This allows to increase the density of the coarse lattice, which in turn translates to smaller compression rates. We also propose and analyze a one-shot version of IF source coding that is simple enough to potentially lead to a new design principle for analog-to-digital converters that can exploit spatial correlations between the sampled signals.

Low-Complexity Integer-Forcing Methods for Block Fading MIMO Multiple-Access Channels

Integer forcing is an alternative approach to conventional linear receivers for multiple-antenna systems. In an integer-forcing receiver, integer linear combinations of messages are extracted from the received matrix before each individual message is recovered. Recently, the integer-forcing approach was generalized to a block fading scenario. Among the existing variations of the scheme, the ones with the highest achievable rates have the drawback that no efficient algorithm is known to find the best choice of integer linear combination coefficients. In this paper, we propose several sub-optimal methods to find these coefficients with low complexity, covering both parallel and successive interference cancellation versions of the receiver. Simulation results show that the proposed methods attain a performance close to optimal in terms of achievable rates for a given outage probability. Moreover, a low-complexity implementation using root LDPC codes is developed, showing that the benefits of the proposed methods also carry on to practice.

Exact SMP Algorithms for Integer-Forcing Linear MIMO Receivers

To obtain the optimal coefficient matrix for the integer-forcing (IF) linear receiver, the successive minima problem (SMP) on lattices needs to be solved. By decomposing SMP into a series of subspace avoiding problems (SAP) and developing a modified sphere-decoding (SD) algorithm based on the Schnorr-Euchner (SE) enumeration strategy to solve each instance of SAP, two practical algorithms are constructed to solve SMP exactly for real and complex lattices respectively. The initial radius and the starting position on the search-tree of the SAP algorithm are optimized by exploiting the intermediate results obtained in previous rounds. As compared to the Minkowski reduction algorithm, the proposed complex SMP algorithm brings not only more freedom in lattice code design, but also computation reduction in finding the coefficient matrix. Moreover, benefiting from the tree-search initialization optimization, the proposed real SMP algorithm can bring even more computation reduction when the system size is large. However, the proposed algorithms achieve practically the same performance as Minkowski reduction for the IF linear receiver, as the Minkowski-reduced basis approximates successive minima very closely.