Compute-and-Forward: Finding the Best Equation

Compute-and-Forward is an emerging technique to deal with interference. It allows the receiver to decode a suitably chosen integer linear combination of the transmitted messages. The integer coefficients should be adapted to the channel fading state. Optimizing these coefficients is a Shortest Lattice Vector (SLV) problem. In general, the SLV problem is known to be prohibitively complex. In this paper, we show that the particular SLV instance resulting from the Compute-and-Forward problem can be solved in low polynomial complexity and give an explicit deterministic algorithm that is guaranteed to find the optimal solution.Comment: Paper presented at 52nd Allerton Conference, October 201

New Shortest Lattice Vector Problems of Polynomial Complexity

The Shortest Lattice Vector (SLV) problem is in general hard to solve, except for special cases (such as root lattices and lattices for which an obtuse superbase is known). In this paper, we present a new class of SLV problems that can be solved efficiently. Specifically, if for an $n$-dimensional lattice, a Gram matrix is known that can be written as the difference of a diagonal matrix and a positive semidefinite matrix of rank $k$ (for some constant $k$), we show that the SLV problem can be reduced to a $k$-dimensional optimization problem with countably many candidate points. Moreover, we show that the number of candidate points is bounded by a polynomial function of the ratio of the smallest diagonal element and the smallest eigenvalue of the Gram matrix. Hence, as long as this ratio is upper bounded by a polynomial function of $n$, the corresponding SLV problem can be solved in polynomial complexity. Our investigations are motivated by the emergence of such lattices in the field of Network Information Theory. Further applications may exist in other areas.Comment: 13 page

Simplified Compute-and-Forward and Its Performance Analysis

The compute-and-forward (CMF) method has shown a great promise as an innovative approach to exploit interference toward achieving higher network throughput. The CMF was primarily introduced by means of information theory tools. While there have been some recent works discussing different aspects of efficient and practical implementation of CMF, there are still some issues that are not covered. In this paper, we first introduce a method to decrease the implementation complexity of the CMF method. We then evaluate the exact outage probability of our proposed simplified CMF scheme, and hereby provide an upper bound on the outage probability of the optimum CMF in all SNR values, and a close approximation of its outage probability in low SNR regimes. We also evaluate the effect of the channel estimation error (CEE) on the performance of both optimum and our proposed simplified CMF by simulations. Our simulation results indicate that the proposed method is more robust against CEE than the optimum CMF method for the examples considered.Comment: Submitted to IET Communications, 29 pages, 7 figures, 1 table, latex, The authors are with the Wireless Research Laboratory (WRL), Department of Electrical Engineering, Sharif University of Technology, Tehran, Ira

Multilevel Coding Schemes for Compute-and-Forward with Flexible Decoding

We consider the design of coding schemes for the wireless two-way relaying channel when there is no channel state information at the transmitter. In the spirit of the compute and forward paradigm, we present a multilevel coding scheme that permits computation (or, decoding) of a class of functions at the relay. The function to be computed (or, decoded) is then chosen depending on the channel realization. We define such a class of functions which can be decoded at the relay using the proposed coding scheme and derive rates that are universally achievable over a set of channel gains when this class of functions is used at the relay. We develop our framework with general modulation formats in mind, but numerical results are presented for the case where each node transmits using the QPSK constellation. Numerical results with QPSK show that the flexibility afforded by our proposed scheme results in substantially higher rates than those achievable by always using a fixed function or by adapting the function at the relay but coding over GF(4).Comment: This paper was submitted to IEEE Transactions on Information Theory in July 2011. A shorter version also appeared in the proceedings of the International Symposium on Information Theory in August 2011 without the proof of the main theore

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

Robust Lattice Alignment for K-user MIMO Interference Channels with Imperfect Channel Knowledge

In this paper, we consider a robust lattice alignment design for K-user quasi-static MIMO interference channels with imperfect channel knowledge. With random Gaussian inputs, the conventional interference alignment (IA) method has the feasibility problem when the channel is quasi-static. On the other hand, structured lattices can create structured interference as opposed to the random interference caused by random Gaussian symbols. The structured interference space can be exploited to transmit the desired signals over the gaps. However, the existing alignment methods on the lattice codes for quasi-static channels either require infinite SNR or symmetric interference channel coefficients. Furthermore, perfect channel state information (CSI) is required for these alignment methods, which is difficult to achieve in practice. In this paper, we propose a robust lattice alignment method for quasi-static MIMO interference channels with imperfect CSI at all SNR regimes, and a two-stage decoding algorithm to decode the desired signal from the structured interference space. We derive the achievable data rate based on the proposed robust lattice alignment method, where the design of the precoders, decorrelators, scaling coefficients and interference quantization coefficients is jointly formulated as a mixed integer and continuous optimization problem. The effect of imperfect CSI is also accommodated in the optimization formulation, and hence the derived solution is robust to imperfect CSI. We also design a low complex iterative optimization algorithm for our robust lattice alignment method by using the existing iterative IA algorithm that was designed for the conventional IA method. Numerical results verify the advantages of the proposed robust lattice alignment method

Signal Alignment: Enabling Physical Layer Network Coding for MIMO Networking

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