Lattices are everywhere

Zamir, Ram

doi:10.1109/ita.2009.5044976

Cited by 118 publications

(105 citation statements)

References 65 publications

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“…We cannot cite or mention all related works on lattices here but we can point the interested reader to an excellent survey by Zamir [3]. By reading this survey, the reader will be able to understand the usefulness of lattices, their definitions, applications and interests.…”

Section: B Compute-and-forward For Multiple Accessmentioning

confidence: 99%

Binary subchannel decomposition of lattice-based multiple access schemes

Fredj

Belfiore

2017

2017 IEEE 13th International Conference on Wireless and Mobile Computing, Networking and Communications (WiMob)

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Abstract-Lattice codes used under the Compute-and-Forward paradigm suggest an alternative strategy for the standard Gaussian multiple-access channel (MAC). It has been proven that decoding an integer linear combination of the transmitted codewords enables higher data rates compared to decoding the messages individually. Recent work of Gasptar et al. proposed a new multiple access channel technique that they called Computeand-Forward Multiple Access Channel (CFMA). The authors proved that, in the case of a two-user MAC and without time sharing, the entire capacity region is achieved using CFMA and a single-user decoder for signal to noise ratios above 1 + √ 2. Throughout this paper, we will work within the Computeand-Forward framework. The strategy relies on lattice codes, here we will construct lattices using a multi-level coding based construction called construction D. To decode the lattice points, a multi-stage decoder (MSD) should be implemented at the receiver part. In this paper, we propose a new decoding strategy that helps to perform the standard MSD algorithm in a less complex way. The new technique is called Binary Subchannel Decomposition (BSCD). Since, we make use of MSD, we need at each level of the decoder to compute soft-inputs: Log-Likelihood ratios. However the calculations are very complex since we must compute infinite sums of exponentials at each stage. Hereinafter we give efficient LLR approximation for multi-stage decoders. Simulation results about the performance of BSCD are included and compared to the performance of the standard multi-stage decoder.

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Section: B Compute-and-forward For Multiple Accessmentioning

confidence: 99%

Binary subchannel decomposition of lattice-based multiple access schemes

Fredj

Belfiore

2017

2017 IEEE 13th International Conference on Wireless and Mobile Computing, Networking and Communications (WiMob)

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“…Hence, the precoder, w l , should be designed such that it is able to minimize z ef f in Equation (12) and maximize the achievable rate in Equation (13). The design of such a precoder is proposed in the next subsection.…”

Section: Problem Formulationmentioning

confidence: 99%

“…This solution maximizes the rate Equation (13), because h z l is the closest integer to the channel, h l . Note that, here, each transmitter precodes the signal by imposing h z l = 0, thereby making the probability of the condition, h z = 0, to become strictly zero.…”

Section: Proposed Integer Forcing Precodermentioning

confidence: 99%

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

Gupta

Vázquez-Castro

2013

Future Internet

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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.

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“…This approach is further extended to the complex channel in [15] for the compound MIMO broadcast channel. In addition to interference channels, lattice codes have also been widely studied in many communication networks [16] such as point-to-point channels and wireless relay networks. Specifically, in [17], the author shows that lattice codes can achieve the capacity of 1 2 log(1+SNR) in the point-to-point additive white Gaussian noise (AWGN) channel.…”

Section: Introductionmentioning

confidence: 99%

Robust Lattice Alignment for $K$-User MIMO Interference Channels With Imperfect Channel Knowledge

Huang

Lau

et al. 2011

IEEE Trans. Signal Process.

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Abstract-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 compared with the TDMA, two-stage ML decoding, generalized Han-Kobayashi (HK), distributive IA and conventional IA methods in the literature.

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Lattices are everywhere

Cited by 118 publications

References 65 publications

Binary subchannel decomposition of lattice-based multiple access schemes

Binary subchannel decomposition of lattice-based multiple access schemes

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

Robust Lattice Alignment for $K$-User MIMO Interference Channels With Imperfect Channel Knowledge

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