Algebraic lattices achieving the capacity of the ergodic fading channel

For ergodic fading, a lattice coding and decoding strategy is proposed and its performance is analyzed for the single-input single-output (SISO) and multiple-input multiple-output (MIMO) point-to-point channel as well as the multiple-access channel (MAC), with channel state information available only at the receiver (CSIR). At the decoder a novel strategy is proposed consisting of a time-varying equalization matrix followed by decision regions that depend only on channel statistics, not individual realizations. Our encoder has a similar structure to that of Erez and Zamir. For the SISO channel, the gap to capacity is bounded by a constant under a wide range of fading distributions. For the MIMO channel under Rayleigh fading, the rate achieved is within a gap to capacity that does not depend on the signal-to-noise ratio (SNR), and diminishes with the number of receive antennas. The analysis is extended to the K-user MAC where similar results hold. Achieving a small gap to capacity while limiting the use of CSIR to the equalizer highlights the scope for efficient decoder implementations, since decision regions are fixed, i.e., independent of channel realizations.

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“…SISO fading channel. Campello et al [25] also proved that algebraic lattices achieve the ergodic capacity of the SISO fading channel.…”

Section: Introductionmentioning

confidence: 96%

Lattice Coding and Decoding for Multiple-Antenna Ergodic Fading Channels

Hindy

Nosratinia

2017

IEEE Trans. Commun.

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“…In this paper, we make a step towards this goal by proving that lattice codes from generalized versions of construction A achieve the capacity of the compound MIMO channel over the entire space of channels (2). This represents an advantage of ideal lattices over the classic Gaussian random codes [10], [11] and standard Construction A [6].…”

Section: Introductionmentioning

confidence: 99%

“…This can be viewed as a compound channel with capacity C. The compound channel model (2) arises in several important scenarios in communications, such as the outage formulation in the open-loop mode and broadcast [3].…”

Section: Introductionmentioning

confidence: 99%

Universal Lattice Codes for MIMO Channels

Campello

Ling

Belfiore

2018

IEEE Trans. Inform. Theory

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We propose a coding scheme that achieves the capacity of the compound MIMO channel with algebraic lattices. Our lattice construction exploits the multiplicative structure of number fields and their group of units to absorb ill-conditioned channel realizations. To shape the constellation, a discrete Gaussian distribution over the lattice points is applied. These techniques, along with algebraic properties of the proposed lattices, are then used to construct a sub-optimal de-coupled coding schemes that achieves a gap to compound capacity by decoding in a lattice that does not depend of the channel realization. The gap is characterized in terms of algebraic invariants of the codes, and shown to be significantly smaller than previous schemes in the literature. We also exhibit alternative algebraic constructions that achieve the capacity of ergodic fading channels.

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“…This paper advances our knowledge of lattice coding theory towards large dimensional random lattices generated from small dimensional ones which are full of structure and can serve as a stepping stone for further research. Recently, there have been some researches that identify the benefits of using lattice codes constructed over rings of algebraic integers other than imaginary quadratic integers (see [20] and [21] for example). The proof techniques used in this paper may be extendable to prove the optimality for those lattices.…”

Section: Introductionmentioning

confidence: 99%

“…researches that identify the benefits of using lattice codes constructed over rings of algebraic integers other than imaginary quadratic integers (see [20] and [21] for example). The proof techniques used in this paper may be extendable to prove the optimality for those lattices.…”

Section: Introductionmentioning

confidence: 99%

Adaptive compute-and-forward with lattice codes over algebraic integers

Huang

Narayanan

Wang

2015

2015 IEEE International Symposium on Information Theory (ISIT)

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We consider the compute-and-forward paradigm with limited feedback. Without feedback, compute-and-forward is typically realized with lattice codes over the ring of integers, the ring of Gaussian integers, or the ring of Eisenstein integers, which are all principal ideal domains (PID). A novel scheme called adaptive compute-andforward is proposed to exploit the limited feedback about the channel state by working with the best ring of imaginary quadratic integers. This is enabled by generalizing the famous Construction A from PID to other rings of imaginary quadratic integers which may not form PID and by showing such the construction can produce good lattices for coding in the sense of Poltyrev and for MSE quantization. Simulation results show that by adaptively choosing the best ring among the considered ones according to the limited feedback, the proposed adaptive computeand-forward provides a better performance than that provided by the conventional compute-and-forward scheme which works over Gaussian or Eisenstein integers solely. Index TermsCompute-and-forward, physical-layer network coding, lattice codes, and algebraic integers. On one hand, since Z[i] and Z[ω] are instances of imaginary quadratic integers, it seems natural to extend the compute-and-forward framework to general rings of imaginary quadratic integers. On the other hand, since the role of the underlying ring of integers can be effectively thought of as being a quantizer of the channel and Z[ω] is already the best quantizer for C where the channel coefficients belong, it seems unnecessary to pursue compute-and-forward over other rings of integers. In this paper, we first seek to better understand the role of rings of algebraic integers in constructing good lattices. We then use an example, namely compute-and-forward with limited feedback, to demonstrate the benefits of performing compute-and-forward over rings of imaginary quadratic integers other than Z[i] and Z [ω].One important difference between a general ring of imaginary quadratic integers and the Gaussian and Eisenstein integers is that the Gaussian integers and Eisenstein integers are not merely rings, they are principal ideal domains (PIDs). Hence, every ideal is generated by a singleton and one can equivalently work with numbers instead of ideals. The constructions of lattices over these two rings [3]- [5] heavily rely on properties of PID. However, a general ring of imaginary quadratic integers is not a PID. Therefore, in

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Algebraic lattices achieving the capacity of the ergodic fading channel

Cited by 8 publications

References 17 publications

Lattice Coding and Decoding for Multiple-Antenna Ergodic Fading Channels

Lattice Coding and Decoding for Multiple-Antenna Ergodic Fading Channels

Universal Lattice Codes for MIMO Channels

Adaptive compute-and-forward with lattice codes over algebraic integers

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