Diversity of low-density lattices

Abstract

The non-ergodic fading channel is a useful model for various wireless communication channels in both indoor and outdoor environments. Building on Poltyrev's work on infinite lattice constellations for the Gaussian channel, we derive a Poltyrev outage limit (POL) for lattices in presence of block fading.We prove that the diversity order of this POL is equal to the number of degrees of freedom in the channel. Further, we describe full-diversity constructions of real lattices defined by their integer-check matrix, i.e., the inverse of their generator matrix. In the first construction suited to maximum-likelihood decoding, these lattices are defined by sparse (low-density) or non-sparse integer-check matrices. Based on a special structure of the lattice binary image, a second full-diversity lattice construction is described for sparse integer-check matrices in the context of iterative probabilistic decoding. Full diversity is theoretically proved in both cases. We also propose a method to construct lattices for diversity order 4 suitable for iterative probabilistic decoding. Computer simulation results for diversity order L = 2 and L = 4 confirm that the proposed low-density lattices attain the maximal diversity order. The newly defined POL is used during this simulation to declare an outage error without decoding, which drastically improves the decoding runtime.

Design and Practical Decoding of Full-Diversity Construction A Lattices for Block-Fading Channels

“…The authors of [25] have employed the decoding algorithm of LDLCs proposed in [16] which has complexity O(n…”

“…Regarding the various parameters involved in estimating the complexity of this decoder that complicates a rigorous comparison, we give a rough comparison idea by replacing the typical values of these parameters and looking at the numerical values. Using the parameters of [25] where Γ(x) := +∞ 0 t x−1 e −t dt denotes the Gamma function and 1 < α ≤ n is a number defined in [55]; for example we can take α = 2. The latter inequality in (60) is obtained by using Stirling's formula for the Gamma function and considering 2πed 2 ≈ n 1+ 1 n in such a way that the probability of the sphere decoder finding a lattice point does not vanish to zero.…”

“…In the next section we modify the proposed algorithm in this section by removing its iterative phase which enables full-diversity decoding of general Construction A lattices without any assumption about their underlying code. We believe that the second part of the proof of Theorem 20 can be provided by using diversity population evolution (DPE) and Density Evolution (DE) techniques similar to the proofs of [25] and [23]. However, going through the details of these techniques pulls us away from our main goal.…”

“…In both cases, the full-diversity property has been proven theoretically. Construction methods in [25] are provided for diversity order at most 4. Using optimal decoders for decoding lattices on BF channels implies exponential complexity in the worst-case.…”

“…Based on Poltyrev's work on infinite lattices for AWGN channels, a Poltyrev outage limit (POL) in presence of block fading has been presented in [24], [25] for lattices. The diversity order of this POL is the same as the number of fading blocks in the channel.…”