Exact SMP Algorithms for Integer-Forcing Linear MIMO Receivers

Ding, Liqin; Kansanen, Kimmo; Wang, Yang; Zhang, Jiliang

doi:10.1109/twc.2015.2462844

Cited by 23 publications

(49 citation statements)

References 30 publications

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“…where log + 2 (x) = max {log 2 (x), 0}. The first step towards maximizing the rates is to set Plug this into (38) and use Woodbury matrix identity for the inverse of a matrix, we have…”

Section: A If and Sbpmentioning

confidence: 99%

Boosted KZ and LLL Algorithms

Lyu

Ling

2017

IEEE Trans. Signal Process.

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Abstract-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-Lovász (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 multiinput multi-output (MIMO) communications. Our simulations confirm their rate and complexity advantages.

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“…where log + 2 (x) = max {log 2 (x), 0}. The first step towards maximizing the rates is to set Plug this into (38) and use Woodbury matrix identity for the inverse of a matrix, we have…”

Section: A If and Sbpmentioning

confidence: 99%

Boosted KZ and LLL Algorithms

Lyu

Ling

2017

IEEE Trans. Signal Process.

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“…Because this problem is equivalent to finding successive minima of a lattice, we can conveniently employ the sphere decoding algorithms [3], [15] or the LLL algorithms [24], [25]. We can also use the recently proposed algorithms specifically for IF-MIMO [8], [26]- [28].…”

Section: B Performance Metricsmentioning

confidence: 99%

“…Even though a brute force for finding the optimal integer matrix A has a high complexity of O(γ M ) [29], some effort has been made to develop more efficient algorithms. For instance, Ding et al [28] developed an optimal algorithm based on SD and Schnorr-Euchner algorithms [17] to find the optimal A with computational complexity of (πe) M +O(log M ) . A similar algorithm with a slightly lower complexity was also proposed in [38].…”

Section: B Decoding Complexitymentioning

confidence: 99%

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Steepest Gradient-Based Orthogonal Precoder for Integer-Forcing MIMO

Hasan

Kurkoski

Sakzad

et al. 2020

IEEE Trans. Wireless Commun.

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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., 64and 256-QAM.

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“…Based on the symmetricity of rings, the SVP algorithm can have a speed-up. For instance, only 1/4 of the points within a Euclidean ball need to be enumerated in Gaussian integers as |Z [i] × | = 4 (as used in [27]), and only 1/6 of the points need to be enumerated in Eisenstein integers as |Z [ω] × | = 6. Although KZ/Minkowski reduction algorithms do not have the constraint due to Lovasz's condition, their basis expansion process [26] still requires the rings to be Euclidean.…”

Section: Beyond Algebraic Lllmentioning

confidence: 99%

Performance Limits of Lattice Reduction over Imaginary Quadratic Fields with Applications to Compute-and-Forward

Lyu

Porter

Ling

2018

2018 IEEE Information Theory Workshop (ITW)

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Complex bases, along with direct-sums defined by rings of imaginary quadratic integers, induce algebraic lattices.In this work, we study such lattices and their reduction algorithms. First, when the lattice is spanned over a two dimensional basis, we show that the algebraic variant of Gauss's algorithm returns a basis that corresponds to the successive minima of the lattice in polynomial time if the chosen ring is Euclidean. Second, we extend the celebrated Lenstra-Lenstra-Lovász (LLL) reduction from over real bases to over complex bases. Properties and implementations of the algorithm are examined. In particular, satisfying Lovász's condition requires the ring to be Euclidean. Lastly, as an application, we use the algebraic algorithms to find the network coding matrices in compute-and-forward.Such lattice reduction-based approaches have low complexity which is not dictated by the signal-to-noise (SNR) ratio. Moreover, such approaches can not only preserve the degree-of-freedom of computation rates, but ensure the independence in the code space as well. Index Termslattice reduction, Gauss's algorithm, LLL, compute-and-forward. Initially coding lattices from ConstructionA over rational integers Z or Gaussian integers Z[i] are the main enabler in showing the achievable information rate in C&F. Recently, however, the Construction A lattices have been extended to over the ring of Eisenstein integers Z[ω] [4], [5] and other rings of imaginary quadratic integers [6], This work was presented in part at the IEEE Information Theory Workshop 2018, Guangzhou, China, and submitted in part to the IEEE Information Theory Workshop 2019, Visby, Gotland, Sweden. S. Lyu is with the College

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Exact SMP Algorithms for Integer-Forcing Linear MIMO Receivers

Cited by 23 publications

References 30 publications

Boosted KZ and LLL Algorithms

Boosted KZ and LLL Algorithms

Steepest Gradient-Based Orthogonal Precoder for Integer-Forcing MIMO

Performance Limits of Lattice Reduction over Imaginary Quadratic Fields with Applications to Compute-and-Forward

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