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On the sphere-decoding algorithm I. Expected complexity

Hassibi, Babak and Vikalo, Haris (2005) On the sphere-decoding algorithm I. Expected complexity. IEEE Transactions on Signal Processing, 53 (8). pp. 2806-2818. ISSN 1053-587X. doi:10.1109/TSP.2005.850352.

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The problem of finding the least-squares solution to a system of linear equations where the unknown vector is comprised of integers, but the matrix coefficient and given vector are comprised of real numbers, arises in many applications: communications, cryptography, GPS, to name a few. The problem is equivalent to finding the closest lattice point to a given point and is known to be NP-hard. In communications applications, however, the given vector is not arbitrary but rather is an unknown lattice point that has been perturbed by an additive noise vector whose statistical properties are known. Therefore, in this paper, rather than dwell on the worst-case complexity of the integer least-squares problem, we study its expected complexity, averaged over the noise and over the lattice. For the "sphere decoding" algorithm of Fincke and Pohst, we find a closed-form expression for the expected complexity, both for the infinite and finite lattice. It is demonstrated in the second part of this paper that, for a wide range of signal-to-noise ratios (SNRs) and numbers of antennas, the expected complexity is polynomial, in fact, often roughly cubic. Since many communications systems operate at noise levels for which the expected complexity turns out to be polynomial, this suggests that maximum-likelihood decoding, which was hitherto thought to be computationally intractable, can, in fact, be implemented in real time - a result with many practical implications.

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Additional Information:© 2005 IEEE. Reprinted with permission. Manuscript received June 25, 2003; revised September 19, 2004. The associate editor coordinating the review of this manuscript and approving it for publication was Prof. Zhi Ding.
Subject Keywords:expected complexity; integer least-squares problem; lattice problems; multiple-antenna systems; NP hard; sphere decoding; wireless communications
Issue or Number:8
Record Number:CaltechAUTHORS:HASieeetsp05
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Official Citation:B. Hassibi and H. Vikalo, "On the sphere-decoding algorithm I. Expected complexity," in IEEE Transactions on Signal Processing, vol. 53, no. 8, pp. 2806-2818, Aug. 2005. doi: 10.1109/TSP.2005.850352
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:1636
Deposited By: Tony Diaz
Deposited On:08 Feb 2006
Last Modified:08 Nov 2021 19:12

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