On the sphere-decoding algorithm I. Expected complexity

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 i...

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Bibliographic Details
Published in:IEEE transactions on signal processing Vol. 53; no. 8; pp. 2806 - 2818
Main Authors: Hassibi, B., Vikalo, H.
Format: Journal Article
Language:English
Published: New York, NY IEEE 01.08.2005
Institute of Electrical and Electronics Engineers
The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Subjects:
Additive noise
Algorithms
Applied sciences
Closed-form solution
Coding, codes
Communication systems
Complexity
Cryptography
Equations
Exact sciences and technology
Expected complexity
Global Positioning System
Information, signal and communications theory
integer least-squares problem
lattice problems
Lattices
Mathematical analysis
Maximum likelihood decoding
multiple-antenna systems
NP hard
Polynomials
Satellite navigation systems
Signal and communications theory
Signal to noise ratio
sphere decoding
Studies
Telecommunications and information theory
Vectors
Vectors (mathematics)
wireless communications
wireless communications
Worst case method
Wireless telecommunication
integer least-squares problem
Implementation
Noise level
sphere decoding
NP hard
Least squares method
Security of data
Cryptography
Antenna array
Additive noise
GPS system
Least squares problem
multiple-antenna systems
Algorithm
Maximum likelihood decoding
lattice problems
Satellite navigation
Linear system
Telecommunication system
NP hard problem
Expected complexity
Signal to noise ratio
ISSN:1053-587X, 1941-0476
Online Access:Get full text
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Summary: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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ISSN:1053-587X
1941-0476
DOI:10.1109/TSP.2005.850352