Speeding up the Sphere Decoder With H^ and SDP Inspired Lower Bounds

• Published in: IEEE transactions on signal processing Vol. 56; no. 2; pp. 712 - 726
• Main Authors: Stojnic, M., Vikalo, H., Hassibi, B.
• Format: Journal Article
• Language: English
• Published: New York, NY IEEE 01.02.2008; Institute of Electrical and Electronics Engineers; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Algorithms; Applied sciences; Branch-and-bound algorithm; Coding, codes; Computational complexity; convex optimization; Digital communication; Estimation theory; Exact sciences and technology; expected complexity; H^{\infty} estimation; Information, signal and communications theory; integer least-squares; Lattices; Maximum likelihood decoding; Maximum likelihood detection; Maximum likelihood estimation; maximum-likelihood (ML) detection; Signal and communications theory; Signal to noise ratio; sphere decoding; Studies; Telecommunications and information theory; Performance evaluation; Worst case method; Convex programming; sphere decoding; Binary modulation; Least squares method; Digital communication; expected complexity; Branch-and-bound algorithm; H infinite optimization; Lower bound; maximum-likelihood (ML) detection; Branch and bound method; integer least-squares; Least squares problem; H∞ estimation; Estimation theory; Algorithm; Computational complexity; Maximum likelihood decoding; Exact solution; NP hard problem; convex optimization; Search tree; Objective function; Maximum likelihood; Signal to noise ratio
• ISSN: 1053-587X; 1941-0476
• DOI: 10.1109/TSP.2007.906697

It is well known that maximum-likelihood (ML) decoding in many digital communication schemes reduces to solving an integer least-squares problem, which is NP hard in the worst-case. On the other hand, it has recently been shown that, over a wide range of dimensions N and signal-to-noise ratios (SNRs), the sphere decoding algorithm can be used to find the exact ML solution with an expected complexity that is often less than N 3 . However, the computational complexity of sphere decoding becomes prohibitive if the SNR is too low and/or if the dimension of the problem is too large. In this paper, we target these two regimes and attempt to find faster algorithms by pruning the search tree beyond what is done in the standard sphere decoding algorithm. The search tree is pruned by computing lower bounds on the optimal value of the objective function as the algorithm proceeds to descend down the search tree. We observe a tradeoff between the computational complexity required to compute a lower bound and the size of the pruned tree: the more effort we spend in computing a tight lower bound, the more branches that can be eliminated in the tree. Using ideas from semidefinite program (SDP)-duality theory and H infin estimation theory, we propose general frameworks for computing lower bounds on integer least-squares problems. We propose two families of algorithms, one that is appropriate for large problem dimensions and binary modulation, and the other that is appropriate for moderate-size dimensions yet high-order constellations. We then show how in each case these bounds can be efficiently incorporated in the sphere decoding algorithm, often resulting in significant improvement of the expected complexity of solving the ML decoding problem, while maintaining the exact ML-performance.