A Branch and Bound Approach to speed up the Sphere Decoder
Proceedings. (ICASSP '05). IEEE International Conference on Acoustics, Speech, and Signal Processing, 2005.
It is well known that maximum-likelihood decoding in many communications applications 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 and SNRs, the sphere decoder can be used to find the exact solution with an expected complexity that is roughly cubic in the dimension of the problem. However, the computational complexity becomes prohibitive if the SNR is too low and/or if
... 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 decoder. The search tree is pruned by computing lower bounds on the possible optimal solution as we proceed to go down the tree. We observe a trade-off between the computational complexity required to compute the 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. Thus, even though it is possible to prune the search tree (and hence the number of points visited) by several orders of magnitude, this may be offset by the computations required to perform the pruning. All of which suggests the need for computationally-efficient tight lower bounds. We present three different lower bounds based on spherical-relaxation, on polytope-relaxation and on duality, simulate their performances and discuss their relative merits.