On the sphere-decoding algorithm II. Generalizations, second-order statistics, and applications to communications

In Part 1, we found a closed-form expression for the expected complexity of the sphere-decoding algorithm, both for the infinite and finite lattice. We continue the discussion in this paper by generalizing the results to the complex version of the problem and using the expected complexity expressions to determine situations where sphere decoding is practically feasible. In particular, we consider applications of sphere decoding to detection in multiantenna systems. We show that, for a wide range of signal-to-noise ratios (SNRs), rates, 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. To provide complexity information beyond the mean, we derive a closed-form expression for the variance of the complexity of sphere-decoding algorithm in a finite lattice. Furthermore, we consider the expected complexity of sphere decoding for channels with memory, where the lattice-generating matrix has a special Toeplitz structure. Results indicate that the expected complexity in this case is, too, polynomial over a wide range of SNRs, rates, data blocks, and channel impulse response lengths

On joint detection and decoding of linear block codes on Gaussian vector channels

Optimal receivers recovering signals transmitted across noisy communication channels employ a maximum-likelihood (ML) criterion to minimize the probability of error. The problem of finding the most likely transmitted symbol is often equivalent to finding the closest lattice point to a given point and is known to be NP-hard. In systems that employ error-correcting coding for data protection, the symbol space forms a sparse lattice, where the sparsity structure is determined by the code. In such systems, ML data recovery may be geometrically interpreted as a search for the closest point in the sparse lattice. In this paper, motivated by the idea of the "sphere decoding" algorithm of Fincke and Pohst, we propose an algorithm that finds the closest point in the sparse lattice to the given vector. This given vector is not arbitrary, but rather is an unknown sparse lattice point that has been perturbed by an additive noise vector whose statistical properties are known. The complexity of the proposed algorithm is thus a random variable. We study its expected value, averaged over the noise and over the lattice. For binary linear block codes, we find the expected complexity in closed form. Simulation results indicate significant performance gains over systems employing separate detection and decoding, yet are obtained at a complexity that is practically feasible over a wide range of system parameters

Near-Optimal Detection in MIMO Systems using Gibbs Sampling

In this paper we study a Markov Chain Monte Carlo (MCMC) Gibbs sampler for solving the integer least-squares problem. In digital communication the problem is equivalent to performing Maximum Likelihood (ML) detection in Multiple-Input Multiple-Output (MIMO) systems. While the use of MCMC methods for such problems has already been proposed, our method is novel in that we optimize the "temperature" parameter so that in steady state, i.e. after the Markov chain has mixed, there is only polynomially (rather than exponentially) small probability of encountering the optimal solution. More precisely, we obtain the largest value of the temperature parameter for this to occur, since the higher the temperature, the faster the mixing. This is in contrast to simulated annealing techniques where, rather than being held fixed, the temperature parameter is tended to zero. Simulations suggest that the resulting Gibbs sampler provides a computationally efficient way of achieving approximative ML detection in MIMO systems having a huge number of transmit and receive dimensions. In fact, they further suggest that the Markov chain is rapidly mixing. Thus, it has been observed that even in cases were ML detection using, e.g. sphere decoding becomes infeasible, the Gibbs sampler can still offer a near-optimal solution using much less computations.Comment: To appear in Globecom 200

On the Proximity Factors of Lattice Reduction-Aided Decoding

Lattice reduction-aided decoding features reduced decoding complexity and near-optimum performance in multi-input multi-output communications. In this paper, a quantitative analysis of lattice reduction-aided decoding is presented. To this aim, the proximity factors are defined to measure the worst-case losses in distances relative to closest point search (in an infinite lattice). Upper bounds on the proximity factors are derived, which are functions of the dimension $n$ of the lattice alone. The study is then extended to the dual-basis reduction. It is found that the bounds for dual basis reduction may be smaller. Reasonably good bounds are derived in many cases. The constant bounds on proximity factors not only imply the same diversity order in fading channels, but also relate the error probabilities of (infinite) lattice decoding and lattice reduction-aided decoding.Comment: remove redundant figure

Joint data detection and channel estimation for OFDM systems

We develop new blind and semi-blind data detectors and channel estimators for orthogonal frequency-division multiplexing (OFDM) systems. Our data detectors require minimizing a complex, integer quadratic form in the data vector. The semi-blind detector uses both channel correlation and noise variance. The quadratic for the blind detector suffers from rank deficiency; for this, we give a low-complexity solution. Avoiding a computationally prohibitive exhaustive search, we solve our data detectors using sphere decoding (SD) and V-BLAST and provide simple adaptations of the SD algorithm. We consider how the blind detector performs under mismatch, generalize the basic data detectors to nonunitary constellations, and extend them to systems with pilots and virtual carriers. Simulations show that our data detectors perform well

Maximum-likelihood decoding and integer least-squares: the 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 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 and show that, for a wide range of noise variances and dimensions, 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