Binary Symbol Recovery Via $\ell _{\infty }$ Minimization in Faster-Than-Nyquist Signaling Systems

The issue of binary symbol detection in faster-than-Nyquist signaling systems (also known as overcomplete frame-modulated digital transmission systems) is addressed in this paper. Through convex relaxation, the original combinatorial optimization problem is transformed to an l∞ minimization problem. Following this idea, we further propose lp approximation algorithms to efficiently tackle such a convex optimization problem. For noiseless case, the recoverability of l∞ minimization is analyzed. It is shown that the binary symbol vector b can be completely recovered via l∞ minimization if and only if there is a vector in the row space of the transmission matrix located in the same quadrant as - b. Otherwise, complete reconstruction via l∞ minimization is hopeless. At the same time, we give an upper bound to the reconstruction probability. For noisy case, the reconstruction probability is analyzed via the probability distribution function of indefinite quadratic form in Gaussian vectors. Numerical results are provided to study the detection performance of l∞ minimization. It is shown that, compared with semidefinite programming algorithm and linear minimum mean-squared error (LMMSE) method, the proposed l∞ minimization detection achieves a better performance-complexity trade-off.