  This paper is concerned with detecting an integer parameter vector inside a
box from a linear model that is corrupted with a noise vector following the
Gaussian distribution. One of the commonly used detectors is the maximum
likelihood detector, which is obtained by solving a box-constrained integer
least squares problem, that is NP-hard. Two other popular detectors are the
box-constrained rounding and Babai detectors due to their high efficiency of
implementation. In this paper, we first present formulas for the success
probabilities (the probabilities of correct detection) of these three detectors
for two different situations: the integer parameter vector is deterministic and
is uniformly distributed over the constraint box. Then, we give two simple
examples to respectively show that the success probability of the
box-constrained rounding detector can be larger than that of the
box-constrained Babai detector and the latter can be larger than the success
probability of the maximum likelihood detector when the parameter vector is
deterministic, and prove that the success probability of the box-constrained
rounding detector is always not larger than that of the box-constrained Babai
detector when the parameter vector is uniformly distributed over the constraint
box. Some relations between the results for the box constrained and ordinary
cases are presented, and two bounds on the success probability of the maximum
likelihood detector, which can easily be computed, are developed. Finally,
simulation results are provided to illustrate our main theoretical findings.
