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.

翻译：本文关注的是从一个框中检测一个线性模型中的整数参数矢量,该模型在Gaussian分布后被一个噪声矢量腐蚀了。 一个常用的检测器是最大概率探测器,这是通过解决一个箱中受限制的整数最小方形问题(NP-hard)获得的。另外两个流行的检测器是箱中受限制的圆形探测器和巴比探测器,因为其执行效率很高。在本文件中,我们首先展示了这三个探测器在两种不同情况下的成功概率(正确检测的概率)的公式:整数参数矢量是确定性的,并且统一分布在制约框中。然后,我们举两个简单的例子分别表明,箱中受限制的圆形探测器的成功概率可能大于箱中受限制的圆形探测器,后者可能大于参数矢量矢量的最大概率。 受限制的圆形探测器的成功概率通常不大于某些受约束的Babai 参数矢量检测器, 当常规测算结果被统一地分布在两个受约束的矩阵中时, 。