Abstract
In this paper, we consider the performance of exclusive-OR (XOR) rule in detecting the presence or absence of a deterministic signal in bivariate Gaussian noise. Signals, when present at the two sensors, are assumed unequal, whereas the noise components have identical marginal distribution but are correlated. The sensors send their one-bit quantized data to a fusion center, which then employs the XOR rule to arrive at the final decision. Here we prove that, in the limit as the correlation coefficient <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$r$ </tex-math></inline-formula> approaches 1, the optimum fusion rule for both parallel and tandem topologies is XOR with identical, alternating partitions (XORAP) of the observations at the sensors. We further quantify the asymptotic decrease of the Bayes error of XORAP towards zero as a constant multiplied by <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\sqrt {1-r}$ </tex-math></inline-formula> , as <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$r$ </tex-math></inline-formula> approaches 1. When compared to the asymptotic Bayes error of CLRT, which decreases to zero exponentially fast, as a function of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$1/(1-r)$ </tex-math></inline-formula> , the Bayes error of XORAP decreases to zero much slower.
Highlights
A collection of sensors are employed in a variety of situations in order to enhance information gathering and processing operations
For several testing points of correlation coefficient r ∈ (0.93, 0.9999), s1 = 0.5, s2 = 1, σ 2 = 1, and η = 1, we studied how one-bit quantization pattern might change if LRT fusion with one-bit quantized data is optimized using the GENETIC ALGORITHM (GA) in the two-sensor parallel topology, like the one used in [16]
For r = 0.99, GA yields the XOR rule which is slightly different from XORAP, where the quantization intervals of the GA alternate over a certain segment around the origin, but at the two ends, only one interval happens for the remaining parts of the real line
Summary
A collection of sensors are employed in a variety of situations in order to enhance information gathering and processing operations. Reference [6] discussed the detection of Gaussian signals in Gaussian noise for the same tandem case It was pointed out in [6] that [7] made an incorrect assumption that the first sensor in a tandem network could employ a simple likelihood ratio test without compromising the global optimality. The alternating partitioning intervals of span D for XOR decision was first mentioned in [14] It showed the optimality of XORAP rule for the case of two-sensor parallel topology, in the sense of achieving perfect decision, when r = 1. It should be noted that the correlation coefficient close to 1 might be rarely seen in practice, investigation of this limiting case has its own theoretical importance that contributes to a better understanding of how the local decisions and the fusion rules would change with respect to the change of correlation coefficient, as well as the corresponding performance
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