# Probability of error of threshold gate networks

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## Abstract

A threshold gate is a logic gate in which the output is determined by a weighted sum of the inputs compared to a threshold. If the weighted sum is less than the threshold, the output is a zero. If the weighted sum is greater than the threshold, the output is a one. In general, the inputs, weights and threshold are correlated random variables. It is possible for the weighted sum and the mean of the weighted sum to lie on opposite sides of the threshold. This will cause an error in the output of the gate. Techniques are derived for calculating the probability of error of single threshold gates and nonsequential threshold gate networks. It is assumed that the means, variances, and correlations of the inputs and weights are known, and that the probability of occurrence of the network input combinations is known. Threshold gates with random inputs and weights are then studied by simulation. Each input and weight is replaced by the appropriately generated random number, thereby generating the density function of the weighted sum. The form of this density function is important in the calculation of the probability of error. The techniques for calculating probability of error are implemented by digital computer programs. These are used as subroutines for an adaptive search technique to minimize the probability of error by adjusting the mean value of the weights. The dependence of probability of error on the variance and correlation of inputs and weights is examined for both optimal and non-optimal realizations.