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The noise produced at the output of combinational logic circuits by individual gate failures is analyzed through the use of Walsh functions. Soft errors are modeled by allowing the output of each gate in a particular realization to fail temporarily, possibly introducing an error in the single binary output. The input variables also are allowed to be stochastically driven. The output probability of error contains the Walsh transform of an extended logic function and the Walsh characteristic functions of the input variables as well as the individual gate failure variables. These results are specialized to the case where the inputs are statistically independent of the soft errors. A discussion of the transform of the extended logic function is included.