In the past years, considerable work has been done in representation and processing of signals by stochastic methods, leading to the development of stochastic computing techniques [3, 4,8, 121. In stochastic computation, variables are represented by means of the probability of pulse occurrence of random pulse trains. This feature leads to the use of switching techniques to handle information in an analoglike organization [ 71. Stochastic computers perform mathematical operations such as addition, multiplication, delay, and integration by simple logical devices. The accuracy and speed of these computers, as well as their organization, are very suitable for process control [5]. The existence of the above interdependence between logic and arithmetic operations through the use of stochastic representation of variables suggested [6] the possibility of establishing a more general theory of the linkage of the two concepts through a mathematical transform. This transform relates bounded real functions with two-valued probabilistic states functions defined on the complex plane, and was called sigma-transform. The direct transform converts each value of the real function into an infinite set of two-valued states distributed along a direction parallel to the imaginary axis, the state probability being the real function value. An infinite set of two-valued functions defined in the complex plane are ,Itransforms of the same real function x(t). This real function can be obtained from any of those transforms by taking CesHro means for every value of c. Hence, this univoque process provides the inverse Z-transform. This paper proves the existence of the Z-transform of a given real function via random noise construction and relates logical operations in the ECtransform domain with arithmetic operations in the source domain. Finally, the extension to signed functions in two different ways is analysed. 224 0022-247)(/83 $3.00