Complex processes for envelopes of normal noise

The paper presents a brief exposition of the technique of complex normal random variables as utilized in the study of the envelopes of Gaussian noise processes. The central concept is the pre-envelope of a real normal process. The pre-envelope of a real function is a complex function whose real part is and whose absolute value is the envelope, in the sense of high-frequency theory, of . The joint probability density for is found and used to get the threshold crossing rate. Consideration of nonstationary processes is included.