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 z( ) of a real normal process. The pre-envelope z( ) of a real function x( ) is a complex function whose real part is x( ) and whose absolute value is the envelope, in the sense of high-frequency theory, of x() . The joint probability density for z(t), z \prime (t) is found and used to get the threshold crossing rate. Consideration of nonstationary processes is included.