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