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.