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.