A detailed discussion of noise theory is given in references 1 and 2.
For any quantity X(t) that exhibits noise, the noise power within a unit bandwidth or power spectral density SX(f) is defined as
where X(f) is the Fourier transform of X(t),
and 
is an ensemble average.
In the simplest case where the transitions that cause the noise are
described by equation (2), where 
is the lifetime
of the fluctuation causing interaction,
the spectral density is given by equation (3).
From simple statistical considerations <( 
)2>
can usually be found, for example in the case of number fluctuations
it is given by Poisson statistics.
This equation is very general with the condition imposed that the
interactions of the electrons are independent. For small
fluctuations this is indeed true and thus the Lorentzian spectrum (equation
(3))
appears often. At
low frequencies ( 
<<1) the spectrum is
white, that is independent of frequency, while at
high frequencies ( >>1) it varies as 1/f2, and its half power
point is at f=1/(2 
).
