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A complex-valued random function  $  X = X( t) $
of a real parameter  $  t $
which may be represented as a [[Stochastic integral|stochastic integral]]:

$$ \tag{* }
X ( t)  =  \int\limits _ {- \infty } ^  \infty  
e ^ {i \lambda t }  d \Phi ( \lambda )  = \ 
\lim\limits  \sum _ { k }
e ^ {i \lambda t }
\Delta _ {k} \Phi ( \lambda ),
$$

where  $  \Phi ( \lambda ) $, 
$  - \infty < \lambda < \infty $, 
is a random process. The increments  $  \Delta _ {k} \Phi ( \lambda ) = \Phi ( \lambda _ {k+} 1 ) - \Phi ( \lambda _ {k} ) $
in (*) define random  "amplitudes"   $  A _ {k} = | \Delta _ {k} \Phi ( \lambda ) | $
and  "phases"   $  \theta _ {k} =  \mathop{\rm arg}  \Delta _ {k} \Phi ( \lambda ) $
of elementary vibrations of the form

$$ 
Ae ^ {i ( \lambda t + \theta ) }  = \ 
e ^ {i \lambda t }
\Delta _ {k} \Phi ( \lambda )
$$

of frequencies  $  \lambda $, 
$  \lambda _ {k} \leq  \lambda \leq  \lambda _ {k+} 1 $, 
the superposition of which yields, in the limit,  $  X = X( t) $. 
The (mean-square) limit in the representation (*) is taken along a sequence of successively-finer subdivisions of the line  $  - \infty < \lambda < \infty $
into intervals  $  \Delta _ {k} = ( \lambda _ {k,\ } \lambda _ {k+} 1 ) $
with  $  \max _ {k} ( \lambda _ {k + 1 }  - \lambda _ {k} ) \rightarrow 0 $. 
It is usually assumed that

$$ 
F ( \Delta _ {1} \times \Delta _ {2} )  = \ 
{\mathsf E} ( \Delta _ {1} \Phi \cdot \Delta _ {2} \overline \Phi \; ) ,
$$

as a function of the sets  $  \Delta _ {1} \times \Delta _ {2} $
in the plane, defines a complex measure of bounded variation; in this case the corresponding process  $  \Phi ( \lambda ) $, 
$  - \infty < \lambda < \infty $(
or, more exactly, the corresponding random measure  $  d \Phi ( \lambda ) $), 
is unambiguously defined by the process  $  X( t) $, 
$  - \infty < t < \infty $, 
itself:

$$ 
\Delta \Phi ( \lambda )  = \ 
\lim\limits _ {T \rightarrow \infty } \ 
{
\frac{1}{2T }
 }
\int\limits _ { - } T ^ { T } 

\frac{e ^ {- i \lambda _ {2} t } - e ^ {- i \lambda _ {1} t } }{- it }

X ( t)  dt
$$

for any interval  $  \Delta = ( \lambda _ {1} , \lambda _ {2} ) $
such that  $  d \Phi ( \lambda _ {1} ) = d \Phi ( \lambda _ {2} ) = 0 $, 
and

$$ 
\Phi ( \lambda )  = \ 
\lim\limits _ {T \rightarrow \infty } \ 
\int\limits _ { - } T ^ { T } 
e ^ {- i \lambda t }
X ( t)  dt
$$

for any point  $  \lambda $, 
$  - \infty < \lambda < \infty $. 
A random process  $  X( t) $, 
$  - \infty < t < \infty $, 
is harmonizable if and only if its covariance is representable in the form

$$ 
B ( s, t)  = \ 
\int\limits _ {- \infty } ^  \infty  
\int\limits _ {- \infty } ^  \infty  
e ^ {i ( \lambda s - \mu t) }
F ( d \lambda \times d \mu ).
$$

===Examples of harmonizable random processes.===

1) A modulated stationary random process. If

$$ 
X _ {0} ( t)  =  \int\limits _ {- \infty } ^  \infty  
e ^ {i \lambda t }  d \Phi _ {0} ( t)
$$

is a stationary random process, a process of the form

$$ 
X ( t)  =  c ( t) X _ {0} ( t),
$$

where  $  c( t) = \int _ {- \infty }   ^  \infty  e ^ {i \lambda t } m ( d \lambda ) $, 
where  $  m ( d \lambda ) $
is a measure on the line, is usually no longer stationary, but will be harmonizable:

$$ 
X ( t)  =  \int\limits _ {- \infty } ^  \infty  
e ^ {i \lambda t }  d \Phi ( \lambda ),
$$

where the random measure  $  d \Phi ( \lambda ) $
is defined by the formula

$$ 
\Delta \Phi ( \lambda )  = \ 
\int\limits _  \Delta 
m ( \Delta - \lambda ) \ 
d \Phi _ {0} ( \lambda ).
$$

2) A process defined by sliding summation (or moving averages)

$$ 
X ( t)  =  \int\limits _ {- \infty } ^  \infty  
c ( t - s)  dZ ( s),
$$

where  $  d Z( t) $
is some random measure on the line and the weight function  $  c( t) $
is of the same type as above:

$$ 
c ( t)  =  \int\limits _ {- \infty } ^  \infty  
e ^ {i \lambda t }
m ( d \lambda ) .
$$

In this case

$$ 
X ( t)  =  \int\limits _ {- \infty } ^  \infty  
e ^ {i \lambda t }  d \Phi ( \lambda ),
$$

where

$$ 
\Delta \Phi ( \lambda )  = \ 
\int\limits _  \Delta  \left [
\int\limits _ {- \infty } ^  \infty  
e ^ {- i \lambda t } \ 
dZ ( t) \right ]
m ( d \lambda ).
$$

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  M. Loève,   "Probability theory" , '''2''' , Springer  (1978)</TD></TR></table>