Lecture
In statistical radio engineering and physics, when studying deterministic signals and random processes, their spectral representation in the form of the spectral density, which is based on the Fourier transform, is widely used.
If the process 
has finite energy and is square integrable (and this is a nonstationary process), then for a single realization of the process one can define the Fourier transform as a random complex function of frequency:
 |
(1) |
However, it turns out to be almost useless for describing the ensemble. A way out of this situation is to discard some of the spectrum parameters, namely the phase spectrum, and to construct a function characterizing the distribution of the process's energy over the frequency axis. Then, according to Parseval's theorem, the energy
 |
(2) |
The function 
thus characterizes the distribution of the realization's energy over the frequency axis and is called the spectral density of the realization. By averaging this function over all realizations one can obtain the spectral density of the process.
Let us now turn to a wide-sense stationary centered random process {\displaystyle x(t)}, whose realizations, with probability 1, have infinite energy and, consequently, have no Fourier transform. The power spectral density of such a process can be found, on the basis of the Wiener-Khinchin theorem, as the Fourier transform of the correlation function:
 |
(3) |
If the direct transform exists, then the inverse Fourier transform also exists, which from the known 
determines 
:
 |
(4) |
If we set in formulas (3) and (4), respectively, 
and 
, we have
 |
(5) |
 |
(6) |
Formula (6), together with (2), shows that the variance determines the total energy of the stationary random process, which is equal to the area under the spectral density curve. The dimensional quantity 
can be interpreted as the fraction of energy concentrated in a small frequency interval from {\displaystyle f-df/2}
to 
. If {\displaystyle x(t)} is understood as a random (fluctuating) current or voltage, then the quantity {\displaystyle S_{x}(f)} will have the dimension of energy [V2/Hz] = [V2s]. For this reason is sometimes called the energy spectrum. In the literature one often encounters another interpretation: 
– is regarded as the average power dissipated by the current or voltage across a 1-Ohm resistance. In this case the quantity is called the power spectrum of the random process.
 . |
(7) |
 . |
(8) |
and the energy spectrum of a wide-sense stationary random process possess all the properties characteristic of a pair of mutual Fourier transforms. In particular, the «wider» the spectrum the «narrower» the correlation function , and vice versa. This result is expressed quantitatively in the form of an uncertainty principle or relation.
Suppose that 
is a moving-average process of uncorrelated random variables
where 
For convergence in the mean of series (7) it is necessary and sufficient that
(corollary 7.6.1). The process 
has spectral density equal to 
and the spectral density of the process 
is
The process 
is called a moving-average process. The covariance function for 
is the quantity
since 
if 
and equal to 0 otherwise.
Conversely, if a stationary process 
has spectral density 
it can be represented in the form (7). The square root of 
can be represented as follows:
where 
Since 
is an even function, 
are real. [Note that in the case when 
does not hold, 
defined by (7), is a complex quantity.] There exists a sequence of uncorrelated random variables 
such that 
can be represented as 
Let us define 
using the spectral representation of the process 
Suppose 
Let 
Then
since
because of 
Thus, from formula (12) we obtain the required equality. For a more detailed exposition, see Doob's book (1953, Ch. X, Sect. 8).
If 
has spectral density 
almost everywhere on 
and
then there exist (real) constants 
and a sequence of random variables 
such that
The sum (16) is, in general, infinite (see Sect. 7.6.3).
Now consider a moving-average process with finite limits of summation
where 
Then
where 
are the roots of the equation 
If 
then all 
roots are nonzero. For finite 
we shall use 
as the standard form in 
As was shown in Sect. 5.7.1, any process with a finite number of nonzero covariances has the same sequence of covariances as a suitably chosen finite moving-average process. Then the spectral density can be written in the form (18). If 
Since 
varies monotonically from —1 to 1 on the interval 
and from 1 to —1 on the interval 
the function 
increases from 
to 
on the interval 
and decreases to 
on the interval 
for 
whereas if 
then 
decreases on the interval 
and increases on 
Thus, if 
the lower frequencies have the greater density, while if 
the upper frequencies do. Since 
the spectral
density can be written as follows:
The latter form corresponds to a process 
has variance 
The covariance functions of this process and of the process defined by formula (20) coincide. If 
, then the latter moving-average process differs from the former one. [The spectral density corresponding to 
can be written in the form 
where 
has variance 
.]
If
If 
then 
is a maximum of the function 
a minimum; if 
then 
is a minimum, 
a maximum. If 
then 
for values of X on the interval 
and at the point 
if 
(implying that 
and the roots of the corresponding polynomial are complex), then 
is a minimum, and 
a relative maximum; if 
(implying that 
and the roots are real), then 
is a maximum, and 
a relative minimum. If 
are the roots of the equation
then
Since 1, the factor 
can be replaced by 
where 
is the conjugate of 
can be replaced by 
Thus, for the function 
any of the following expressions holds:
If 
are real, then each of the expressions given for 
corresponds to the spectral density of a moving-average process. The four moving-average processes are distinct if 
three moving-average processes are distinct if 
two moving-average processes are distinct if 
or if 
finally, there is only one moving-average process if 
The spectral density is the product of two densities of the indicated type for 
If 
are complex conjugates, say 
then 
are not real if 
and the first two expressions of the function 
in (27) cannot 
correspond to a moving-average process with real coefficients. Two moving-average processes with real coefficients are distinct. All expressions for 
and the moving-average processes coincide if 
(i.e. 
When the roots are complex conjugates,
If y is close to 1, then the minimum value of 
is attained for values of X close to ±0. Indeed, the minimum of function (28) is attained at 
if the latter expression is less than 1 in absolute value.
For arbitrary 
the spectral density is a product analogous to formulas (21) and (28). Let 
(If 
then 
For some 
Then
If 
is close to 1 (i.e. if 
lies close to the unit circle in the complex plane), then
will be close to 0. Thus, frequencies near 
will have low intensity.
In the general case, the factor 
as shown in formula (18), can be rewritten as follows:
where 
is the complex conjugate of 
If all the roots are real, distinct, and different from ±1, then there exist 
distinct representations of the function 
corresponding to different moving-average processes. The number of distinct moving-average processes in the general case depends on the number of roots whose absolute values equal 1, as well as on the multiplicity of the various roots and the number of complex conjugate roots. We shall not enumerate all the possibilities for the case 
It will be convenient for us to represent the moving-average process in such a form that no root of formula (19) is greater than unity in absolute value. (Note that a root whose absolute value is 1 is permitted for a moving-average process.)
The moving-average process (17) can be written in the form
where the operators and 
are defined so that 
If the roots in (19) are less than 1 in absolute value, then from (32) it follows that
If
then (33) can be rewritten as
or
Thus,
is the best forecast of the quantities 
in terms of the values 
in the sense that the mean-square error is minimized.
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