ABSTRACT
Covariance functions of wide-sense stationary stochastic processes can be repre sented by their spectral densities. These spectral representations of covariance functions have proved a useful analytic tool in many technical and physical ap plications. The mathematical treatment of spectral representations and the application of the results, particularly in electrotechnics and electronics, is facilitated by introducing the concept of a complex stochastic process. i e R } is a complex stochastic process if X(t) is given by
E rg o d ic ity If the process {A(t)> t e R} is strictly stationary, then one anticipa tes that, for any of its sample paths *(/) = y(t) + iz(t), its constant trend function can be obtained from
(8.3)
This representation of the trend as an improper integral uses the full information which is contained in one sample path of the process. On the other hand, if N sam ple paths of the process *2(0 > — > *jv(0 3X6 each only scanned at time point *0 and if these values are obtained independently of each other, then m = E(X(to)) can be obtained from
(8.4)
The equivalence of formulas (8.3) and (8.4) allows a simple physical interpreta tion: the mean of a stochastic process at a given time point is equal to its mean over the whole observation period. With respect to their practical application, this is the most important property of the ergodic stochastic processes. Besides the re presentation (8.2), for any sample path x = x(t), the covariance function of an er godic process can be obtained from
(8.5)
The exact definition of ergodic stochastic processes cannot be given here. In the technical literature, the ergodicity of stationary processes is frequently simply defi ned by properties (8.3) and (8.5). The application of formula (8.5) is usefol if the sample path of an ongoing stochastic process is being recorded continuously. The estimated value of C(t) becomes the better the larger the time span of observation ί-τ,+τ\. Assumption This chapter deals only with wide-sense stationary processes. Hence, the at tribute "wide-sense" is generally omitted. Moreover, without of loss of generality, the trend function of all processes considered is identically zero. In view of this assumption, representation (8.2) of the covariance function simpli fies to
Solving for sin* and cos* yields
(8.7)
In this section the general structure of stationary stochastic processes with discrete spectra is developed. Next the simple stochastic process {Ar(/)> f ^ 0} with
(8.9)
are necessary. Moreover, because of (8.5), the function
where o(f) is a real function. Substituting (8.11) into (8.10) shows that the differ ence ω(ί + r) - ω(/) must not depend on t. Thus, if ω(ή is assumed to be differen tiable, then ω(0 satisfies equation d [ω(ί+ τ) - ω(ί)]/<Λ = 0, or, equivalently,
Hence, ω(ί) = ω f + φ, where ω and φ are constants. (Note that for proving this re sult it is only necessary to assume the continuity of ω(0·) Thus,
If in (8.9) the random variable X is multiplied by |a |e '^ and \a\e^X is again de noted as X, then the desired result assumes the following form:
The real part { Y(t), tZ 0} of the stochastic process (Yff), i e R } given by (8.12) describes a cosine oscillation with random amplitude and phase. Its sample paths are, therefore, given by
(8.14) where
Generalizing equation (8.13) leads to
(8.15)
(8.16)
where
In particular, (8.17)
The oscillation X(t) given by (8.15) is an additive superposition of n harmonic oscillations. According to (8.17), its mean power is equal to the sum of die mean powers of these n harmonic oscillations. Now let Χγ,Χ2,... be a countably infinite sequence of uncorrelated complex ran dom variables with E(X^) = 0; k= 1 ,2, . . . ; and
(8.18)
Under these assumptions, the equation
(8.19)
defines a stationary process {ΛΊ(ί), / e R} with covariance function
The sets {ω|,α>2, ..., (on } and {oj, g>2, ...} are said to be the spectra of the stochas tic processes {X(t), i e R ) defined by (8.15) and (8.19), respectively. If all ω* are sufficiently close to a single value ω, then {Υ(ί), i e R } is called a narrow-band process (Figure 8.1), otherwise it is called a wide-bandprocess (Figure 8.2). The covariance function of stationary processes with discrete spectrum does not tend to zero as |τ| -> oo. However, it can be shown that, with respect to convergence in mean-square, any stationary process (Αχ/), i> 0} can be sufficiently closely ap proximated by a stationary process of structure (8.15) in any finite time interval
Using property (7.60) of the delta function δ(ί), the covariance function (8.20) can be written in the form
Hence, symbolically,
where (8.22)
Real stationary processes In contrast to a stochastic process with structure (8.12), a stationary process {AXD, i e R } with structure (8.13), i.e.
