3.2 Proofs for Section 2.2.3 “Second-Order Spectral Characterization”

3.2.1 The μ Functional

In this section, the μ functional is introduced. It is defined as the functional whose value is the infinite-time average of the test function. The μ functional is formally characterized as the limit of approximating functions similar to the characterization of the Dirac delta as the limit of delta-approximating functions (Zemanian 1987, Section 1.3).

Let μ be the functional that associates to a test function ϕ its infinite-time average value. That is,

(3.16)

provided that the limit exists (and, hence, is independent of t).

In the following, the μ functional is heuristically characterized through formal manipulations. Let it be

(3.17)

where rect(t) = 1 for |t| ≤ 1/2 and rect(t) = 0 otherwise. For any finite T one has

(3.18)

Thus, in the limit as T→ ∞, we (rigorously) have

(3.19)

and we can formally write

(3.20)

with rhs independent of t, where μ(t) is formally defined as (see also (Silverman 1957))

(3.21)

In the sense of the ordinary functions, the limit in the rhs of (3.21) is the identically zero function. However, observing that for any finite T it results

(3.22)

then, in the limit as T→ ∞, we (rigorously) have

(3.23)

and we can formally write

(3.24)

That is, μ(t) can be interpreted as the limit of a very tiny and large rectangular window with unit area. This limit, of course, is the identically zero function in spaces of ordinary functions. However μ(t) can be formally managed as an ordinary function satisfying (3.20) and (3.24) similar to the formal manipulations of the Dirac delta.

The Fourier transform of μ(t) can be formally derived by the following “limit passage.” Accounting for the Fourier transform pair

(3.25)

we can formally write (Figure 3.1)

(3.26)

where δ_f is the Kronecker delta, that is, δ_f = 1 for f = 0 and δ_f = 0 for f ≠ 0. Thus, μ(t) cannot be expressed as ordinary inverse Fourier transform (Lebesgue integral) since the Kronecker delta δ_f is zero a.e.

Figure 3.1 (Top) Function μ_T(t) and (bottom) its Fourier transform for increasing values of T (form thin line to thick line)

The following properties of the μ functional can be formally proved.

1. For every t₀ 0, from (3.20) it follows that

(3.27)

where the fact that t₀ is finite has been accounted for. The case t₀ < 0 is similar.

2. If x(t) contains the finite-strength additive sinewave component x_αe^j2παt (Section 6.3.1), then

(3.28)

3. From (3.26) and using the modulation theorem for Fourier transform we have

(3.29)

4. From (3.29) and using the duality theorem for Fourier transform we have

(3.30)

3.2.2 Proof of Theorem 2.2.22 Loève Bifrequency Spectrum of GACS Processes

From Theorem 2.2.7 (t + τ = t₁ and t = t₂ in (2.18)) we have

(3.31)

Thus, we formally have

(3.32)

where, in the third equality, the variable change t₁ = t + τ and t₂ = t is made and in the fourth equality (2.71) and (2.72) are used. Then, (2.70) immediately follows.