SOME BASIC STATISTICAL CONCEPTS IN WIND ENGINEERING - Theory of Bridge Aerodynamics - page 42

This shows that the auto spectral density is the Fourier transform of the auto
covariance function. Vice versa, it follows that the auto covariance function, which is the
Fourier constant to the spectral density, is given by

+∞

i

() ( )

ωτ

∫

Cov

τ =

S

ω ⋅

e d

ω (2.84)

x x

−∞

Similarly, the cross covariance function together with the cross spectral density will also
constitute a pair of Fourier transforms:

+∞ +∞

1
2

−

i

ωτ i ωτ

() () () ( )

S

∫

Cov e d Cov

∫

S e d ω = τ ⋅ τ and τ = ω ⋅ ω (2.85)

xy xy

xy xy

π

−∞ −∞

2.8 Coherence function and normalized co-spectrum

The coherence function is defined by

2

()
()

S ω

S

xy

()

Coh

ω = (2.86)

xy

()

ωω ⋅

x y

() () () ()

x t and y t are realisations of the same process, then S S

If ω = ω and the

x y

() ()

S S

cross-spectrum ω = ω is given by

xy xx

()

i xx

() () ()

e ϕ

ω

S S Coh

ω = ω ⋅ ω ⋅

(2.87)

xx x xx

()

Coh

ϕ is the phase spectrum (see
Eq. 2.79) . In the practical use of cross-spectra all imaginary parts will cancel out, and
thus it is only the co-spectrum that is of interest. Therefore, a normalised co-spectrum is
defined

ω is called the root-coherence function and

xx xx

()
()

Re

⎡

S

ω ⎤

⎣ ⎦

ˆ

xy

()

Co

ω = (2.88)

xy

()

SS

ωω ⋅

x y