SOME BASIC STATISTICAL CONCEPTS IN WIND ENGINEERING - Theory of Bridge Aerodynamics - page 39

In a continuous format, i.e. in the limit of both N and T approaching infinity, the double-
sided cross-spectral density is defined by

*

( ) ( )

E Xt

⎡

, Yt ,

⎤

ω ⋅ ω

⎣ ⎦

( )

S lim lim

±= ω

xy

Δ ω

TN

→∞ →∞

(2.73)

1

*

() ()

lim lim a a

= ωω ⋅

X Y

2 T

π

TN

→∞ →∞

The single sided version is then simply

1

() ( )

*

() ()

S

ω =⋅

2 S

± = ω

lim lim a

ω ⋅

a

ω (2.74)

xy xy X Y

π

T

TN →∞ →∞

while the corresponding single-sided version using frequency f ( Hz ), is defined by

2

() ( )

*

() ()

S f

=⋅

2

π

S

ω →∞ =

lim lim

⋅

afaf

⋅ (2.75)

xy xy x y

T

TN →∞

Thus, the covariance between the two processes may be calculated from

+∞ ∞ ∞

( ) ( ) ( )

Cov

∫

S d

∫

S d

∫

S f df

= ± ωω = ωω = (2.76)

xy xy xy xy

0 0

−∞

The cross-spectrum will in general be a complex quantity. With respect to the frequency
argument, its real part is an even function labelled the co-spectral density

()

Co

ω ,

xy

()

Qu

while its imaginary part is an odd function labelled the quad-spectrum ω , i.e.

xy

() () ()

S Co i Qu ω = ω − ⋅ ω

(2.78)

xy xy xy

()

S

as illustrated in Fig. 2.12. Alternatively, ω may be expressed by its modulus and

xy

phase, i.e.

()

i

e ϕ

⋅ ω

() ()

xy

S

ω =

S

ω ⋅ (2.79)

xy xy

() () ()

arc tan

⎡

Qu Co

⎤

where the phase spectrum ϕ ω = ω ω

⎦ .

⎣

xy xy xy