Page 459 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

446 Random Vibrations Chap. 13

Substituting this into the expression for the autoeorrelation, we obtain
1
R ( t ) = lim y / x ( t ) x ( t + t ) d t
T-^oo
= lim y r x { t ) r d f d t

= r lim ^ f x { t ) e ‘ ^ ^ f ' d t X{f)e'^^f^df
J_^T^oo ^ —00

= r [ lim ].X*if )X(f) d f
ooL T ’ —» 0 0 ^
By substituting from Eq. (13.7-8) the preceding equation becomes

R(r) = r S{f)e'^^f^df (13.7-9)

—00
The inverse of the preceding equation is also available from the Fourier transform:
/ oo dr (13.7-10)
- 00
Because R{r) is symmetric about r = 0, the last equation can also be written as
.oc
S(f) = 2l R(r)cos27rfrdr (13.7-11)
•'o
These are the Wiener-Khintchine equations, and they state that the spectral density
function is the FT of the autocorrelation function.
As a parallel to the Wiener-Khintchine equations, we can define the cross
correlation between two quantities jc(r) and y{t) as
Rxyi'^) ={x{t)y{t + t)> = lim y x{t)y{t + t) dt
T^cc ‘ J - t/2
= r lim jX*(f )Y(f )e jl n f ' ^df (13.7-12)

»
—00 7" —00 ^

l^xyir) = r S,^{f)e‘^-f^df
•' —ao
where the cross-spectral density is defined as
1
5 , / / ) = lim y 2 T * (/)y (/) - o o < /<
»
T’ —00 ■*

1
= lim j X ( f ) Y * i f )
T —> oo ^
= 5,%(/) = 5 , , ( - / ) (13.7-13)
Its inverse from the Fourier transform is
.o o
Sxy(f) = /_ Rxy{T)e-^-f^ d r (13.7-14)