Page 458 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

Sec. 13.7 Fourier Transforms 445

we obtain

—fTico^ + i COc k)Y(^ CO) — X(^co^

where Xico) and y(io) are the FT of x{t) and y{t), respectively.
Parseval’s theorem. ParsevaTs theorem is a useful tool for converting
time integration into frequency integration. If X^(f) and X 2i f ) are Fourier
transforms of real time functions x^(t) and ^ 2(0 , respectively, ParsevaPs theorem
states that
r x,(t)x2(t)dt = r x , ( f ) x n f ) d f
— dc — 00
(13.7-7)
= f x n f ) x , { f ) d f
— oc
This relationship may be proved using the Fourier transform as follows:
X\{t)x2{t) =Xj {t ) ( X^{f)e'-^"df
^ ^ 00
/ X^(t)x2{t) dt = j X.(1)J X,(f)e'-^>'dfdt

r X, {f) [X,(t)e'^^>'dt df

f X , ( f ) X * { f df
)
All the previous formulas for the mean square value, autocorrelation, and cross
correlation can now be expressed in terms of the Fourier transform by ParsevaPs
theorem.
Example 13.7-4
Express the mean square value in terms of the Fourier transform. Letting x,(i) =
JC2(0 = xU), and averaging over F, which is allowed to go to 00, we obtain

f
A'“ = lim x^{t) dt ^ f lim ^X( f ) X*( ) df

Ioc ^ ■^—7 /2 —X/^ ^ ^

Comparing this with Eq. (13.6-6), we obtain the relationship
S U \) = lim j X { f ) X * { f ) (13.7-8)
/ - . X i
where S(f^) is the spectral density function over positive and negative frequencies.
Example 13.7-5
Express the autocorrelation in terms of the Fourier transform. We begin with the
Fourier transform of x{t + r):
x(t + r) = f