Page 460 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

Sec. 13.7 Fourier Transforms 447

which is the parallel to Eq. (13.7-10). Unlike the autocorrelation, the cross-correlation
and the cross-spectral density functions are, in general, not even functions; hence, the
limits -00 to -hoc are retained.
Example 13.7-6
Using the relationship

S(f) = 2Î R ( t ) cos 27rfr dr
and the results of Example 13.5-1,
R ( t ) = A ^ { T - t )
find Sif) for the rectangular pulse.
Solution: Because Rir) = 0 for r outside ± 7, we have
CT
S(f) = 2Î A^(T - r)cos27rfrdr
•'()
= 2 A ^ T c o s 2 ttf 7 d r — 2 A ^ j T C O s 2 T r f r d T

= 2A^T- - 2A‘ cos 27t/ t ---r sin 27t/'t
2 v f 277/
(2'n-/)
2A^
r ( l - 0 0 8 2 7 7 / 7 ) =
{27TfY
Thus, the power spectral density of a rectangular pulse using Eq. (13.7-11) is

\
Note from Example 13.7-3 that this is also equal to Xif)X*(f) = X(f )\^.
Example 13.7-7
Show that the frequency response function Hioj) is the Fourier transform of the
impulse response function hit).
Solution: From the convolution integral, Eq. (4.2-1), the response equation in terms of the
impulse response function is

x ( 0 - r f i O K t - O d ^
d — ro
where the lower limit has been extended to - oo to account for all past excitations. By
letting T = (i - ^), the last integral becomes

^ ( 0 = f f { t - r ) h { r ) d T
•'n