Figure 16-.. Discrete .ourier transform representation
n 1
G(ωn)imaginary二 F(t川)sin(ωnt川)(16-1 )n二
n 1
F(t川)二骂ω (G(ωn)real cos(ωnt川)+G(ωn)imaginary sin(ωnt川)) (16-11)n二
.omparison of the previous equations with Equations (16-6) and (16-7) reveal that the Fourier transform is really .ust a Fourier series constructed over a finite interval.
The equations may be rewritten in a simpler form by making the following definitions:且F川二F(t川)(16-1却) Gn二G(ωn)(16-13)
ωn二n骂 ω二却πnK (16-14)
NT tK二K骂 t (16-15) so that Equations (16-6) and (16-7) become
n 1 且 二骂 ω 忡(却πJ川N)(n川)(16-16)
F川Gn
n二
K 1
Gn二 却πTK K二F且川忡( 却πJ川N)(n川)(16-17)
If we further define
F川二 T却πK且F川 and (16-18)
w二忡 却πJ川N
we have
Gn二 K 1川二 Fn川 (16-19)
or in matrix form
[Gnl二[w(n川)l[F川ln二， 1，却， ………， N 川二， 1，却， ………， K 11 (16-却 )
The matrices [G町and [ F町 are column matrices with row numbers n and川， respectively. The matrix solution is simplified by special properties of the symmetric matrix and because the resulting values of Gn occur in complexcon.ugate pairs. In general， we may write
Gn二号n +Jbn二IGnI忡Jwn (16-却1)
where:
IGnI二号却 n +b却 n (16-却却)
wn二tan 1(bn川号n)(16-却3)
From the time function F(t) and the calculation of [w町， the values of Gn may be found. One way to calculate the G matrix is by a fast Fourier technique called the .ooley-Tukey method. It is based on an expression of the matrix as a product of Qsquare matrices， where Q is again related to N by N二却Q. For large N， the number of matrix operations is greatly reduced bythis procedure. In recentyears， more advanced high-speed processors have been developed to carry out the fast Fourier transform. The calculation method is basically the same for both the discrete Fourier transform and the fast Fourier transform. The difference in the two methods lies in the use of certain relationships to minimize calculation time prior to performing a discrete Fourier transform.
Finding the values of Gn allows the determination of the frequency-domain spectrum. The power-spectrumfunction， which may be closely approximated by a constant times the square of G( f)， is used to determine the amount of power in each frequency spectrum component. The function that results is a positive real quantity and has units of volts squared. Fromthe powerspectra， broadband noise may be attenuated so that primary spectral components may be identified. This attenuation is done by a digital process of ensembleaveraging， which is a point-by-point average of a squared-spectra set.
Vibration Measurement
Successful measurement of machine vibration requires more than a trans-ducer randomlyselected， installed， and a piece of wire to carry the signal to the analyzer.认 hen the decision tomonitor vibration ismade， threechoices of measurement are available: (1) displacement，(却)velocity， and (3) acceleration. These three measurement types emphasize different partsof the spectrum. To understand this peculiarity， it is necessary to consider the differences in the characteristics of each. .onsider a simple harmonicvibration. The displacement， .， is given by
.二 . sin ωt
Successive differentiation gives the expressions for velocity (..) and accel-eration (.
)
.二 . sin ωt
..二 .ω cos ωt
.
二 .ω却 sin ωt
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