4.1. PART 1: PRELIMINARY MEASURED DATA ANALYSIS 69
4.1.6 CROSS-SPECTRUM OR CROSS-POWER SPECTRAL DENSITY
FUNCTION
Closely related to the autospectrum is the cross-spectrum, which will be featured in Part 2 of
this chapter. In a manner similar to Equation (4.13), the digital cross-spectrum is defined as [1]
G
yx
.f / D
2
.n
d
T /
n
d
X
iD1
Y
i
.f / X
i
.f /: (4.14)
Guidelines and practices for estimation of a digital cross-spectrum and autospectrum are
identical.
4.1.7 THE SPECTROGRAM FUNCTION
An extremely informative application of the autospectrum, namely the spectrogram, is defined
as a “running” autospectrum, which is a function of individual or “grouped” autospectra that are
computed at successive time segments of a data record. e value of spectrograms and all of the
above described preliminary measured data analysis functions is demonstrated in the following
discussion.
A series of illustrative examples are provided to demonstrate application of preliminary
data analysis fundamentals in various situations. Overall classification of individual time history
records is effected through calculation and display of the time history, spectrogram, autospec-
trum and probability density function. Data quality and content is evaluated by closer examina-
tion of the autospectrum (including the spectrogram), probability density and total probability,
and response spectrum.
4.1.8 ILLUSTRATIVE EXAMPLE: SINUSOIDAL TIME HISTORY WITH
BACKGROUND RANDOM NOISE
Consider a measured time history record consisting of a 20 Hz sinusoidal signal, contaminated
by broadband random noise, and sampled at dt D :005 s over a duration of 50 s. e composite
preliminary data analysis display shown in Figure 4.1 includes a color spectrogram (upper left),
time history trace (lower left), autospectrum (upper right), and probability density (lower right).
e dashed curve in the probability density plot indicates the ideal Gaussian probability density
function. In addition, the title indicates the window length .N
W
/, frequency bandwidth (f ),
and number of distinct averages (N
av
) of autospectrum sub-records, as well as the mean and
standard deviation (Std) values for the data record.
Based on general uniformity of the spectrogram and time history plots with respect to
time, the data record is judged stationary. Both the spectrogram and autospectrum indicate that
the record is dominated by a 20 Hz sinusoidal signal. e record, in spite of the presence of
broad band noise, may be classified as predominantly deterministic due to general consistency
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