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10.5 Analysis of Random Signals

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Random (stochastic) signals are generated by some random mechanism. Each time the mechanism is

operated a new time history (sample function) is generated. The chance of any two sample functions

becoming identical is governed by some probabilistic law. If all sample functions are identical (with unity

probability), then the corresponding signal is a deterministic signal. A randomprocess is denoted by X~ ðtÞ;

while any sample function of it is denoted by xðtÞ: No numerical computations can be performed on X~ ðtÞ

because it is not known for certain. Its Fourier transform, for instance, can be written down as an

analytical expression, but cannot be numerically computed. However, once the signal is generated,

numerical computations can be performed on that sample function xðtÞ because it is a completely known

function of time.

10.5.1 Ergodic Random Signals

At any given time t1; X~ ðt1Þ is a random variable which has a certain probability distribution. Consider a

well-behaved function f {X~ ðtÞ} of this random variable (which is also a random variable). Its expected

value (statistical mean) is E½ f {X~ ðtÞ}􀀉: This is also known as the ensemble average because it is equivalent

to the average value at t of a collection (ensemble) of a large number of sample functions xðtÞ:

Consider the function f {xðtÞ} of one sample function xðtÞ: Its temporal (time) mean is expressed by

lim

T!1

ðT

f {xðtÞ}dt

Now, if

E½ f {X~ ðt1Þ}􀀉 ¼ lim

T!1

ðT

f {xðtÞ}dt ð10:29Þ

10-18 Vibration and Shock Handbook

then the random signal is said to be ergodic. It should be noted that the right-hand side of Equation 10.29

does not depend on time. Consequently, the left-hand side should also be independent of the time

point t1:

For analytical convenience, random vibration signals are usually assumed to be ergodic (the ergodic

hypothesis). Using this hypothesis, the properties of a random signal could be determined by performing

computations on a sufficiently long record (sample function) of the signal. Since the ergodic hypothesis is

not exactly satisfied for vibration signals, and since it is impossible to analyze infinitely long data records,

the accuracy of the numerical results depends on various factors such as the record length, sampling rate,

frequency range of interest, and the statistical nature of the random signal (e.g., closeness to a

deterministic signal, frequency content, periodicity, damping characteristics). Accuracy can be improved

in general, by averaging the results for more than one data record.

10.5.2 Correlation and Spectral Density

If for a random signal X~ ðtÞ; the joint statistical properties of X~ ðt1Þ and X~ ðt2Þ depend on the time difference

ðt2 2 t1Þ and not on t1 itself, then the signal is said to be stationary. Consequently, the statistical

properties of a stationary X~ ðtÞ will be independent of t: It is noted from Equation 10.29 that ergodic

random signals are necessarily stationary. However, in general the converse is not true.

The cross-correlation function of two random signals X~ ðtÞ and Y~ðtÞ is given by E½ X~ ðtÞ Y~ðt þtÞ􀀉: If the

signals are stationary, this expected value is a function of t (not t) and is denoted by fxy ðtÞ: In view of the

ergodic hypothesis, the cross-correlation function may be expressed as

fxy ðtÞ ¼ lim

T!1

ðT

xðtÞyðt þ tÞdt

􀀒 􀀓

ð10:30Þ

The FIT of fxy ðtÞ is the cross-spectral density function which is denoted by Fxy ð f Þ: When the two signals

are identical, we have the autocorrelation function fxx ðtÞ in the time domain and the power spectral

density (PSD) Fxx ðf Þ in the frequency domain. The continuous and the discrete versions of the

correlation theorem are given in the first row of Table 10.5. It follows that the cross-spectral density may

be estimated using the DFT (FFT) of the two signals, as ½Xn􀀉pYn

􀀋

T in which T is the record length and

½Xn􀀉p is the complex conjugate of ½Xn􀀉:

Parseval’s theorem (second row of Table 10.5) follows directly from the correlation theorem.

Consequently, the mean square value of a random signal may be obtained from the area under the PSD

curve. This suggests a hardware-based method of estimating the PSD as illustrated by the functional

diagram in Figure 10.14(a). Alternatively, a software-based digital Fourier analysis could be used

(Figure 10.14(b)). A single sample function would not give the required accuracy, and averaging is

usually needed. In real-time digital analysis, the running average as well as the current estimate is usually

computed. In the running average, it is desirable to give a higher weighting to the more recent estimates.

The fluctuations in the PSD estimate about the local average could be reduced by selecting a large filter

TABLE 10.5 Some Useful Fourier Transform Results

Description Continuous Discrete

Correlation theorem If zðtÞ ¼

1 xðtÞyðt þ tÞdt zm ¼ DT

PN21

r¼0 xr yrþm

Then Zð f Þ ¼ ½Xð f Þ􀀉p Y ð f Þ Zn ¼ ½Xn 􀀉p Yn

Parseval’s theorem If yðtÞ !FIT

Y ð f Þ {ym } ! DFT

{Yn }

Then

1 y2 ðtÞdt ¼

1 lY ð f Þl2 df DT

PN21

m¼0 y2

m ¼ DF

PN21

n¼0 lYn l2

Convolution theorem If yðtÞ ¼

1 hðtÞuðt 2 tÞdt ¼

1 hðt 2 tÞuðtÞdt ym ¼ DT

PN21

r¼0 hr um2r ¼ DT

PN21

r¼0 hm2r ur

Then Y ð f Þ ¼ Hð f ÞU ð f Þ Yn ¼ Hn Un

Vibration Signal Analysis 10-19

bandwidth Df and a large record length T: A measure of these fluctuations is given by

1 ¼

1 ffiffiffiffiffiffi

Df T

p ð10:31Þ

It should be noted that a large Df results in reduction of the precision of the estimates while improving

the appearance. To offset this, T has to be increased further.

10.5.3 Frequency Response Using Digital Fourier Transform

Vibration test programs usually require a resonance search type pretesting. In order to minimize the

damage potential, it is carried out at a much lower intensity than the main test. The objective of such

exploratory tests is to determine the significant frequency-response functions of the test specimen. These

provide the natural frequencies, damping ratios, and mode shapes of the test specimen. Such frequencyresponse

data are useful in planning and conducting the main test. For example, more attention is

required when testing in the vicinity of the resonance points (slower sweep rates, larger dwell periods,

etc.). Also, the frequency-response data are useful in determining the most desirable test input directions

and intensities. The degree of nonlinearity and time variance of the test object can be determined by

conducting more than one frequency-response test at different input intensities. If the deviation of the

frequency-response function is sufficiently small, then linear, time-invariant analysis is considered to be

satisfactory. Often, frequency-response tests are conducted at full test intensity. In such cases, it is

considered as a part of the main test rather than a prescreening test. Other uses of the frequency-response

function include the following: it can be employed as a system model (experimental model) for further

analysis of the test specimen (experimental modal analysis). Most desirable frequency range and sweep

rates for vibration testing can be estimated by examining frequency-response functions.

The time response hðtÞ to a unit impulse is known as the impulse-response function. For each pair of

input and output locations (A,B) of the test specimen, a corresponding single response function would be

obtained (assuming linearity and time-invariance). Entire collection of these responses would determine

the response of the test specimen to an arbitrary input signal. The response yðtÞ at B to an arbitrary input

uðtÞ applied at A, is given by

yðtÞ ¼

ð1

hðtÞuðt 2 tÞdt ¼

ð1

hðt 2 tÞuðtÞdt ð10:32Þ

The right-hand side of Equation 10.32 is the convolution integral of hðtÞ and uðtÞ and is denoted by

hðtÞ p uðtÞ: By substituting the inverse FIT relations (Table 10.2) in Equation 10.32, the frequencyresponse

function (frequency-transfer function) Hð f Þ is obtained as the ratio of the (complex) FITs of

the output and the input. It exists for physically realizable (casual) systems even when the individual FITs

of the input and output signals do not converge. The continuous convolution theorem and the discrete

Signal Approx. psd

(a)

Tracking

Filter

Bandwidth Δf

Squaring

Hardware

Averaging

Network Δf

Signal Approx. psd

(b)

Analog-to-Digital

Conversion

(ADC)

Digital

Correlation

Function

Digital

Fourier

Transform

Averaging

Software

FIGURE 10.14 Power spectral density computation. (a) Narrow-band filtering method; (b) correlation and Fourier

transformation method.

10-20 Vibration and Shock Handbook

counterpart are given in the last row of Table 10.5. The discrete convolution can be interpreted as the

trapezoidal integration of Equation 10.32. Frequency-response function is a valid representation (model)

for linear, time-invariant systems. It is related to the system transfer function GðsÞ (ratio of the Laplace

transforms of output and input with zero initial conditions) through

Hðf Þ ¼

Y ð f Þ

Uð f Þ ¼ Gð2pfjÞ ð10:33Þ

But for notational convenience, the frequency-response function corresponding to GðsÞ may be denoted

by either Gð f Þ or GðvÞ; where v is the angular frequency and f is the cyclic frequency.

Using Fourier transform theory, three methods of determining Hðf Þ can be established. First, using

any transient excitation signal to a system at rest and the corresponding output, Hð f Þ is determined from

their FITs (Table 10.5). Second, if the input is sinusoidal, the signal amplification of the steady-state

output is the magnitude lHð f Þl at the input frequency, and the phase lead of the steady-state output is the

corresponding phase angle /Hð f Þ: Third, using a random input signal and the corresponding input and

output spectral density functions, Hðf Þ is determined as the ratio

Hð f Þ ¼ Fuy ðf Þ=Fuuð fÞ ð10:34Þ

10.5.4 Leakage (Truncation Error)

In digital processing of vibration signals (e.g., accelerometer signals), sampled data are truncated to

eliminate less significant parts. This is of course essential in real-time processing because, in that case,

only sufficiently short segments of continuously acquisitioned data are processed at a time. The computer

memory (and buffer) limitations, the speed and cost of processing, the frequency range of importance,

the sampling rate, and the nature of the signal (level of randomness, periodicity, decay rate, etc.) should

be taken into consideration in selecting the truncation point of data.

The effect of direct truncation of a signal xðtÞ on its Fourier spectrum is shown in Figure 10.15. In the

time domain, truncation is accomplished by multiplying xðtÞ by the box-car function bðtÞ: This is

equivalent to a convolution ðXð f Þ p Bð f ÞÞ in the frequency domain. This procedure introduces ripples

(side lobes) into the true spectrum. The resulting error ðXðf Þ 2 Xð f Þ p Bð f ÞÞ is known as leakage or

truncation error. Similar leakage effects arise in the time domain, as a result of truncation of the

frequency spectrum. The truncation error may be reduced by suppressing the side lobes, which requires

modification of the truncation function (window) from the box-car shape bðtÞ to a more desirable shape.

Commonly used windows are the Hanning, Hamming, Parzen, and Gaussian windows.

10.5.5 Coherence

Random vibration signals X~ ðtÞ and Y~ðtÞ are said to be statistically independent if their joint probability

distribution is given by the product of the individual distributions. A special case of this is the

uncorrelated signals which satisfy

E½ X~ ðt1Þ Y~ðt2Þ􀀉 ¼ E½ X~ ðt1Þ􀀉E½ Y~ðt2Þ􀀉 ð10:35Þ

In the stationary case, the means mx ¼ E½ X~ ðtÞ􀀉 and my ¼ E½ Y~ðtÞ􀀉 are time independent. The

autocovariance functions are given by

cxxðtÞ ¼ E½{X~ ðtÞ 2mx}{X~ ðt þtÞ 2mx}􀀉 ¼ fxxðtÞ 2m2

x ð10:36Þ

cyyðtÞ ¼ E½{Y~ðtÞ 2my}{Y~ðt þtÞ 2mx}􀀉 ¼ fyyðtÞ 2m2y

ð10:37Þ

and the cross-covariance function is given by

cxyðtÞ ¼ E½{X~ ðtÞ 2mx}{Y~ðt þtÞ 2my}􀀉 ¼ fxyðtÞ 2mxmy ð10:38Þ

Vibration Signal Analysis 10-21

For uncorrelated signals fxy ðtÞ ¼ mxmy and cxy ðtÞ ¼ 0: The correlation function coefficient is defined by

rxy ðtÞ ¼

cxy ðt ffiffiffiffiffiffiffiffiffiÞffiffiffiffiffiffi

cxx ð0Þcyy ð0Þ

p ð10:39Þ

which satisfies 21 # rxy ðtÞ # 1:

For uncorrelated signals we have rxy ðtÞ ¼ 0: This function measures the degree of correlation of the

two signals. In the frequency domain, the correlation is determined by its (ordinary) coherence function

xy ð f Þ ¼

lFxy ð f Þl2

Fxx ð f ÞFyy ð f Þ ð10:40Þ

which satisfies the condition 0 # g2

xy ð f Þ # 1: In this definition, the signals are assumed to have zero

means. Alternatively, the FIT of the covariance functions may be used. If the signals are uncorrelated (or

better, independent) the coherence function vanishes. On the other hand, if Y~ðtÞ is the response of a

linear, time-invariant system to an input X~ ðtÞ; then

Fxy ð f Þ ¼ Fxx ðf ÞHð fÞ ð10:41Þ

Fyy ðf Þ ¼ Fxx ð f ÞlHð f Þl2 ð10:42Þ

Consequently, the coherence function becomes unity for this ideal case. In practice, however, the

coherence function of an excitation and the corresponding response is usually less than unity. This is due

to deviations such as measurement noise, system nonlinearities, and time-variant effects. Consequently,

the coherence function is commonly used as a measure of the accuracy of frequency-response estimates.

TRUE

SIGNAL

(a)

x(t)

0 T t

|X( f )|

0 f

TRUNCATION

WINDOW

(c)

b(t)

0 T t

|B( f )|

1/T f

−1/T

RESULT

(b)

x(t)b(t)

0 T t

|X( f )*B( f )|

0 f

TIME FREQUENCY

FIGURE 10.15 Illustration of truncation error. (a) Signal and its frequency spectrum; (b) a rectangular (box-car)

window and its frequency spectrum; (c) truncated signal and its frequency spectrum.

10-22 Vibration and Shock Handbook

10.5.6 Parseval’s Theorem

For a pair of rapidly decaying (aperiodic) deterministic signals xðtÞ and yðtÞ; the cross-correlation

function is given by

fxy ðtÞ ¼

ð1

xðtÞyðt þ tÞdt ð10:43Þ

This is equivalent to Equation 10.30, for a pair of ergodic, random (stochastic) signals xðtÞ and yðtÞ: By

using the definition of the inverse FIT (see Table 10.2) in Equation 10.43, and by following

straightforward mathematical manipulation, it may be shown that

Fxy ð f Þ ¼ ½Xð f Þ􀀉pY ð fÞ ð10:44Þ

in which the cross-spectral density Fxy ð f Þ is the FIT of fxy ðtÞ; as given by

Fxy ðf Þ ¼

ð1

fxy ðtÞ expð2j2pf tÞdt ð10:45Þ

and [ ]p denotes the complex conjugation operation. This result, which is known as the correlation

theorem (see Table 10.5) has applications in the evaluation of the correlation functions and PSD

functions of finite-record-length data.

The inverse FIT relation corresponding to Equation 10.45 is

fxy ðtÞ ¼

ð1

Fxy ð f Þ expð j2pf tÞdf ð10:46Þ

From Equation 10.44, we have

fxy ðtÞ ¼

ð1

21½Xðf Þ􀀉pY ð f Þexpðj2pf tÞdf ð10:47Þ

If we set t ¼ 0 and x ¼ y in Equation 10.47, we get

fxy ð0Þ ¼

ð1

lY ðf Þl2df ð10:48Þ

Similarly, from Equation 10.43, we get

fxy ð0Þ ¼

ð1

y2ðtÞdt ð10:49Þ

By comparing Equation 10.48 and Equation 10.49, we obtain Parseval’s theorem and thus

ð1

y2ðtÞdt ¼

ð1

lY ð f Þl2df ð10:50Þ

By using the discrete correlation theorem is an analogous manner, we can establish the discrete version of

Equation 10.50:

NX21

m¼0

m ¼ DF

NX21

n¼0

lYnl2 ð10:51Þ

These results are listed in the second row of Table 10.5.

10.5.7 Window Functions

Consider the unit box-car function wðtÞ; defined as

wðtÞ ¼

1 for 2 0 # t , T

0 otherwise

(

ð10:52Þ

Vibration Signal Analysis 10-23

This is shown in Figure 10.15(b). The FIT of wðtÞ is

W ðf Þ ¼

j2pf ½1 2 cos 2p f T þ j sin 2p f T􀀉 ð10:53Þ

Clearly, this (rectangular) window function produces side lobes (leakage) in the frequency domain.

In spectral analysis of vibration signals, it is often required to segment the time history into several parts,

and then perform spectral analysis on the individual results to observe the time development of

the spectrum. If segmenting is done by simple

truncation (multiplication by the box-car window),

the process would introduce rapidly fluctuating

side lobes into spectral results. Window

functions, or smoothing functions other than the

box-car function, are widely used to suppress the

side lobes (leakage error). Some common smoothing

functions are defined in Table 10.6.

A graphical comparison of these four window

types is given in Figure 10.16. Hanning windows are

very popular in practical applications. A related

window is the Hamming window, which is simply a

Hanning window with rectangular cut-offs at the

two ends. A Hamming window will have characteristics

similar to those of a Hanning window,

except that the side lobe fall-off rate at higher

frequencies is less in the Hamming window.

From Figure 10.16(b), we observe that the

frequency-domain weight of each window varies

with the frequency range of interest. Obviously,

the box-car window is the worst. In practical

applications, the performance of any window

could be improved by simply increasing the

window length T. In addition, a window in the

shape of a Gaussian function may be used when a

rapid roll-off without side lobes is desired.

(a)

w(t)

T t

Time

Box-Car

Hanning

Bartlett

Parzen

(b)

W( f )

Box-Car

Hanning & Bartlett

Parzen

Frequency

FIGURE 10.16 Some common window functions.

(a) Time-domain function; (b) frequency spectrum.

TABLE 10.6 Some Common Window Functions

Function Name Time-Domain Representation ½wðtÞ􀀉 Frequency-Domain Representation ½W ðf Þ􀀉

Box-car

1 for 0 # t , T

0 otherwise

j2p f ½1 2 cos 2p f T þ j sin 2p f T􀀉

Hanning

2 þ

cos

for ltl , T

0 otherwise

T sin 2p f T

2p f T½1 2 ð2 f TÞ2 􀀉

Parzen

1 2 6

􀀑 t

􀀜2

þ 6

􀀑ltl

􀀜3

for ltl ,

􀀑

1 2

ltl

􀀜3

for

, ltl # T

0 otherwise

sin p f T=2

p f T

664

775

Bartlett

1 2

ltl

for ltl # T

0 otherwise

􀀑sin p f T

p f T

􀀜4

(

10-24 Vibration and Shock Handbook

Characteristics of the signal that is being analyzed and also the nature of the system that generates the

signal should be considered in choosing an appropriate truncation window. In particular, the Hanning

window is recommended for signals generated by heavily damped systems and the Hamming window is

recommended for use with lightly damped systems. Table 10.7 lists some useful signal types and

appropriate window functions.

10.5.8 Spectral Approach to Process Monitoring

In mechanical systems, component degradation

may be caused by vibrating excitations, which can

result in malfunction or failure. In this

sense, continuous monitoring during testing of

mechanical deterioration in various critical components

of a vibratory system is of prime

importance. This usually cannot be done by simple

visual observation, unless malfunction is detected

by operability monitoring of the system. However,

since mechanical degradation is always associated

with a change in vibration level, by continuously

monitoring the development of Fourier spectra in

time (during system operation) at various critical

locations of the system, it is possible to conveniently

detect any mechanical deterioration and

impending failure. In this respect, real-time Fourier analysis is very useful in process monitoring and

failure detection and prediction. Special-purpose real-time analyzers with the capability of spectrum

comparison (often done by an external command) are available for this purpose.

Various mechanical deteriorations manifest themselves at specific frequency values. A change in

spectrum level at a particular frequency (and its multiples) would indicate a specific type of mechanical

degradation or component failure. An example is given in Figure 10.17, which compares the Fourier

spectrum at a monitoring location of a vibratory system at the start of test with the Fourier spectrum

after some mechanical degradation has taken place. To facilitate spectrum comparison within a narrowfrequency

band, it is customary to plot such Fourier spectra on a linear frequency axis. It is seen that the

overall spectrum levels have increased as a result of mechanical degradation. Also, a significant change

has occurred near 30 Hz. This information is useful in diagnosing the cause of degradation or

malfunction. Figure 10.17 might indicate, for example, impending failure of a component having

resonant frequency close to 30 Hz.

TABLE 10.7 Signal Types and Appropriate Windows

Signal Type Window

Periodic with period ¼ T Rectangular

Rapid transients within ½0; T􀀉 Gaussian

Periodic with period – T Flat-top cosine

Quasi-periodic Hamming

Slow transients beyond ½0; T􀀉 Hanning

Nonstationary random Gaussian

Beat-like signals with period < T Bartlett (triangular)

Narrow-band random Rectangular

Stationary random Hamming

Important low-level components mixed with

widely spaced high-level spectral components

Parzen

Broad-band random (white noise, pink noise, etc.) Gaussian

Frequency (Hz)

Fourier Spectrum

Magnitude 0

10 20 30 40 50 f

At Start

With Degradation

FIGURE 10.17 Effect of mechanical degradation on a

monitored Fourier spectrum.

Vibration Signal Analysis 10-25

10.5.9 Cepstrum

A function known as the cepstrum is sometimes used to facilitate the analysis of Fourier spectrum in

detecting mechanical degradation. The cepstrum (complex) CðtÞ of a Fourier spectrum Y ð f Þ is defined

CðtÞ ¼ F21 log Y ðfÞ ð10:54Þ

The independent variable t is known as quefrency, and it has the units of time.

An immediate advantage of cepstrum arises from the fact that the logarithm of the Fourier spectrum is

taken. From Equation 10.33 it is clear that, for a system having frequency-transfer function Hð f Þ; and

excited by a signal having Fourier spectrum U ðf Þ; the response Fourier spectrum Y ð f Þ could be expressed

in the logarithmic form:

log Y ð f Þ ¼ log Hð f Þ þ log U ðfÞ ð10:55Þ

Since the right-hand side terms are added rather than multiplied, any variation in Hð f Þ at a particular

frequency will be less affected by a possible low-spectrum level in the excitation Uð f Þ at that frequency,

when considering log Y ð f Þ rather than Y ðf Þ: Consequently, any degradation will be more conspicuous in

the cepstrum than in the Fourier spectrum. Another advantage of cepstrum is that it is better capable of

detecting variation in phenomena that manifest themselves as periodic components in the Fourier spectrum

( for example, harmonics and sidebands). Such phenomena, which appear as repeated peaks in the

Fourier spectrum, occur as a single peak in the cepstrum, and so any variations can be detected more easily.

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