Vibration and Shock Handbook 10

Vibration and Shock Handbook 10 Every so often, a reference book appears that stands apart from all others, destined to become the definitive work in its field. The Vibration and Shock Handbook is just such a reference. From its ambitious scope to its impressive list of contributors, this handbook delivers all of the techniques, tools, instrumentation, and data needed to model, analyze, monitor, modify, and control vibration, shock, noise, and acoustics. Providing convenient, thorough, up-to-date, and authoritative coverage, the editor summarizes important and complex concepts and results into “snapshot” windows to make quick access to this critical information even easier. The Handbook’s nine sections encompass: fundamentals and analytical techniques; computer techniques, tools, and signal analysis; shock and vibration methodologies; instrumentation and testing; vibration suppression, damping, and control; monitoring and diagnosis; seismic vibration and related regulatory issues; system design, application, and control implementation; and acoustics and noise suppression. The book also features an extensive glossary and convenient cross-referencing, plus references at the end of each chapter. Brimming with illustrations, equations, examples, and case studies, the Vibration and Shock Handbook is the most extensive, practical, and comprehensive reference in the field. It is a must-have for anyone, beginner or expert, who is serious about investigating and controlling vibration and acoustics.

10 Vibration Signal

Analysis

Clarence W de Silva

The University of British Columbia

10.1 Introduction 10-110.2 Frequency Spectrum 10-2

Frequency † Amplitude Spectrum † Phase Angle † Phasor Representation of Harmonic Signals † RMS Amplitude Spectrum † One-Sided and Two-Sided Spectra † Complex Spectrum

10.3 Signal Types 10-710.4 Fourier Analysis 10-7

Fourier Integral Transform † Fourier Series Expansion

† Discrete Fourier Transform † Aliasing Distortion † Another Illustration of Aliasing

10.5 Analysis of Random Signals 10-18

Ergodic Random Signals † Correlation and Spectral Density † Frequency Response Using Digital Fourier Transform † Leakage (Truncation Error) † Coherence

† Parseval’s Theorem † Window Functions † Spectral Approach to Process Monitoring † Cepstrum

10.6 Other Topics of Signal Analysis 10-26

Bandwidth † Transmission Level of a Band-Pass Filter

† Effective Noise Bandwidth † Half-Power (or 3 dB) Bandwidth † Fourier Analysis Bandwidth † Resolution

in Digital Fourier Results

10.7 Overlapped Processing 10-28

Order AnalysisSummary

This chapter considers the nature and analysis of vibration signals Both time-domain techniques and domain techniques are investigated, which are related through Fourier transform Topics covered include signaltypes, signal sampling, aliasing error, truncation error, window functions, spectral analysis, bandwidth issues, andorder analysis Several applications of signal analysis in mechanical vibration are discussed

frequency-10.1 Introduction

Numerous examples can be drawn from engineering applications for vibrating (dynamic) systems

A steam generator of a nuclear power plant that undergoes flow-induced vibration; a high-rise buildingsubjected to seismic motions at its foundation; an incinerator tower subjected to aerodynamicdisturbances; an airplane excited by atmospheric turbulence; a gate valve under manual operation; and aheating, ventilating, and air conditioning (HVAC) control panel stressed due to vibrations in its supportstructure are such examples

10-1

Consider an aircraft in flight, as schematically shown in Figure 10.1 There are many excitations on thisdynamic system For example, jet engine forces and control surface movements are intentionalexcitations, whereas aerodynamic disturbances are unintentional (and unwanted) excitations.The primary response of the aircraft to these excitations will be the motions in various degrees offreedom (DoF), including rigid-body and flexible (vibratory) mode motions.

Even though the inputs and outputs (excitations and responses) are functions of time, they can also berepresented as functions of frequency, through Fourier transformation The resulting Fourier spectrum of asignal can be interpreted as the set of frequency components which the original signal contains Thisfrequency-domain representation of a signal can highlight many salient characteristics of the signal andalso those of the corresponding system For this reason, frequency-domain methods, particularly Fourieranalysis, are used in a wide variety of applications such as data acquisition and interpretation,experimental modeling and modal analysis, diagnostic techniques, signal/image processing and patternrecognition, acoustics and speech research, signal detection, telecommunications, and dynamic testingfor design development, quality control, and qualification of products Many such applications involvethe study of mechanical vibrations

10.2 Frequency Spectrum

Excitations (inputs) to a dynamic system progress

with time, thereby producing responses (outputs)

which themselves vary with time These are signals

that can be recorded or measured A measured

signal is a time history Note that in this case the

independent variable is time and the signal is

represented in the time domain A limited

amount of information can be extracted by

examining a time history As an example, consider

the time-history record shown in Figure 10.2

It can be characterized by parameters such as the

following:

ap¼ peak amplitude

Tp¼ period in the neighborhood of the peak ¼ 2 £ interval between successive zero crossings nearthe peak

Te¼ duration of the record

Ts¼ duration of strong response (i.e., the time interval beyond which no peaks occur that are largerthan ap=2Þ

Nz¼ number of zero crossings within TsðNz¼ 14 in Figure 10.2)

It is obviously cumbersome to keep track of so many parameters and, furthermore, not all of them areequally significant in a given application Note, however, that all the parameters listed above are directly

AerodynamicExcitations ExcitationsEngine Control Surface

Excitations

AerodynamicExcitations

BodyResponse

FIGURE 10.1 In-flight excitations and responses of an aircraft.

T p/2

Ts(Nz= 14)

FIGURE 10.2 A time-history record.

or indirectly related to either the amplitude or the frequency of zero crossings within a given time interval.This signifies the importance of a frequency variable in representing a time signal This is probably thefundamental motivation for using frequency-domain representations In this context, however, morerigorous definitions are needed for the parameters: amplitude and frequency A third parameter, known

as phase angle, is also needed for unique representation of a signal in the frequency domain

10.2.1 Frequency

Let us further examine the basis of frequency-domain analysis Consider the periodic signal of period Tthat is formed by combining two harmonic (or sinusoidal) components of periods T and T=2 andamplitudes a1and a2as shown in Figure 10.3 The cyclic frequency (cycles/sec, or hertz, or Hz) of the twocomponents are f1¼ 1=T and f2¼ 2=T: Note that in order to obtain the angular frequency (radians/sec),the cyclic frequency has to multiplied by 2p:

10.2.2 Amplitude Spectrum

An alternative graphical representation of the periodic signal shown in Figure 10.3 is given in Figure 10.4

In this representation, the amplitude of each harmonic component of the signal is plotted against thecorresponding frequency This is known as the amplitude spectrum of the signal, and it forms the basis ofthe frequency-domain representation Note that this representation is often more compact, and can befar more useful than the time-domain representation Note further that in the frequency-domainrepresentation, the independent variable is frequency

10.2.3 Phase Angle

In its present form, Figure 10.4 does not contain all

the information of Figure 10.3 For instance, if the

high-frequency component in Figure 10.3 is

shifted through half its period ðT=4Þ; the resulting

signal is shown inFigure 10.5 This signal is quite

different from that in Figure 10.3 but since the

amplitudes and the frequencies of the two

harmonic components are identical for both

signals, they possess the same amplitude spectrum

So, what is lacking in Figure 10.3 in order to make

FIGURE 10.3 Time-domain representation of a periodic signal.

it a unique representation of a signal, is the information concerning the exact location of the harmoniccomponents with respect to the time reference or origin ðt ¼ 0Þ: This is known as the phase information.For example, the distance of the first positive peak of each harmonic component from the time origin can

be expressed as an angle (in radians) by multiplying it by 2p=T: This is termed the phase angle of theparticular component In both signals (shown inFigure 10.3and Figure 10.5) the phase angle of the firstharmonic component is the same and equalsp=2 according to the present convention The phase angle ofthe second harmonic component isp=2 in Figure 10.3 and zero in Figure 10.5

10.2.4 Phasor Representation of Harmonic Signals

A convenient geometric representation of a harmonic signal of the form

is possible by means of a phasor This representation is illustrated inFigure 10.6 Specifically, consider arotating arm of radius a; rotating in the counter-clockwise (ccw) direction at an angular speed ofvrad/sec Suppose that the arm starts (i.e., at t ¼ 0) at an angular positionf with respect to the y-axis(vertical axis) in the ccw sense Then, it is clear from Figure 10.6(a) that the projection of the rotating arm

on the y-axis gives the time signal yðtÞ: This is the phasor representation, where we have

Signal amplitude ¼ length of the phasor

Signal frequency ¼ angular speed of the phasor

Signal phase angle ¼ initial position of the phasor with respect to the y-axis

It should be clear that a phase angle makes practical sense only when two or more signals arecompared This is so because for a given harmonic signal we can pick any point as the time reference

ðt ¼ 0Þ: However, when two harmonic signals are compared, as in Figure 10.6(b), we may consider one

of those signals that starts (at t ¼ 0) at its position peak as the reference signal This will correspond to

a phasor whose initial configuration coincides with the positive y-axis As is clear from Figure 10.6(b),for this reference signal we have,f ¼ 0: Then the phase angle f of any other harmonic signal wouldcorrespond to the angular position of its phasor with respect to the reference phasor Note that, inthis example, the time shift between the two signals isf=v; which is also a direct representation of thephase It should be clear, then, that the phase difference between two signals is also a representation ofthe time lead or time lag (delay) of one signal with respect to the other Specifically, the phase that isahead of the reference phasor is considered to “lead” the reference signal In other words, thesignal a cosðvt þ fÞ has a phase lead of f or a time lead of f=v with respect to the signal of a cos vt:

FIGURE 10.5 A periodic signal with an identical amplitude spectrum as for Figure 10.2.

Another important observation may be made with regard to the phasor representation of a harmonicsignal A phasor may be expressed as the complex quantity

yðtÞ ¼ a ejðvtþfÞ¼ a cosðvt þ fÞ þ ja sinðvt þ fÞ ð10:2Þwhose real part is a cosðvt þ fÞ; which is in fact the signal of interest It is clear from Figure 10.6 that, if

we take the y-axis to be real and the x-axis to be imaginary, the complex representation 10.2 is indeed acomplete representation of a phasor By using the complex representation 10.2 for a harmonic signal,significant benefits of mathematical convenience could be derived in vibration analysis It suffices toremember that practical vibrations are “real” signals, and regardless of the type of mathematical analysisthat is used, only the real part of a complex signal of the form 10.2 will make physical sense

10.2.5 RMS Amplitude Spectrum

If a harmonic signal yðtÞ is averaged over one period T; the negative portion cancels out the positiveportion, giving zero Consider a harmonic signal of angular frequencyv (or cyclic frequency f ), phaseanglef; and amplitude a; as given by

yðtÞ ¼ a cosðvt þ fÞ ¼ a cosð2pft þ fÞ ð10:3ÞIts average (mean) value is

f w

(a)

y a

y = a cos (wt+f)

t

y a

which can be verified by direct integration, while

noting that

T ¼ 1=f ¼ 2p=v ð10:5ÞFor this reason, mean value is not a measure of the

“strength” of a signal in general Now let us define

the root mean square (RMS) value of a signal This

is the square root of the mean value of the square

of the signal By direct integration, it can be shown

that for a sinusoidal (or harmonic) signal; the

10.2.6 One-Sided and Two-Sided Spectra

Mean squared amplitude spectrum of a signal (sometimes called power spectrum because the square of avariable such as voltage and velocity is a measure of quantities such as power and energy, even though it isnot strictly the spectrum of power in the conventional sense) is obtained by plotting the mean squaredamplitude of the signal against frequency Note that these are one-sided spectra because only the positivefrequency band is considered This is a realistic representation because one cannot talk aboutnegative frequencies for a real system But, from a mathematical point of view, we may consider negativefrequencies as well In a spectral representation it is at times convenient to consider the entire frequencyband (consisting of both negative and positive values of frequency) It then becomes a two-sided spectrum

In this case the spectral component at each frequency value should be equally divided between thepositive and the negative frequency values (hence, the spectrum is symmetric), such that the overall meansquared amplitude (or power or energy) remains the same

We have seen that for a harmonic signal component of amplitude a and frequency f (e.g.,

a cosð2pf þ fÞ) the RMS amplitude is a2=2 at frequency f ; whereas the two-side spectrum has amagnitude of a2=4 at both the frequency values 2f and þf :

Note that, even though it is possible to interpret the meaning of a negative time (which represents thepast, previous to the starting point), it is not possible to give a realistic meaning to a negative frequency.This concept is introduced primarily for analytical convenience

10.2.7 Complex Spectrum

We have shown that for unique representation of a signal in the frequency domain, both amplitude andphase information should be provided for each frequency component Alternatively, the spectrum can beexpressed as a complex function of frequency, having a real part and an imaginary part For instance, for aharmonic component given by a cosð2pfiþfÞ the two-sided complex spectrum can be expressed as

Yð fiÞ ¼ ai

2ðcosfiþ j sinfiÞ ¼ ai

2ejfiand,

the basis of (complex) Fourier series expansion (FSE), which we shall consider in detail in a later section.

It should be clear that the complex conjugate of a spectrum is obtained by changing either j to 2j or

v to 2v (or f to 2f )

10.3 Signal Types

Signals can be classified into different types

depending on their characteristics Note that the

signal itself is a time function, but its

frequency-domain representation can bring up some of its

salient features Signals particularly important to us

here are the excitations and responses of vibrating

systems These can be divided into two broad

classes: deterministic signals and random signals

depending on whether we are dealing with

deterministic vibrations or random vibrations

Consider a damped cantilever beam that is

subjected to a sinusoidal base excitation of

frequency v and amplitude u0 in the lateral

direction (Figure 10.8) In the steady state, the tip

of the beam will also oscillate at the same frequency,

but with a different amplitude y0and, furthermore, there will be a phase shift by an anglef: For a givenfrequency and known beam properties, the quantities y0andf can be completely determined Under theseconditions the tip response of the cantilever is a deterministic signal in the sense that when the experiment

is repeated, the same response is obtained Furthermore, the response can be expressed as a mathematicalrelationship in terms of parameters whose values are determined with 100% certainty, and probabilities arenot associated with these parameters (such parameters are termed deterministic parameters) Randomsignals are nondeterministic (or stochastic) signals Their mathematical representation requires probabilityconsiderations Furthermore, if the process were to be repeated there would always be some uncertainty as

to whether an identical response signal could be obtained again

Deterministic signals can be classified as periodic, quasi-periodic, and transient Periodic signals repeatexactly at equal time periods The frequency (Fourier) spectrum of a periodic signal constitutes a series ofequally spaced impulses Furthermore, a periodic signal will have a Fourier series representation Thisimplies that a periodic signal can be expressed as the sum of sinusoidal components whose frequencyratios are rational numbers (not necessarily integers) Quasi-periodic (or almost periodic) signals alsohave discrete Fourier spectra, but the spectral lines are not equally spaced Typically, a quasi-periodicsignal can be generated by combining two or more sinusoidal components, provided that at least two ofthe components have as their frequency ratio an irrational number Transient signals have continuousFourier spectra These types of signals cannot be expressed as a sum of sinusoidal components (or aFourier series) All signals that are not periodic or quasi-periodic can be classified as transient Mostoften, highly damped (overdamped) signals with exponentially decaying characteristics are termedtransient, even though various other forms of signals such as exponentially increasing (unstable)responses, sinusoidal decays (underdamped responses), and sinesweeps (sinewaves with variablefrequency) also fall into this category.Table 10.1gives examples for these three types of deterministicsignals The corresponding amplitude spectra are sketched in Figure 10.9 A general classification ofsignals, with some examples is given in Box 10.1

of the Fourier transform is that, through its use, differential operations (differentiation and integration)

in the time domain are converted into simpler algebraic operations (multiplication and division).Transform techniques are quite useful in mathematical applications For example, a simple, yet versatiletransformation from products into sums is accomplished through the use of the logarithm Three versions

of Fourier transform are important to us The Fourier integral transform (FIT) can be applied to anygeneral signal, whereas the FSE is applicable only to periodic signals, and the discrete Fourier

Frequency f

0(a)

Magnitude

|Y( f )|

0(c)

Magnitude

|Y( f )|

FIGURE 10.9 Magnitude spectra for three types of deterministic signals (a) Periodic; (b) quasi-periodic; (c) transient.

TABLE 10.1 Deterministic Signals

Quasi-periodic Discrete and irregularly spaced y0 sin v t þ y1 sinðpffiffi2 v t þ f Þ

transform (DFT) is used for discrete signals As we shall see, all three versions of transform areinterrelated In particular, we have to use the DFT in digital computation of both FIT and FSE.

10.4.1 Fourier Integral Transform

The Fourier spectrum Xð f Þ of a time signal xðtÞ is given by the forward transform relation

Xð f Þ ¼ð1

with j ¼pffiffiffiffi21and f the cyclic frequency variable When Equation 10.8 is multiplied by expð j2pf tÞ andintegrated with respect to f using the orthogonality property (which can be considered as a definition ofthe Dirac delta functiond)

* Machine tool vibration

* Jet engine noise

* Response of avariable-speed rotor

we get the inverse transform relation

According to the present definition, the Fourier spectrum is defined for negative frequency values aswell as positive frequencies (i.e., a two-sided spectrum) The complex conjugate of a complex value isobtained by simply reversing the sign of the imaginary part; in other words, replacing j with 2j

By noting that replacing j with 2j in the forward transform relation is identical to replacing f with 2f ;

it should be clear that the Fourier spectrum (of real signals) for negative frequencies is given by thecomplex conjugate Xpð f Þ of the Fourier spectrum for positive frequencies As a result, only the positive-frequency spectrum needs to be specified and the negative-frequency spectrum can be convenientlyderived from it, through complex conjugation

The Laplace transform is similar to the FIT Laplace transform is defined by the forward and inverserelations

10.4.2 Fourier Series Expansion

For a periodic signal xðtÞ of period T; the FSE is given by

xðtÞ ¼ DF X1

with DF ¼ 1=T: Strictly speaking (see FIT relations) this is the inverse transform relation The scalingfactor DF is not essential but is introduced so that the Fourier coefficients Anwill have the same units asthe Fourier spectrum The Fourier coefficients are obtained by multiplying the inverse transform relation

by expð2j2pmt=TÞ and integrating with respect to t from 0 to T using the orthogonality condition

1T

ð10:16Þ

for integer values of m and n: The forward

transform that results is given by

An¼ðT

0xðtÞ expð2j2pnt=TÞdt ð10:17ÞNote that Anare complex quantities in general

It can be shown that for periodic signals, FSE is a

special case of FIT, as expected Consider a Fourier

spectrum consisting of a sum of equidistant

impulses separated by the frequency interval

DF ¼ 1=T:

Xð f Þ ¼ DF X1

n¼21Andð f 2 n · DFÞ ð10:18ÞThis is shown in Figure 10.10 (only the magnitudes lAnl can be plotted in this figure because An iscomplex in general) If we substitute this spectrum into the inverse FIT relation given earlier, we get theinverse FSE relation 10.14 Furthermore, this shows that the Fourier spectrum of a periodic signal is aseries of equidistant impulses

10.4.3 Discrete Fourier Transform

The DFT relates an N-element sequence of sampled (discrete) data signal

1N

and leakage (or truncation error) should be considered This subject will be addressed later The threetransform relations, corresponding inverse transforms, and the orthogonality relations are summarized

in Table 10.2

The link between the time-domain signals and models and the corresponding frequency-domainequivalents is the FIT Table 10.3 provides some important properties of the FIT and the correspondingtime-domain relations that are useful in the analysis of signals and system models These properties may

be easily derived from the basic FITrelations (Equation 10.8 through Equation 10.10) It should be notedthat, inherent in the definition of the DFT given in Table 10.2 is the N-point periodicity of the twosequences; that is, Xn¼ XnþiN and xm¼ xmþiN; for i ¼ ^1; ^2; …:

The definitions given in Table 10.2 may differ from the versions available in the literature by amultiplicative constant However, it is observed that according to the present definitions, the DFTmay beinterpreted as the trapezoidal integration of the FIT The close similarity between the definitions ofthe FSE and the DFT is also noteworthy Furthermore, according to the last row in Table 10.2, the FSEcan be expressed as a special FIT consisting of an equidistant set of impulses of magnitude An T located

at f ¼ n=T:

10.4.4 Aliasing Distortion

Recalling that the primary task of digital Fourier analysis is to obtain a discrete approximation to the FIT

of a piecewise continuous function, it is advantageous to interpret the DFT as a discrete (digitalcomputer) version of the FIT rather than an independent discrete transform Accordingly, the resultsfrom a DFT must be consistent with the exact results obtained if the FIT were used The definitions given

in Table 10.2 are consistent in this respect because the DFT is given as the trapezoidal integration of theFIT However, it should be clear that if Xð f Þ is the FIT of xðtÞ; then the sequence of sampled values{Xðn · DFÞ} is not exactly the DFT of the sampled data sequence {xðm · DTÞ}: Only an approximaterelationship exists

A further advantage of the definitions given in Table 10.2 is apparent when dealing with the FSE As wehave noted, the FIT of a periodic function is a set of impulses We can avoid dealing with impulses by

TABLE 10.2 Unified Definitions for Three Fourier Transform Types

Relation Name Fourier Integral Transform Discrete Fourier Transform (DFT) Fourier Series Expansion (FSE) Forward

£ expð j2 p nt=TÞ Orthogonality Ð121 exp½ j2 p f ð t 2 tÞ df

¼ d ð t 2 tÞ

1 N

PN21n¼0 exp½ j2 p nðr 2 mÞ=N ¼ d rm 1

TABLE 10.3 Important Properties of the Fourier Transform

relating the complex Fourier coefficients to the DFTsequence of sampled data from the periodic functionvia the present definitions.

Aliasing distortion is an important consideration when dealing with sampled data from a continuoussignal This error may enter into computation in both the time domain and the frequency domain,depending on the domain in which the results are presented We will address this issue next

10.4.4.1 Sampling Theorem

The basic relationships between the FIT, the DFT, and the FSE are summarized in Table 10.4 By means ofstraightforward mathematical procedures, the relationship between the FIT and the DFT can beestablished Even though {Xðn · DFÞ} is not the DFT of {xðm · DTÞ}; the results in Table 10.4 show that{ ~Xðn · DFÞ} is the DFT of {~xðm · DTÞ} where the periodic functions ~Xð f Þ and ~xðtÞ are as defined as inTable 10.4 This situation is illustrated in Figure 10.11 It should be recalled that Xð f Þ is a complexfunction in general, and as such it cannot be displayed as a single curve in a two-dimensional coordinatesystem Both the magnitude and the phase angle variations with respect to frequency f are needed Forbrevity, only the magnitude lXð f Þl is shown in Figure 10.11(a) Nevertheless, the argument presentedapplies to the phase angle /Xð f Þ as well

It is obvious that in the time interval ½0; T ; xðtÞ ¼ ~xðtÞ and xm¼ ~xm: However, ~Xðn · DFÞ is onlyapproximately equal to Xðn · DFÞ in the frequency interval ½0; F : This is known as the aliasing distortion

in the frequency domain As DT decreases (i.e., as F increases) ~Xð f Þ will become closer to Xð f Þ in thefrequency interval ½0; F=2 ; as is clear from Figure 10.11(c) Furthermore, due to the F-periodicity of

~Xð f Þ; its value in the frequency range ½F=2; F will approximate Xðf Þ in the frequency range ½2F=2; 0 :

It is clear from the preceding discussion that if a time signal xðtÞ is sampled at equal steps of DT; noinformation regarding its frequency spectrum Xð f Þ is obtained for frequencies higher than fc¼ 1=ð2DTÞ:This fact is known as Shannon’s sampling theorem, and the limiting (cut-off) frequency is called theNyquist frequency In vibration signal analysis, a sufficiently small sample step DT should be chosen inorder to reduce aliasing distortion in the frequency domain, depending on the highest frequency ofinterest in the analyzed signal This however, increases the signal processing time and the computerstorage requirements, which is undesirable, particularly in real-time analysis It can also result in stabilityproblems in numerical computations The Nyquist sampling criterion requires that the sampling rateð1=DTÞ for a signal should be at least twice the highest frequency of interest Instead of making thesampling rate very high, a moderate value that satisfies the Nyquist sampling criterion is used in practice,together with an antialiasing filter to remove the distorted frequency components It should be noted thatthe DFT results in the frequency interval ½fc; 2fc are redundant because they merely approximate thefrequency spectrum in the negative frequency interval ½2fc; 0 which is known for real signals This fact isknown as the Hermitian property

The last column of Table 10.4 presents the relationship between the FSE and the DFT It is noted thatthe sequence { ~An} rather than the sequence of complex Fourier series coefficients {An} represents the DFT

TABLE 10.4 Unified Fourier Transform Relationships

k¼21 AnþkN

~Xð f Þ ¼ P 1 k¼21 Xð f þ kFÞ

of the sampled data sequence {xðm · DTÞ}: In practice, however, An! 0 as n ! 1: Consequently, ~Anis agood approximator to Anin the range ½2N=2 # n # N=2 for sufficiently large N: This basic result isuseful in determining the Fourier coefficients of a periodic signal using discrete data that are sampled attime steps of DT ¼ 1=F; in which F is the fundamental frequency of the periodic signal Again the aliasingerror ð ~An2 AnÞ may be reduced by increasing the sampling rate (i.e., by decreasing DT or increasing N).

)

( f

X

0(b)

~x(t)

0(a)

10.4.4.2 Aliasing Distortion in the Time Domain

In vibration applications it is sometimes required to reconstruct the signal from its Fourier spectrum.Inverse DFT is used for this purpose and is particularly applicable in digital equalizers in vibrationtesting Due to sampling in the frequency domain, the signal becomes distorted The aliasing errorð~xm2 xðmDTÞÞ is reduced by decreasing the sample period DF: It should be noted that no informationregarding the signal for times greater than T ¼ 1=DF is obtained from the analysis

By comparingFigure 10.11(a)with (c), or (e) with ( f), it should be clear that the aliasing error in ~X

in comparison with the original spectrum X is caused by “folding” of the high-frequency segment

of X beyond the Nyquist frequency into the low-frequency segment of X: This is illustrated inFigure 10.12

10.4.4.3 Antialiasing Filter

It should be clear from Figure 10.12 that, if the original signal is low-pass filtered at a cut-off frequencyequal to the Nyquist frequency, then the aliasing distortion would not occur due to sampling A filter ofthis type is called an antialiasing filter In practice, it is not possible to achieve perfect filtering Hence,some aliasing could remain even after using an antialiasing filter Such residual errors may be reduced byusing a filter cut-off frequency that is slightly less than the Nyquist frequency The resulting spectrumwould then only be valid up to this filter cut-off frequency (and not up to the theoretical limit of Nyquistfrequency)

Example 10.1

Consider 1024 data points from a signal, sampled at 1 msec intervals

Sample rate fs¼ 1=0:001 samples=sec ¼ 1000 Hz ¼ 1 kHzNyquist frequency ¼ 1000=2 Hz ¼ 500 HzDue to aliasing, approximately 20% of the spectrum (i.e., spectrum beyond 400 Hz) will be distorted.Here we may use an antialiasing filter

FoldedHigh-FrequencySpectrum(a)

SpectralMagnitude

Suppose that a digital Fourier transform computation provides 1024 frequency points of data up to

1000 Hz Half of this number is beyond the Nyquist frequency and will not give any new informationabout the signal

Spectral line separation ¼ 1000=1024 Hz ¼ 1 Hz ðapproximatelyÞKeep only the first 400 spectral lines as the useful spectrum

Note: Almost 500 spectral lines may be retained if an accurate antialiasing filter is used

Some useful information on signal sampling is summarized in Box 10.2

10.4.5 Another Illustration of Aliasing

A simple illustration of aliasing is given in Figure 10.13 Here, two sinusoidal signals of frequency,

f1¼ 0:2 Hz and f2¼ 0:8 Hz; are shown (Figure 10.13(a)) Suppose that the two signals are sampled at therate of fs¼ 1 sample/sec The corresponding Nyquist frequency is fc¼ 0:5 Hz: It is seen that, at thissampling rate, the data samples from the two signals are identical In other words, the high-frequencysignal cannot be distinguished from the low-frequency signal Hence, a high-frequency signal component

of frequency 0.8 Hz will appear as a low-frequency signal component of frequency 0.2 Hz This is aliasing,

as is clear from the signal spectrum shown in Figure 10.13 Specifically, the spectral segment of the signalbeyond the Nyquist frequency ð fcÞ cannot be recovered

It is apparent from Figure 10.11(e) that the aliasing error becomes increasingly prominent forfrequencies of the spectrum closer to the Nyquist frequency With reference to the expression for ~Xð f Þ in

Table 10.4, it should be clear that when the true Fourier spectrum Xð f Þ has a steep roll-off prior to F=2ð¼ fcÞ; the influence of the Xð f 2 nFÞ segments for n $ 2 and n # 21 is negligible in the discretespectrum in the frequency range ½0; F=2 : Hence the aliasing distortion in the frequency band ½0; F=2comes primarily from Xð f 2 FÞ; which is the true spectrum shifted to the right through F: Therefore,

Box 10.2

S IGNAL S AMPLING C ONSIDERATIONS

The maximum useful frequency in digital Fourier results is half the sampling rate

Nyquist frequency or cut-off frequency or computational bandwidth:

fc¼ 1

2 £ sampling rateAliasing distortion:

High-frequency spectrum beyond Nyquist frequency folds on to the useful spectrum, therebydistorting it

Summary:

1 Pick a sufficiently small sample step DT in the time domain, to reduce the aliasing distortion

in the frequency domain

2 The highest frequency for which the Fourier transform (frequency-spectrum) informationwould be valid, is the Nyquist frequency fc¼ 1=ð2DTÞ:

3 DFT results that are computed for the frequency range ½fc; 2fc merely approximate thefrequency spectrum in the negative frequency range ½2fc; 0 :