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10.4 Fourier Analysis

Back

Fourier analysis is the key to frequency analysis of vibration signals. The frequency-domain

representation of a time signal is obtained through the Fourier transform. One immediate advantage

y0 sin (wt+f)

u0 sin wt

FIGURE 10.8 Response to base excitations of a tall

structure (cantilever).

Vibration Signal Analysis 10-7

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 versatile

transformation 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 any

general signal, whereas the FSE is applicable only to periodic signals, and the discrete Fourier

Frequency f

(a)

Magnitude

|Y( f )|

Frequency f

(b)

Magnitude

|Y( f )|

(c)

Magnitude

|Y( f )|

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

TABLE 10.1 Deterministic Signals

Primary Classification Nature of the Fourier Spectrum Example

Periodic Discrete and equally spaced y0 sin vt þ y1 sin

vt þ f

Quasi-periodic Discrete and irregularly spaced y0 sin vt þ y1 sinð

ffiffi

2 p vt þ fÞ

Transient Continuous y0 expð2ltÞsinðvt þ fÞ

10-8 Vibration and Shock Handbook

transform (DFT) is used for discrete signals. As we shall see, all three versions of transform are

interrelated. 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 Xðf Þ of a time signal xðtÞ is given by the forward transform relation

Xðf Þ ¼

ð1

xðtÞ expð2j2p ftÞdt ð10:8Þ

with j ¼

ffiffiffiffi

21 p and f the cyclic frequency variable. When Equation 10.8 is multiplied by expð j2pf tÞ and

integrated with respect to f using the orthogonality property (which can be considered as a definition of

the Dirac delta function d)

ð1

exp½j2p f ðt 2 tÞ􀀉dt ¼ dðt 2 tÞ ð10:9Þ

Box 10.1

SIGNAL CLASSIFICATION

Signal Types

Deterministic Random

(Future not precisely known through

finite observations or analysis)

Periodic Transient

E.g., E.g., E.g.,

* Blade-passing signal

of a turbine at constant

speed

* Shock wave generated

from an impact test with

known impulse

* Machine tool vibration

* Jet engine noise

* Aerodynamic gusts

* Road irregularity disturbances

* Atmospheric temperature

* Earthquake motions

* Electrical line noise

* Counter-rotating-mass

shaker signal at constant

speed

* Step response of a

damped oscillator

* Response of a

* Step response of an variable-speed rotor

undamped oscillator * Excitation of a variable*

Steady-state response of frequency shaker

a damped system to a

sine excitation

Vibration Signal Analysis 10-9

we get the inverse transform relation

xðtÞ ¼

ð1

Xð f Þ expð2jp ftÞdf ð10:10Þ

The forward transform is denoted by the operator F and the inverse transform by F21. Hence, the

Fourier transform pair is given by

Xðf Þ ¼ F xðtÞ and xðtÞ ¼ F21Xð fÞ ð10:11Þ

Note that for real systems, xðtÞ is a real function but Xð f Þ is a complex function in general. Hence, the

Fourier spectrum of a signal can be represented by the magnitude lXð f Þl and the phase angle /Xð f Þ of the

(complex) Fourier spectrum Xð f Þ: Alternatively, the real part Re Xð f Þ and the imaginary part Im Xð f Þ

together can be used to represent the Fourier spectrum.

According to the present definition, the Fourier spectrum is defined for negative frequency values as

well as positive frequencies (i.e., a two-sided spectrum). The complex conjugate of a complex value is

obtained by simply reversing the sign of the imaginary part; in other words, replacing j with 2 j.

By noting that replacing j with 2 j 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 the

complex conjugate Xpð f Þ of the Fourier spectrum for positive frequencies. As a result, only the positivefrequency

spectrum needs to be specified and the negative-frequency spectrum can be conveniently

derived from it, through complex conjugation.

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

relations

XðsÞ ¼

ð1

xðtÞ expð2stÞdt ð10:12Þ

and

xðtÞ ¼

2p j

ðsþj1

s2j1

XðsÞ expðstÞds ð10:13Þ

Since the signal itself is zero for t , 0; it is seen that for all practical purposes, Fourier transform results

can be deduced from the Laplace transform analysis, simply by substituting s ¼ j2pf ¼ jv and s ¼ 0:

10.4.2 Fourier Series Expansion

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

xðtÞ ¼ DF

n¼21

An expð j2pnt=TÞ ð10:14Þ

with DF ¼ 1=T: Strictly speaking (see FIT relations) this is the inverse transform relation. The scaling

factor DF is not essential but is introduced so that the Fourier coefficients An will have the same units as

the 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

ðT

exp½j2pðn 2 mÞt=T􀀉dt ¼ dmn ð10:15Þ

Note that the Kronecker delta dmn is defined as

dmn ¼

1 for m ¼ n

0 for m – n

(

ð10:16Þ

10-10 Vibration and Shock Handbook

for integer values of m and n: The forward

transform that results is given by

An ¼

ðT

xðtÞexpð2j2pnt=TÞdt ð10:17Þ

Note that An are 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:

Xðf Þ ¼ DF

n¼21

Andð f 2 n · DFÞ ð10:18Þ

This is shown in Figure 10.10 (only the magnitudes lAnl can be plotted in this figure because An is

complex in general). If we substitute this spectrum into the inverse FIT relation given earlier, we get the

inverse FSE relation 10.14. Furthermore, this shows that the Fourier spectrum of a periodic signal is a

series of equidistant impulses.

10.4.3 Discrete Fourier Transform

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

{xm} ¼ ½x0; x1; …; xN21􀀉 ð10:19Þ

to an N-element sequence of spectral results

{Xn} ¼ ½X0; X1; …; XN21􀀉 ð10:20Þ

through the forward transform relation

Xn ¼ DT

NX21

m¼0

xm expð2j2pmn=NÞ ð10:21Þ

with n ¼ 0; 1; …; N 2 1: The values Xn are called the spectral lines. It can be shown that these quantities

approximate the values of the Fourier spectrum (continuous) at the corresponding discrete frequencies.

Let us identify DT as the sampling period (i.e., the time step between two adjacent points of sampled data).

The inverse transform relation is obtained by multiplying the forward transform relation by

expð j2pnr=NÞ and summing over n ¼ 0 to N 2 1; using the orthogonality property

NX21

n¼0

exp½j2pnðr 2 mÞ=N􀀉 ¼ drm ð10:22Þ

Note that this orthogonality relation can be considered as a definition of Kronecker delta. The inverse

transform is

xm ¼ DF

NX21

n¼0

Xn expðj2pmn=NÞ ð10:23Þ

The data record length is given by

T ¼ N · DT ¼ 1=DF ð10:24Þ

The DFT is a transform in its own right, independent of the FIT. It is possible, however, to interpret this

transformation as the trapezoidal integration approximation of FIT. We have deliberately chosen

appropriate scaling factors DT and DF in order to maintain this equivalence, and it is very useful in

computing the Fourier spectrum of a general signal or the Fourier coefficients of a periodic signal using a

digital computer. Proper interpretation of the digital results is crucial, however, in using DFT to compute

(an approximate) Fourier spectrum of a (continuous) signal. In particular, two types of error: aliasing

| An|

−3ΔF −2ΔF −ΔF 0 ΔF 2ΔF 3ΔF

FIGURE 10.10 Fourier spectrum of a periodic signal

and its relation to Fourier series.

Vibration Signal Analysis 10-11

and leakage (or truncation error) should be considered. This subject will be addressed later. The three

transform 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-domain

equivalents is the FIT. Table 10.3 provides some important properties of the FIT and the corresponding

time-domain relations that are useful in the analysis of signals and system models. These properties may

be easily derived from the basic FIT relations (Equation 10.8 through Equation 10.10). It should be noted

that, inherent in the definition of the DFT given in Table 10.2 is the N-point periodicity of the two

sequences; that is, Xn ¼ XnþiN and xm ¼ xmþiN ; for i ¼ ^1; ^2; …:

The definitions given in Table 10.2 may differ from the versions available in the literature by a

multiplicative constant. However, it is observed that according to the present definitions, the DFT may be

interpreted as the trapezoidal integration of the FIT. The close similarity between the definitions of

the FSE and the DFT is also noteworthy. Furthermore, according to the last row in Table 10.2, the FSE

can 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 (digital

computer) version of the FIT rather than an independent discrete transform. Accordingly, the results

from 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 the

FIT. However, it should be clear that if Xðf 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 approximate

relationship exists.

A further advantage of the definitions given in Table 10.2 is apparent when dealing with the FSE. As we

have 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

transform

Xð f Þ ¼

1 xðtÞexpð2j2p ftÞdt Xn ¼ DT

PN21

m¼0 xm expð2j2pnm=NÞ;

n ¼ 0; 1; …; N 2 1

An ¼

ÐT

0 xðtÞ expð2j2pnt=TÞdt

n ¼ 0; 1; …

Inverse

transform

xðtÞ ¼

1 Xð f Þexpð j2p ftÞdf xm ¼ DF

PN21

n¼0 Xn expð j2pnm=NÞ;

m ¼ 0; 1; …; N 2 1

xðtÞ ¼ DF

¼21 An

£ expð j2pnt=TÞ

Orthogonality

1 exp½ j2p f ðt 2 tÞ􀀉df

¼ dðt 2 tÞ

PN21

n¼0 exp½ j2pnðr 2 mÞ=N􀀉 ¼ drm

ÐT

0 exp½ j2pðr 2 nÞt=T􀀉dt ¼ drn

Notes T ¼ NDT DF ¼ 1=T Xðf Þ ¼ DF

¼21 Andð f 2 n=TÞ

TABLE 10.3 Important Properties of the Fourier Transform

Function of Time Fourier Spectrum

xðtÞ Xð f Þ

k1 x1 ðtÞ þ k2 x2ðtÞ k1 X1 ð f Þ þ k2 X2 ðf Þ

xðtÞ expð2j2p taÞ Xð f þ aÞ

xðt þ tÞ Xð f Þ expð j2p f tÞ

dn xðtÞ

dtn ð j2p f Þn Xð f Þ

Ðt2

1 xðtÞdt

Xð f Þ

j2p f

10-12 Vibration and Shock Handbook

relating the complex Fourier coefficients to the DFT sequence of sampled data from the periodic function

via the present definitions.

Aliasing distortion is an important consideration when dealing with sampled data from a continuous

signal. 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 of

straightforward mathematical procedures, the relationship between the FIT and the DFT can be

established. Even though {Xðn · DFÞ} is not the DFT of {xðm · DTÞ}; the results in Table 10.4 show that

{X~ ðn ·DFÞ} is the DFT of {x~ðm·DTÞ} where the periodic functions X~ ð f Þ and x~ðtÞ are as defined as in

Table 10.4. This situation is illustrated in Figure 10.11. It should be recalled that Xð f Þ is a complex

function in general, and as such it cannot be displayed as a single curve in a two-dimensional coordinate

system. Both the magnitude and the phase angle variations with respect to frequency f are needed. For

brevity, only the magnitude lXð f Þl is shown in Figure 10.11(a). Nevertheless, the argument presented

applies to the phase angle /Xðf Þ as well.

It is obvious that in the time interval ½0; T􀀉; xðtÞ ¼ x~ðtÞ and xm ¼ x~m: However, X~ ðn ·DFÞ is only

approximately 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) X~ ðf Þ will become closer to Xðf Þ in the

frequency interval ½0; F=2􀀉; as is clear from Figure 10.11(c). Furthermore, due to the F-periodicity of

X~ ð 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; no

information regarding its frequency spectrum Xðf Þ is obtained for frequencies higher than fc ¼ 1=ð2DTÞ:

This fact is known as Shannon’s sampling theorem, and the limiting (cut-off) frequency is called the

Nyquist frequency. In vibration signal analysis, a sufficiently small sample step DT should be chosen in

order to reduce aliasing distortion in the frequency domain, depending on the highest frequency of

interest in the analyzed signal. This however, increases the signal processing time and the computer

storage requirements, which is undesirable, particularly in real-time analysis. It can also result in stability

problems in numerical computations. The Nyquist sampling criterion requires that the sampling rate

ð1=DTÞ for a signal should be at least twice the highest frequency of interest. Instead of making the

sampling 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 that

the DFT results in the frequency interval ½fc; 2fc􀀉 are redundant because they merely approximate the

frequency spectrum in the negative frequency interval ½2fc; 0􀀉 which is known for real signals. This fact is

known as the Hermitian property.

The last column of Table 10.4 presents the relationship between the FSE and the DFT. It is noted that

the sequence {A~ n} rather than the sequence of complex Fourier series coefficients {An} represents the DFT

TABLE 10.4 Unified Fourier Transform Relationships

Description Relationship

DFT and FIT DFT and FSE

Given xðtÞ !FIT

Xð f Þ xðtÞ !FSE

{An }

Form x~ðtÞ ¼

¼21 xðt þ kTÞ A~ n ¼

¼21 AnþkN

X~ ð f Þ ¼

¼21 Xð f þ kFÞ

Then {x~m} !DFT

{X~ n} {x~m} !DFT

{A~ n}

Where x~m ¼ x~ðm·DTÞ; X~ n ¼ X~ ðn ·DFÞ xm ¼ xðm · DTÞ

F ¼ 1=DT; T ¼ 1=DF N¼ T=DT

Vibration Signal Analysis 10-13

of the sampled data sequence {xðm · DTÞ}: In practice, however, An !0 as n!1: Consequently, A~ n is a

good approximator to An in the range ½2N=2 # n # N=2􀀉 for sufficiently large N: This basic result is

useful in determining the Fourier coefficients of a periodic signal using discrete data that are sampled at

time steps of DT ¼ 1=F; in which F is the fundamental frequency of the periodic signal. Again the aliasing

error ð A~ n 2 AnÞ may be reduced by increasing the sampling rate (i.e., by decreasing DT or increasing N).

X( f )

(b) 0

~x(t)

−2T −T T 2T t

(a) 0

x(t)

T Time t −F/2 0 F/2 Frequency f

(c)

−3F/2 −F −F/2 0 F/2 F 3F/2 f

(d) 0

xm ~

−2T −T T = NΔT 2T t

(e)

−3F/2 −F = −NΔF −F/2

−F/2

(−fc) (fc)

F/2

F = 1/ΔT 3F/2 f

(f) f

Xn = X(n.ΔF)

ΔT

ΔF

X ( f )

FIGURE 10.11 Relationship between FIT and DFT, with an illustration of aliasing error. (a) Fourier integral

transformation (FIT) of a signal; (b) periodicity arranged time signal; (c) periodicity of the frequency spectrum;

(d) sampled time signal; (e) sampled frequency spectrum (with aliasing error); (f) sampled original spectrum

(no aliasing error).

10-14 Vibration and Shock Handbook

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 vibration

testing. Due to sampling in the frequency domain, the signal becomes distorted. The aliasing error

ðx~m 2 xðmDTÞÞ is reduced by decreasing the sample period DF: It should be noted that no information

regarding the signal for times greater than T ¼ 1=DF is obtained from the analysis.

By comparing Figure 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 in

Figure 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 frequency

equal to the Nyquist frequency, then the aliasing distortion would not occur due to sampling. A filter of

this 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 by

using a filter cut-off frequency that is slightly less than the Nyquist frequency. The resulting spectrum

would then only be valid up to this filter cut-off frequency (and not up to the theoretical limit of Nyquist

frequency).

Example 10.1

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

Sample rate fs ¼ 1=0:001 samples=sec ¼ 1000 Hz ¼ 1 kHz

Nyquist frequency ¼ 1000=2 Hz ¼ 500 Hz

Due to aliasing, approximately 20% of the spectrum (i.e., spectrum beyond 400 Hz) will be distorted.

Here we may use an antialiasing filter.

Folded

High-Frequency

Spectrum

(a)

Spectral

Magnitude

fc Frequency f

(b)

Spectral

Magnitude

fc Frequency f

Aliasing

fc = Nyquist frequency

Original Spectrum

FIGURE 10.12 Aliasing distortion of frequency spectrum. (a) Original spectrum; (b) distorted spectrum due to

aliasing.

Vibration Signal Analysis 10-15

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 information

about the signal.

Spectral line separation ¼ 1000=1024 Hz ¼ 1 Hz ðapproximatelyÞ

Keep 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,

f1 ¼ 0:2 Hz and f2 ¼ 0:8 Hz; are shown (Figure 10.13(a)). Suppose that the two signals are sampled at the

rate of fs ¼ 1 sample/sec. The corresponding Nyquist frequency is fc ¼ 0:5 Hz: It is seen that, at this

sampling rate, the data samples from the two signals are identical. In other words, the high-frequency

signal 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 signal

beyond the Nyquist frequency ð fcÞ cannot be recovered.

It is apparent from Figure 10.11(e) that the aliasing error becomes increasingly prominent for

frequencies of the spectrum closer to the Nyquist frequency. With reference to the expression for X~ ð 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=2

ð¼ fcÞ; the influence of the Xð f 2 nFÞ segments for n $ 2 and n # 21 is negligible in the discrete

spectrum in the frequency range ½0; F=2􀀉: Hence the aliasing distortion in the frequency band ½0; F=2􀀉

comes primarily from Xð f 2 FÞ; which is the true spectrum shifted to the right through F: Therefore,

Box 10.2

SIGNAL SAMPLING CONSIDERATIONS

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

Nyquist frequency or cut-off frequency or computational bandwidth:

fc ¼

2 £ sampling rate

Aliasing distortion:

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

distorting 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) information

would be valid, is the Nyquist frequency fc ¼ 1=ð2DTÞ:

3. DFT results that are computed for the frequency range ½fc; 2fc􀀉 merely approximate the

frequency spectrum in the negative frequency range ½2fc; 0􀀉:

10-16 Vibration and Shock Handbook

a reasonably accurate expression for the aliasing error is

en ¼ Xð2ðF 2 n · DFÞÞ ¼ Xðn · DF 2 FÞ ¼ X2ðN2nÞ ¼ Xn2N n ¼ 0; 1; 2; …; N=2 ð10:25Þ

Note from Equation 10.8 that the spectral value obtained when f in the complex exponential is replaced

by 2 f is the same as the spectral value obtained when jis replaced by 2 j. Since the signal xðtÞ is real, it

follows that the Fourier spectrum for the negative frequencies is simply the complex conjugate of the

Fourier spectrum for the positive frequencies; thus

Xð2f Þ ¼ Xp ðfÞ ð10:26Þ

or, in the discrete case

Xn2N ¼ Xp

N2n ð10:27Þ

It follows from Equation 10.25 that the aliasing distortion is given by

en ¼ Xp

N 2n for n ¼ 0; 1; 2; …; N=2 ð10:28Þ

This result confirms that aliasing can be interpreted as folding of the complex conjugate of the true

spectrum beyond the Nyquist frequency fcð¼ F=2Þ over to the original spectrum. In other words, due to

aliasing, frequency components higher than the Nyquist frequency appear as lower frequency

components (due to folding). These aliasing components enter into the digital Fourier results in the

useful frequency range ½0; fc􀀉:

Aliasing reduces the valid frequency range in digital Fourier results. Typically, the useful frequency

limit is fc

􀀋

1:28 so that the last 20% of the spectral points near the Nyquist frequency should be neglected.

(a)

Signal

Time (s)

(b)

Amplitude

Spectrum

Frequency (Hz)

Sampling rate fs = 1 sample/s

Nyquist frequency fc = 0.5 Hz

1 2 3 4 5

f1 = 0.2 Hz

f2 = 0.8 Hz

0.2 0.5 0.8

f1 f2

FIGURE 10.13 A simple illustration of aliasing. (a) Two harmonic signals with identical sampled data;

(b) Frequency spectra of the two harmonic signals.

Vibration Signal Analysis 10-17

It should be clear that if a low-pass filter with its cut-off frequency set at fc is used on the time signal prior

to sampling and digital Fourier analysis, the aliasing distortion can be virtually eliminated. Analog

hardware filters may be used for this purpose. They are the antialiasing filters. Note that sometimes

􀀋

1:28ðø 0:8fcÞ is used as the filter cut-off frequency. In this case, the computed spectrum is accurate up

to 0:8fc and not up to fc:

The buffer memory of a typical commercial Fourier analyzer can store N ¼ 210 ¼ 1024 samples of

data from the time signal. This is the size of the data block analyzed in each digital Fourier

transform calculation. This will result in N=2 ¼ 512 spectral points (spectral lines) in the frequency

range ½0; fc􀀉: Out of this only the first 400 spectral lines (approximately 80%) are considered free of

aliasing distortion.

Example 10.2

Suppose that the frequency range of interest in a particular vibration signal is 0 – 200 Hz. We are

interested in determining the sampling rate (digitization speed) and the cut-off frequency for the

antialiasing (low-pass) filter.

The Nyquist frequency fc is given by fc

􀀋

1:28 ¼ 200

Hence; fc ¼ 256 Hz

The sampling rate (or digitization speed) for the time signal that is needed to achieve this range of

analysis is F ¼ 2fc ¼ 512 Hz: With this sampling frequency, the cut-off frequency for the antialiasing

filter could be set at a value between 200 and 256 Hz.

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