Signal Processing in Rotating Machinery Iitg

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CHAPTER 16

SIGNAL PROCESSING IN ROTATING MACHINERIES

In previous chapter, we studied various kinds of transducers, and other measuring and analyses

equipments used in the vibration and acoustics. These transducers produce the signals in the form of

charge or voltage. Through amplifiers these signals are fed to the data acquisition system to digitize

these signals into a computer. In the present chapter, the main focus would be to study methods

involved in the processing these signals in various required form in time and frequency domains (fast

Fourier Transform function, FFT; etc.) or time-frequency domain (wavelet transform, WT; etc.).

Various graphical forms of display of these measurements are presented. Error involved in the

measurement (mechanical, electrical or digital) itself would be addressed. These include error due tothe aliasing effect, error in digitizing, leakage error, tuning of sampling and signal periodicity, effects

due to windowing, etc. Fourier series, Fourier transform, discrete Fourier transform, FFT and

complex-FFT would be covered in some detail. The uncertainty in the estimated parameters due to

error in measurements would be addressed. In the subsequent chapter, various fault conditions and its

effects on the vibration signature are looked in.

16.1 Display of Vibration Measurements

Vibration signal collected has to be measured and displayed. The quantities that have to be measured

are usually vibration amplitude, its frequency and its phase. The history of the vibration measurement

in the form of digital display (through vibration meter) gives only the trend of vibration level (peak-

to-peak, rms, etc.), which might give only indication about the condition of the machine whether

machine is in good or bad condition. However, the cause of the increased vibration level at the

component level is difficult to assign with such lone measurement of vibration level by vibration

meter. To investigate the causes of vibration in component level, one needs to investigate the

relationship between the vibration signal frequencies and the rotational speed of the shaft. In

principle, every fault has a unique frequency (or frequencies) to contribute in the overall vibration

signal, especially in rotating machinery different faults at the component level might contribute

vibration frequencies in integer multiples (or divisions) of rotational frequency of component speeds.

This can be done by the spectrum analysis with the help of FFT-analyzer (fast Fourier transformation)

equipments. Spectrum analysers have various convenient functions, such as FFT, tracking analysis

(Bode plot, Nyquist plot), waterfall diagram, Campbell diagram, etc.

A tracking filter is a device that accepts two input signals, one being the vibration signal under

consideration and the other being a phase reference signal. In the tracking analysis, dynamic


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characteristics of a rotating machine are investigated by measuring vibrations for the run-up or

rundown of the machinery, which is a practical way of getting dynamic behaviour of the machinery at

various operational speeds. The tracking of the vibration signal is done normally at 1 rotational

speed of the rotor. The tracking filter removes from the vibration signal any components, which are

not of the same frequency as the reference signal. The amplitude and phase (relative to the reference

signal) of the remaining signal are then displayed, usually in digital form. The output from a tracking

filter can be used to construct the Bode and Nyquist diagrams used, for example, during balancing

operations. This can be done several ways for example (i) by manually tuning the narrow bandpass

filter to the required central frequency (1 or integer multiples of rotational speed of rotors) and

measuring the vibration level by vibration meter, (ii) tuning the central frequency as in method (i),

however, displaying the signal in the oscilloscope to take required measurement, and (iii) measuring

the vibration signal continuously for the total run-up or rundown period and processing the signal

afterwards in computer. Apart from the amplitude of vibration often it is required to measure the

phase of the vibration signal with respect to some fixed reference on the rotating component, which is

called the reference signal. Phase meter can be used for method (i) which requires two signals, i.e.,

first the vibration signal and the other is the reference signal. For method (ii) both the signals could be

displayed on the same screen and relative phase can be measured. In method (iii) vibration signals and

the reference signal have to recaptured at various small time period and amplitude and phase of the

vibration signal can be measured by displaying it on the computer screen or by advance signal

processing software.

16.1 Bode plot (the variation of the amplitude and

phase with the rotational frequency of rotor)

Fig. 16.2 Nyquist plot


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The vibration amplitude and the phase so obtained, which is called the frequency response function,

can be displayed as the Bode and Nyquist plots as shown in Figures 16.1 and 16.2 (the Bode plot as

shown is for multi-DOF system whereas the Nyquist plot is plot of one of the resonance peak or for

single-DOF system). The amplitude of the Bode plots should be plotted in semi-logarithmic scale

when the peak at the resonance is very sharp with very high magnitudes. The Nyquist plot to be

considered as blown view of the resonance peak and this plot is very useful for obtaining the modal

damping (Ewins, 1984). In the Nyquist plot, it is shown that both the amplitude and phase information

of the Bode plot is merged in a single Nyquist plot. The frequency information is indicated along the

circumference of the circle of the Nyquist plot. It can easily be seen that the Nyquist plot is a complex

plane with the horizontal axis as real axis and the vertical axis as the imaginary axis. This will be

more clear when we see the Bode plot information can also be plotted as the real and imaginary part

of the signal with respect to the frequency (i.e.,j cos jsin j

r iXe X X X

= + = + ; where X is the

amplitude, is the phase,rX is the real part and iX is the imaginary part).

Figure 16.3 Spectrum diagram (FFT of a capture signal)

Figure 16.4 Waterfall diagram (FFT of captured signals at different speeds)


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Both the diagrams (Bode and Nyquist plots) should be plotted using readings, which have been

nulled at slow speed and between critical speeds, thereby removing the effects of runout and the

effects of vibration modes which do not relate to the critical speed under consideration. Depending

upon the vibration parameters, i.e., the displacement, velocity and acceleration, the FRF (frequency

response function, i.e, the ratio of the response to the force) is called respectively the receptance,

mobility and inertance. The inverse of FRF corresponding to the displacement, velocity and

acceleration measurements are called respectively the dynamic stiffness, the mechanical impedance

and the apparent mass.

The spectrum diagram is the plot of captured unfiltered vibration signal in frequency domain (or may

be filtered with the broad bandpass filter to remove the unnecessary noise). It is obtained by

performing the FFT of the captured signal and one such spectrum diagram is shown in Figure 16.3.

The spectrum analyser is used to separate out the incoming vibration signal into all of frequencies

from which the total signal is composed. The amplitude and phase, relative to some reference signal,

of all the frequency components is displayed in the form of a graph of amplitude (or phase) against

frequency. The data is thus said to be displayed in thefrequency domain.

Figure 16.5 Campbell diagram

Many spectrum analyzer have the facility to plot several such graphs in cascade or waterfall

diagram, as shown in Figure 16.4. Such diagrams help to determine the relationship between the

vibration signal and machine running speed, which in turn help in the monitoring of the machine

condition. A waterfall diagram is a 3-dimensional plot of spectra at various speeds. Vibration signals

are captures at different speeds and FFT of the signals are cascade as shown in Figure 16.4. Another

version of the waterfall diagram in 2-dimension is shown in Figure 16.5, where the amplitudes of the

vibrations are depicted by a circle with radius corresponding to the amplitude of vibration at particular

frequency (i.e., 1or integer multiples/divisions). The axis is chosen similar to the Campbell diagram,i.e., spin speed of the rotor as abscissa and frequency of whirl as the ordinate.


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representation. The vibration of the rotor is a whirling motion and therefore not only the frequencies

but also the directions of the whirling motions are important enough to pursue their causes. Some

oscilloscopes enable a phase-indicating pulse of extra-bright beam to be displaced on the orbit once

per revolution of the shaft, which indicates the direction of whirl. The shape of the orbit, or Lissajous

figure, can itself be a useful tool in monitoring machine health. A typical whirl orbit for a machine

subject to a small amount of unbalance is shown in Figure 16.7 (as displayed on an oscilloscope).

Figure 16.7 Lissajous plot

The orbit plot can be seen clearly for simple rotor vibration when a single frequency is present in the

measured signals. However, using manual tracking filter the orbit plot can be obtained for 1 or

integer multiples/divisions of the rotational frequency of the rotor, which is a tedious process. In

frequency domain , however, since the usual fast Fourier transform (FFT) theory gives information

about magnitudes of frequencies and phases only, we cannot know the whirl direction using the

conventional FFT-analyser. For this purpose, Ishida (1997) and Lee (2000) proposed a signal

processing method where the whirling plane of a rotor is overlapped to the complex plane. This

method is called the complex-FFT (or directional-FFT) method, enables us to know the directions of

whirling motion besides the magnitudes of the frequencies. They also used this method to extract a

component form non-stationary time histories obtained numeric simulations and experimented data

and depicted the amplitude variation of the component. We will discuss fundamental ideas necessary

to understand the signal processing by computer. In addition, applications of the complex-FFT

method in studying stationary and non-stationary vibrations are explained.

Once the vibration signals have been collected and measured, they are used to judge whether or not

the machine in question is operating properly. These judgments are made on the basis of whether

there are unusual features of vibration signal which are not normally present, or which may usually be

present under particular fault conditions. However, to generate such plots and use various functions

correctly, we must have some background knowledge of signal processing. Moreover, if we have to

construct a specific data analysis system that fits our research for example for system parameter

estimation (i.e., the crack parameter, bearing and seal dynamic parameters, residual unbalance,


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misalignment, etc.), we must have sufficient understanding of the fundamental of the signal

processing.

16.2 Accuracy of Vibration Measurements

A basic measurement system consists of a transducer with a conditioner (which senses the signal, and

converts to a measurable quantity like voltage or charge with some amplification), an analog-to-

digital converter, and a processing and display unit as shown in Figure 16.8. We shall focus our

attention in error involved in transforming the analog signal to the digital signal.

Figure 16.8 A basic measurement system

Vibration signal from the vibrating structure is detected and converted into a measurable physical

quantity by transducers. In a rotating machine, rotor displacements in two orthogonal directions and a

rotating speed are detected as voltage variations. The output signal x(t) from the transducer is an

analog signal that is continuous with time (it is similar to a theoretical function that is defined for all

possible values of its variables) as shown in Figure 16.9(a). However, the signal is discretised when it

is acquired by computer through an A/D interface as shown in Figure 16.9(b), where the solid line

represent the actual signal and circles represent the digitized signal data at grid nodes. It can be seen

that depending upon the resolution of the instrument the accuracy of the acquired data will depend,

finer the grid more the accuracy and vice a versa, since the digitized data will be aquired only at the

grid points. A digital signal is discretised in both time and magnitude. This digital signal is a series of

discrete data { }nx obtained by measuring (called sampling) an analog signal instantly at every time

interval t and is given as )( tnxxn = , where nis an integer. This interval t is called a sampling

interval, which is generally constant. Discretization in magnitude is called quantization, and the

magnitude is represented by binary numbers (unit: bit. A bit (short for binary digit) is the smallest unit

of data in a computer. A bit has a single binary value, either 0 or 1. Although computers usually

provide instructions that can test and manipulate bits, they generally are designed to store data and

execute instructions in bit multiples called bytes. In most computer systems, there are eight bits in a

byte). Binary describes a numbering scheme in which there are only two possible values for each

digit: 0 and 1. In binary numbers the digits' weight increases by powers of 2, rather than by powers of

10 as in the more familiar decimal numbers. In a digital numeral, the digit furthest to the right is the

"ones" digit; the next digit to the left is the "twos" digit; next comes the "fours" digit, then the "eights"digit, then the "16s" digit, then the "32s" digit, and so on. The decimal equivalent of a binary number

Transducer and

conditioner

(signal detection)

Interface

(A/D-transform)

Personal Computer

(Processing and display)


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can be found by summing all the digits. For example, the binary 10101 is equivalent to the decimal 1

+ 0 + 4 + 0 +16 = 21.

(a) Analog signal (b) Digitised signal

Figure 16.9 The analog and digital signals

Now we shall highlight some other errors which may be involved in measuring the vibration signal.

When an analog signal x(t) is changed into a sequence of digital data { }nx (n= 0, 1, 2, , N) a

virtual (or imaginary) wave is obtained if a fast signal is sampled slowly. For example, when a signal

illustrated by the full line is sampled as shown in Figure 16.10, a virtual signal wave illustrated by the

dashed line appears, although it is not contained in the original signal.

Figure 16.10 Aliasing effect on a fast signal while slow sampling

)(tx

t

nx

t


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(a)

(b)

(c)

0 0.2 0.4 0.6 0.8 1

-1

-0.5

0

0.5

1

0 0.1 0.2 0.3 0.4 0.5

-1

-0.5

0

0.5

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

-1

-0.5

0

0.5

1

=10 Hz

sf = 2 Hz

sf = 17 Hz

Time, s

x(t)

Time, s

Time, s

x(t)

x(t)


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(d)

(e)

(f)

0 0.2 0.4 0.6 0.8 1

-1

-0.5

0

0.5

1

0 0.2 0.4 0.6 0.8 1

-1

-0.5

0

0.5

1

0 0.2 0.4 0.6 0.8 1

-1

-0.5

0

0.5

1

sf = 20 Hz

sf = 24 Hz

Time, s

Time, s

Time, s

x(t)

x(t)

x(t)

sf = 30 Hz


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Figure 16.11 Illustration of aliasing effect and sampling frequency in a sinusoidal signal

This phenomenon is called the aliasing. This phenomenon could be seen in old movies where the

wheel of a horse-cart appears rotating in the opposite direction even when the cart moves forward.

This is due to the fact that the picture frame recorded per second by the camera is much slower than

the rotational speed of the wheel. Another example could be that when we look an accelerating fan it

appears to us in between as if it is rotating in the backward direction (or in the forward direction with

very slow speed). It is again due to the fact that the human sight can capture pictures at a certain rate

and when the fan speed is just slightly higher (or lower) than the rate of picture frame captured by the

sight, then the fan appears as if it is rotating in opposite direction (or at different speed then it is

actually rotating). It is more obvious when we put stroboscope light on a rotating shaft. In stroboscope

the light flashes (switches on and off) at a particular rate, which can be adjustable. When the flashing

rate is same as the shaft speed (or integer multiples) the shaft appears as if it is stationary. It due to the

fact that whenever light flashes the shaft occupies the same orientation and if we see a mark (or

0 0.2 0.4 0.6 0.8 1

-1

-0.5

0

0.5

1

0 0.2 0.4 0.6 0.8 1

-1

-0.5

0

0.5

1

Time, s

Time, s

x(t)

x(t)

sf = 40 Hz

sf = 50 Hz


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keyway) on the shaft, it will appear stationary. A slight decease in the flashing rate makes the shaft to

appear as rotating slowly in the same direction as the actual direction (when flashing rate is slightly

lower than the shaft speed the mark on the shaft will shift in forward direction in the subsequent

flashes and hence the shaft will appear as it is rotating in forward direction with slower speed i.e. with

a speed equal to the difference between the actual shaft speed and flashing rate), however, a slight

increase in the flashing rate makes the shaft to appear as if rotating slowing in backward direction as

compared to the actual direction of rotation. It is obvious that we must sample with a smaller

sampling interval as the signal frequency increases. This suggests that aliasing effects not only

changes the amplitude and frequency but also the whirl direction of a rotor. We can determine

whether or not we have this aliasing by following the sampling theorem. It says: when a signal is

composed of the components whose frequencies are all smaller than cf , we must sample it with a

frequencies higher than cf2 for the sake of not loosing the original signals information. The

frequency cf2 is called the Nyquist frequency. For example, if a sine wave with period Tis sampled

whenever 0)( =tx , that is, with sampling interval T/2, we have 0=nx (i.e., a signal with a constant

amplitude of zero). Moreover, if it is sampled whenever ( )x t A= (whereAis the amplitude), that is,

with sampling interval T/2, we havenx A= (i.e., a constant amplitude signal).Therefore, two

samplings in a period are clearly insufficient. However, this theorem teaches us that digital data with

more than two points during one period can express the original signal correctly. For example, if we

sample the signal having components of 1, 2 and 7 kHz with a sampling frequency of 10 kHz. Then 1

and 2 kHz signal will be measured without aliasing effect since the Nyquist frequency is 10/2 = 5 kHz

and is more than these vibration signal frequencies. However, we have an imaginary spectrum of 3

kHz (10 kHz - 7 kHz = 3 kHz), which does not exist practically. But, if we sample it with a frequency

of more than 14 kHz ( 2 7 kHz), such an aliasing problem does not occur. In practical

measurements, we do not commonly determine the sampling frequency by trial measurement. Instead,

we use a low-pass filter to eliminate the unnecessary high-frequency components in the signal and

sample with the frequency higher than twice the cutoff frequency. By such a procedure, we can

prevent aliasing.

Example 16.1 If we sample the signal having components of 120, 200, 460, 700, 800 and 900 Hz

with a sampling frequency of 1000 Hz. Whether aliasing effect will be present in the measurement?

What are the frequency that will appear in the captured signal?

Solution: The sampling frequency is 1000 Hz, hence the Nyquist frequency will be 1000/2 = 500 Hz.

Hence, we can be able to measure frequency below 500 Hz accurately that means frequency 120, 200

and 460 Hz will be measured accurately (Figs. 16.12-16.14). However, frequency 700 Hz will appear


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as 1000-700 = 300 Hz (Fig. 16.15) and frequency 800 Hz will also appear as 1000-800 = 200 Hz (Fig.

16.16). Frequency 900 Hz will appear as 1000-900 = 100 Hz signal (Fig. 16.17). So the captured

signal will contain erroneous high amplitude of 200 Hz signal with an additional frequency of 100 Hz

which is actually not present at all in the actual signal.

(a) Original signal

(b) Sampled signal

Fig. 16.12 Signal of 120 Hz sampled with sampling frequency 1000 Hz

(a) Original signal

(b) Sampled signalFig. 16.13 Signal of 200 Hz sampled with sampling frequency1000 Hz

0 0.02 0.04

-1

-0.5

0

0.5

1

0 100 200 3000

2

4

6

8x 10

5

0 0.02 0.04

-1

-0.5

0

0.5

1

0 100 200 3000

200

400

600

0 0.02 0.04

-1

-0.5

0

0.5

1

0 200 400 6000

5

10

15x 10

5

0 0.02 0.04

-1

-0.5

0

0.5

1

0 200 400 6000

200

400

600

Time, s

Time, s

Time, s

Time, s Frequency, Hz

Frequency, Hz

Frequency, Hz

Frequency, Hz

x(t)

x(t)

x(t)

x(t)

FFT[x(t)]

FFT[x(t)]

FFT[x(t)]

FFT[x(t)]


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(a) Original signal


(a) Original signal


0 0.005 0.01 0.015 0.02

-1

-0.5

0

0.5

1

0 500 10000

1

2

3x 10

6

0 0.005 0.01 0.015 0.02

-1

-0.5

0

0.5

1

0 200 4000

200

400

600

0 0.005 0.01 0.015 0.02

-1

-0.5

0

0.5

1

0 500 1000 15000

1

2

3

4x 10

6

0 0.005 0.01 0.015 0.02

-1

-0.5

0

0.5

1

0 200 400 6000

200

400

600

Time, s

Time, s

Time, s

Time, s

Frequency, Hz

Frequency, Hz

Fre uenc , Hz

Frequency, Hz

x(t)

x(t)

x(t)

x(t)

FFT

[x(t)]

FFT[x(t)]

FFT[x(t)]

FFT[x(t)]


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Example 16.2A shaft is rotating at 30 Hz, what will be the speed of shaft which will appear to the

observer when the stroboscope flashing frequency is (i) 30 Hz (ii) 60 Hz (iii) 25 Hz (iv) 34 Hz (v) 54

Hz, (vi) 76 Hz and (vii) 600 Hz (viii) 15 Hz and (ix) 45 Hz (x) 13 Hz and (xi) 18 Hz. Give a plot of

such variation.

Solution: The following speed of the shaft will be observed (i) stationary (ii) stationary (iii) 25 30 =

- 5 Hz in the opposite direction as the actual shaft rotation (iv) 34 30 = 4 Hz in the same direction

the actual shaft rotation (v) 54 -60 = - 6 Hz in the opposite direction as the actual shaft rotation (vi) 76

60 = 16 Hz or 76- 90 = - 14 Hz; we will see 14 Hz in the opposite direction as the actual shaft

rotation (vii) stationary (viii) 15 -30 = - 15 Hz, since it is half the frequency of the shaft if we mark

two different colours on the diagonally opposite on the shaft surface we will see them in alternatively,

however, shaft will appear as rotating with the half actual speed of the shaft. (ix) 45-30 = 15 Hz, in

this also case if we mark two different colours on the diagonally opposite we will see them in

alternatively, however, shaft will appear as rotating with the half actual speed of the shaft (x) 13- 30 =

- 17 Hz in the opposite direction as the actual shaft rotation (xi) 18 -30 = -12 Hz, in the opposite

direction as the actual shaft rotation.

In general ifsf is the shaft frequency and sbf is the frequency of the stroboscope then

(a) the Nyquist frequency would be 2 sf

(b) when sb sf f< then the shaft will appear as if it is rotating at frequency of sb sf f and the virtual

rotation of the shaft will be in the opposite to the actual rotation. However, when 0.5sb sf f= if we

mark two different colours on the diagonally opposite we will see them in alternatively and the shaft

will appear as rotating with the half of actual speed of the shaft.

(c) whensb sf f= the shaft will appear as stationary.

(d) when 1.5s sb sf f f< < the shaft will appear to rotate at frequency sb sf f in the same direction as

the actual.

(e) when 1.5sb sf f= if we mark two different colours on the diagonally opposite we will see them in

alternatively and the shaft will appear as rotating with the half actual speed of the shaft.

(f) when 1.5 2s sb sf f f< < the shaft will appear to rotate at frequency 2sb sf f in the opposite

direction as the actual.

(g) when 2sb sf f= the shaft will again appear as stationary.

(h) In general, when sb sf nf= ( 1,2,3,n = ) the shaft will appear as stationary and 0.5sb sf nf= if we

mark two different colours on the diagonally opposite we will see them in alternatively and the shaft

will appear as rotating with the half of actual speed of the shaft. When 0.5(2 1)s sb s

nf f n f < < + the

shaft will appear to rotate at frequencysb sf nf in the same direction as the actual. When


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0.5(2 1) 2s sb sn f f nf + < < the shaft will appear to rotate at frequency 2sb sf nf in the opposite

direction as the actual. Even when we flashing frequency is higher than the Nyquist frequency the

actual shaft speed may not be seen by the stroboscope flashing because of the cyclic nature of the

shaft rotation. At 0.5sb sf nf= the shaft will appear to rotate with the half of actual speed and at

flashing frequency it will be lower than this (but the direction of rotation would change). Hence only

at the 0sbf = we will observed the actual motion of the shaft.

16.3 Fourier Series

In mathematics, a Fourier series decomposes a periodic function into a sum of simple oscillating

functions, namely sines and cosines. The study of Fourier series is a branch of Fourier analysis.

Fourier series were introduced by Joseph Fourier (17681830) for the purpose of solving the heat

equation in a metal plate (Fourier, 1822). In data processing, we must first know the frequency

components contained in a signal. The fundamental knowledge necessary for it is the Fourier series.

We will briefly summaries it from the point of view of signal processing. On type of Fourier series is

expressed by real numbers, while the other is by complex number.

(i)Real Fourier Series:

A periodic function )(tx with period Tcan be expanded by trigonometric functions which belong to

the orthogonal function systems as follows

( ) ( )01

cos sin2

n n

n

ax t a n t b n t

=

= + + (16.1)

where T/2 = . This series is called the Fourier series or real Fourier series. Its coefficients are

given by

/ 2

/ 2

2

( )cos

T

nT

a x t n tdt T

=

,

/ 2

/ 2

2

( )sin

T

nT

b x t n tdt T

=

(16.2)

The mathematical conditions for the convergence of equation (16.1) are extremely general and cover

practically every conceivable engineering situation (Churchill, 1941). The only important restriction is

that, when ( )x t is discontinuous, the series gives the average value of ( )x t at the discontinuity.

(ii) Complex Fourier Series:


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Fourier series can be expressed by complex numbers using Eulers formulas: sincos jej +=

andj cos sine j = this gives

j -j

cos2

e e

+= and

j -j

sin2j

e e

= . Complex numbers make it

easier to treat the expressions. The complex representation also makes it possible to represent a

whirling motion of a rotor on the complex plane. Substituting the Eulers formula into equation (16.1)

, we have

( ) ( ) ( ){ }

( ) ( ) ( ) ( ) ( )

j -j j -jj -j0 0

1 1

1j -j j j j0

1 1 0 1

0.5 j 0.5 +j2 2 2j 2

2

n t n t n t n t n t n t

n n n n n n

n n

n t n t n t n t n t

n n n n n

n n n n n

a ae e e ex t a b a b e a b e

ac e c e c e c e c e

= =

= = = = =

+ = + + = + +

= + + = + +

which finally gives

=

=

n

tjn

nectx )( (16.3)

with

2

nn

n

jbac

= ,

2

00

ac = ,

2

nn

n

jbac

+=

(16.4)

Between the real and complex Fourier coefficients, the relationships. where the complex coefficients

are given by

=

2/

2/

)(1

T

T

ntj

n dtetxT

c

( ),2,1,0 =n (16.5)

Equation (16.3) is called the complex Fourier series. From equation (16.4), we know the followingrelationship

nn cc = (16.6)

which tells that when the real part of complex Fourier coefficients is plotted with respect to the

( ),2,1,0 =nn , it is symmetric about the n= 0. Similarly, when the imaginary part of complex

Fourier coefficients is plotted with respect to the ( ),2,1,0 =nn , it is skew-symmetric about the

n= 0. These complex Fourier coefficients can also be represented by


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nj

nn ecc

= (16.7)

where the absolute value 222 nnn bac += is called an amplitude spectrum, the angle

( )nnnn abc1tan == a phase spectrum and

2

nc a power spectrum.

Example 16.3Consider a wave defined as

1)( =tx for 10 t and 87 t

0= for 71 t

The time period of the wave is T= 8. Obtain the complex Fourier coefficients of the square wave and

plot them with respect to gradually increasing Fourier series order (or harmonics) n. It should

illustrate how by considering gradual increase in harmonics of the Fourier series, it actually converges

to the real signal.

Solution: The given square wave is plotted in Figure 16.18.

Fig. 16.18 The time history of a square signal

For this square wave, we can obtain complex Fourier coefficients from equation (16.5), as follows

[ ]/ 2 1 0 1 4

0/ 2 4 1 0 1

1 1 1 2 2( ) 0 1 1 0 0 (1 1) (1 0) 0 0.25

8

T

Tc x t dt dt dt dt dt

T T T T

= = + + + = + + + + = = =

(a)

and


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( ) ( ) [ ]

0 1j j1 0 1 4

j j

4 1 0 11 0

j j

1 10 0

j j

1 11 1 cos jsin cos jsin

j j

2sinj 0

nt nt nt nt

n

n nt

e ec dt e dt e dt dt

T T nt nt

e e n n n nnT nT

n

Tn

= + + + = +

= + = +

= +

for 0n (b)

Complex Fourier series coefficients (i.e., the line spectrum) given by equations (a) and (b) are plotted

in Figure 16.19 for real part with respect to n. The abscissa can be taken as frequency (for 1n= ,

2 /f T= = 0.784 Hz; similarly for n k= , 2 /f k T= = 0.784k Hz;). This diagram is called the

spectrum of the time signal. Since given square wave is even function ( ) ( )x t x t= , imaginary part of

n

c is zero. For odd function we will have condition as ( ) ( )x t x t= . These are illustrated in Figure

16.20.

Figure 16.19 (a) The line diagram with actual data of a square wave (complex form)

Figure 16.19 (b) The line spectrum and an envelope (dashed line) of a square wave (complex form)

-15 -10 -5 0 5 10 15-0.05

0

0.05

0.1

0.15

0.2


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(a) (b)

Figure 16.20(a) An odd function (b) An even function

Fig. 16.21 shows a square wave and its various harmonic sine waves, by adding more and more these

harmonic waves the square wave could be obtained. Figures 16.22(a)-(g) show how by gradually

adding more terms in the Fourier series, it approaches the actual signal.

Fig. 16.21 A square wave and its various harmonics

-10 -5 0 5 10

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

Time, s

( )x t


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(a)

(b)

(c)

(d)

-9 -7 -5 -3 -1 1 3 5 7 9

0

0.5

1

-9 -7 -5 -3 -1 1 3 5 7 9

0

0.5

1

-9 -7 -5 -3 -1 1 3 5 7 9

0

0.5

1

-9 -7 -5 -3 -1 1 3 5 7 9

0

0.5

1

( )x t

( )x t

( )x t

( )x t

Time, s

Time, s

Time, s

Time, s

n= 1

n = 2

n = 3

n = 5


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(e)

(f)

(g)

Fig. 16.22 Comparison between the continuous square wave in time domain and the corresponding

complex Fourier series up to different harmonics (nis the number of harmonics included)

16.4 Fourier Transform and Fourier Integral

In previous case we have seen the spacing between adjacent harmonics is 2 /T = and it will be

seen that, when the period T becomes large, the frequency spacing becomes smaller, and the

Fourier components become correspondingly tightly packed in Figure 16.19. In the limit when

T , they will in fact actually merge together. Since in this case )(tx no longer represents a

periodic phenomenon we can then no longer analyse it into discrete frequency components. For

-9 -7 -5 -3 -1 1 3 5 7 9

0

0.5

1

-9 -7 -5 -3 -1 1 3 5 7 9

0

0.5

1

-9 -7 -5 -3 -1 1 3 5 7 9

0

0.5

1

( )x t

( )x t

( )x t

Time, s

Time, s

Time, s

n = 10

n = 50

n = 100


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example, when )(tx is an isolated pulse, it cannot be converted to a discrete spectrum since it is not

periodic. Subject to certain conditions, we can however still follow the same line of thought except

that the Fourier series (16.1) (or (16.3)) turns into a Fourier integral and the Fourier coefficients (16.2)

(or (16.5)) turn into continuous functions of frequency called Fourier Transforms. Let us consider that

this interval is extended to infinity. Then the spectra obtained will represent the spectra of the isolated

pulse. Substituting equation (16.5) into equation (16.3), we get

0 0

/ 2

j j

/ 2

1 2( ) ( )

2

T

n t n t

n T

x t x t e dt eT

=

=

(16.8)

where the frequency T/2= of the fundamental wave is denoted by 0 and is called

fundamental harmonics. Here we represent of the nth order by nn =0 and the difference in

frequencies between the adjacent components by ===+

Tnn /201 . If we make

T , we have

j j1( ) ( )

2

t tx t x t e dt e d

=

(16.9)

where ,n

and are replaced by , d and , respectively. This can be expressed in separate

forms as follows

j( ) ( ) tx t X e dt

= (16.10)

with

( ) j1

( )2

tX x t e dt

= (16.11)

Equation (16.11) is called the Fourier transform of )(tx and equation (16.10) is called the inverse

Fourier transform (or the Fourier integral) of )(X . The classical Fourier analysis theory (Churchill,

1941)considers the conditions that ( )x t must satisfy equations (16.10) and (16.11) to be true. For

engineering applications, the important condition is usually expressed in the form

( )x t dt

< (16.12)


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It means that classical theory applies only to functions which decay to zero when t . This

condition may be relaxed when impulse functions are introduced in the generalized theory of Fourier

analysis. As for the discrete Fourier series, when there is a discontinuity in ( )x t , equation (16.10)

gives average value of ( )x t at the discontinuity.

Example 16.4Consider a square pulse defined as

x(t) = 1 for 11 t

= 0 for all other t

Obtain the continuous spectrum for the pulse and compare the same with the square wave of example

16.3.

Solution: Figure 16.23(a) shows the square pulse given in the problem. From equation (16.11), i.e., by

the Fourier transformation, we get

1j j j

1j

11

1 1 1 1 2 jsin sin( )

2 2 -j 2 -j 2 -j

t t tt e e e

X e dt

= = = = = (a)

Now, let us compare the line spectrum of a square wave of period Tas shown in Figure 16.19 and that

of a square pulse shown in Figure 16.23(b) in the form of a continuous spectrum. From equation (b)

of example 16.3, we have the Fourier coefficients, nc as

( )

( )0

00

0 0

sin2sin2sin

2

nn

nn Tcnc

Tn Tn n

=

= = = (b)

On comparing equation (b) with equation (a) for 0 = , the Fourier coefficients, nc , and the Fourier

transform, )(X , can be related as

)(2

0

nXTcn

= (c)

where 0 is the fundamental frequency. Therefore, the envelope of the quantities obtained by

multiplying 2/T to the line spectra of the Fourier coefficients, nc , of the square wave gives the


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continuous spectra of the Fourier transform )(X of the square pulse. For example at 0= from

equation (a), we have

( )

( )0 00

sinsin cos 1( 0)

d

dXd

d

= =

=

= = = = = (d)

From equation (a) of example 16.3, we have0

0.25c = . Hence, 08 0.25 1

2 2

Tc

= = , which is same as

equation (d). Similarly, at any other harmonics it can be verified that the Fourier coefficients, nc , and

the Fourier transform )(X are related as in equation (c).

(a) Square pulse in time domain (b) Fourier transform of the square pulse

Figure 16.23 A square pulse and its continuous spectrum

16.5 Discrete Fourier Transform

Till now, we have discussed the Fourier series, Fourier transform and Fourier Integral on the

assumption that we know a continuous signal wave (it is also called the continuous time series) in the

infinite time domain. However, in practical experiments, the data acquired, converted from the data

measured by an analog-to-digital converter, are sequences of data { }nx (n= 0, 1, 2, , n-1, n) that are

discrete and with finite number. To perform the spectrum analysis using these finite numbers of

discrete data (it is also called discrete time series), we must use the discrete Fourier transform (DFT).

In order to estimate spectra from discrete Fourier series the obvious method is to estimate the

appropriate correlation function (e.g., the auto- and cross- correlation functions) first and then to

Fourier transform this function to obtain the required spectrum. Until the late 1960s, this was the basis

of practical calculation procedures which followed the formal mathematical route by which spectra

are defined as Fourier transforms of correlation functions. The assumptions and approximations


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involved were studied in detail and there is an extensive literature on the classical method (Bendat, et

al. 1966). However the position was changed by the advent of the fast Fourier transform (or FFT).

This is a remarkably efficient way of calculating the Fourier transform of a discrete time series.

Instead of estimating spectra by first determining correlation of a time series and then calculating their

Fourier transforms, it is now quicker and more accurate to calculate spectral estimates directly from

discrete time series by a method which we shall describe in detail.

This DFT is defined as follows: Given N data sampled with a constant interval t , the DFT is

defined as a series expansion on the assumption that the original signal is a periodic function with the

period tN (although the original signal is not necessary periodic). However, various problems occur

in the course of this processing. On performing FT in a discrete environment introduces artificial

effects, like aliasing effects, spectral leakages, scalloping losses, etc.

If the sampling rate, in the time domain, is lower than the Nyquist rate the aliasing occurs. Two

signals are said to alias if the difference of their frequencies falls in the frequency range of interest,

which is always generated in the process of sampling (aliasing is not always bad; it is called mixing or

heterodyning in analog electronics, and is commonly used in tuning radios and TV channels). It

should be noted that, although obeying the Nyquist sampling criterion is sufficient to avoid aliasing, it

does not give high quality display in time domain record. If a sinusoid existing in the time signal not

bin-centered (i.e., if its frequency is not equal to any of the frequency samples) in the frequencydomain spectral leakage occurs. In addition, there is a reduction in coherent gain if the frequency of

the sinusoid differs in value from the frequency samples, which is termed scalloping loss.

(a) The first is the aliasingproblem. When the signal is sampled with a interval t , the information

about the components with frequencies higher than1

2 t is lost, which is the Nyquist frequency.

Therefore, we must only consider to the valid range of the spectra obtained, i.e., below the Nyquist

frequency.

(b) The second is the challenge of the coincidence of periods. It is impossible to know the correct

period of the original signal before the measurement. Therefore, the period of the original signal and

the period of DFT do not coincide, and this difference produces the leakage error. We will discuss this

leakage error and its countermeasure subsequently (e.g., by window functions).

(c) The third is the problem about the length of measurement. In the case of an isolated signal x(t), we

cannot have data in an infinite time range. However, since the Fourier coefficients nc and Fourier


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transform ( )X coincide at discrete points as explained in previous section, we can obtain ( )X by

connecting the values of nc smoothly.

Now the procedure of computing the DFT is explained in the subsequent section. Let us assume that

we obtained discrete time series 1210 ,,,, Nxxxx by the sampling. These data are extended

periodically to make a virtual periodic signal, as shown by the dashed curve in Figure 16.24.

Figure 16.24 Formation of virtual continuous period signal with sampled signal sequences

The fundamental period is tNT = and the fundamental frequency is T 20 == . If this

dashed curve is given as a continuous time function, its Fourier series expansion is given by the

expressions obtained by replacing with in equation (16.3) and (16.5). However, in the case of

a discrete signal, the integral of equation (16.5) must be calculated by replacing t, T,x(t) and with

, ,k

k t N t x and respectively. By such replacements, we have

1 1j j (2 / ) ( / )

0 0

1 1N Nn k t n T k T N n k k

k k

c x e t x eT N

= =

= = (16.13)

We represent the right-hand side of this expression by nX , that is

1j2 /

0

1 N nk Nn k

k

X x eN

=

= ; (n= 0, 1, 2, ,N-1) (16.14)

and call this descrete Fourier transform of the discrete time signal 110 ,,, Nxxx . Paired with this is

the following expression, called the inverse discrete Fourier transform (IDFT)


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1j2 /

0

Nnk N

k n

n

x X e

=

= ; (k= 0, 1, 2, , (N-1)) (16.15)

These transformations map the discrete signal of a finite number on the time axis to the discrete

spectra of a finite number on the frequency axis, or vice versa. These expressions using complex

numbers are called the complex discrete Fourier transform and the complex inverse discrete Fourier

transform. We also have transformation using only real numbers. One is the real discrete Fourier

transform, given by

1

0

1

0

1 2cos

1 2sin

N

n k

k

N

n k

k

nkA x

N N

nkB xN N

=

=

=

=

(n= 0, 1, 2, ,N-1) (16.16)

where nA and nB are quantities defined by nnn jBAX += . Further, the inverse real discrete

Fourier transform is given by

1

0

2 2cos sin

N

k n n

n

nk nk x A B

N N

=

=

; (n= 0, 1, 2, ,N-1) (16.17)

We explain the characteristics of the spectra obtain by the DFT using an example subsequently.

Before that let us examine the aliasing effect on the DFT. We have seen that the DFT of the series

110 ,,, Nxxx is defined by

12 /

0

1 N nk Nn k

k

X x eN

=

= ; (n= 0, 1, 2, ,N-1) (16.18)

Suppose we try to calculate values ofnX for the case when nis greater than (N-1). Let for example,

1n N= + . Then,

1 1 1j(2 / )( 1) j(2 ) j(2 / ) j(2 / )

1 1

0 0 0

1 1 1N N Nk N N k k N k N N k k k

k k k

X x e x e e x e XN N N

+

+

= = =

= = = = (16.19)

Sincej(2 )ke is always equal to 1 whatever the value of k. Hence, we have


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1 1NX X+ = , 2 2 ,NX X+ = , (16.20)

The coefficients nX therefore just repeat themselves for ( 1)n N> , so that if we plot the magnitudes

nX along a frequency axis 2 /n n N t =

, the graph repeats itself periodically. Furthermore, it is

also easy to see that from equation (16.18) that, provided terms in the { }kx series are real, we have

1j2 / *

1 1

0

1 N k Nk

k

X x e XN

=

= = (16.21)

where*

1X is the complex conjugate of 1X . Hence

1 1X X = (16.22)

so the spectra diagram for the nX would be symmetrical about the zero frequency position. The

unique part of the graph occupies the frequency range / t . Higher frequencies just show

spurious Fourier coefficients which are repetitions of those apply at frequencies below / t . We can

therefore see that the coefficients nX calculated by the DFT are only correct Fourier coefficients for

frequencies up to

2

2n

n Nn

N t t

= = =

(16.23)

that is for nin the range 0,1,2, , / 2n N= . Moreover, if there are frequencies above / t present in

the original signal, these introduce a distortion of the graph called aliasing. The high frequency

components contribute to the { }kx series and falsely distort the Fourier coefficients calculated by the

DFT for frequencies below / t . If max is the maximum frequency component present in ( )x t ,

then aliasing can be avoided by ensuring that the sampling interval t is small enough that

maxt

>

or max

1

2f

t>

(16.24)

with

maxmax

2f

= (16.25)


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The frequency 1/ 2 is called the Nyquist frequency (or some times the folding frequency) and is the

maximum frequency that can be detected from data sampled at time spacing . The phenomenon of

aliasing is most important when analyzing practical data. The sampling frequency 1/ 2 must be high

enough to cover the full frequency range of the continuous time series. Otherwise the spectrum from

equally spaced sampled will differ from the true spectrum because of aliasing. In some cases the only

way to be certain that this condition is met may be to filter the time series to remove intentionally all

frequency components higher than 1/ 2 before beginning the analysis.

Example 16.5A square wave with period T= 8 sec and sixteen sampled data: 140 === xx and

0155 === xx . (i) Obtain the spectrum of the discrete time series. (ii) Compare the spectrum of

the same square wave for thirty two sampled data:0 8 1x x= = = and 9 31 0x x= = = . It should be

noted that the signal is sampled intentionally in the range that coincides with the period of the original

square wave to avoid the leakage error.

Solution: (i) Sixteen sampled points: The sampling of the square wave for sixteen sampled data is

shown in Figure 16.25. The period of the signal is 8 sec, hence the fundamental frequency would be

0 1/ 1/8 0.125f T= = = Hz. The sampling interval is / 8 /16 0.5t T N = = = sec. The Nyquist

frequency would be ( )1/ 2 1/ 2 0.5 1cf t= = = Hz. Hence, the maximum harmonics which will be

valid is 0/ 1/ 0.125 8cn f f= = = . Total number of harmonics are same as number of sampled points,

i.e., 16.

Figure 16.25 Discrete time series of a square wave

On using equation (16.14), the spectrum,nX , can be calculated by simple substitution and addition of

various terms. For the present case 16N = . In general, nX is a complex quantity and it can be plotted

in various form, e.g., (1) the real ( )Ren nA X= and imaginary ( )Imn nB X= parts versus harmonic

number, n, (or frequency,f), (Figure 16.26(a, b)) and (2) the magnitude nX and phase nX versus

harmonic number, n, (or frequency,f) (Figure 16.26(c, d)).


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(a) Real part of the spectrum (b) Imaginary part of the spectrum

(c) Magnitude part of the spectrum (d) Phase part of the spectrum

Figure 16.26 Spectrum of the discrete time series for a square wave with sixteen sampled points

Figure 16.27 Spectrum of the discrete time series for a square wave with thirty-two sampled points

These spectra have the following characteristics: (a) The spectra is periodic with periodN(this is also

observed in equation (16.20)). (b) The same spectra as those of the negative order n= -N/2, , -1 also

appear in the range n= N/2, , (N-1). (c) The spectra of the real part and those of the amplitude are

both symmetric about n=N/2. (d) The spectra of the imaginary part and those of the phase are skew-

symmetric about n=N/2. (e) The spectra in the left half of the zone n= 0, , (N-1) are valid. The

spectra in the right half are virtual and are too high compared to the sampling frequency (this is alsoobserved while discussing the aliasing effect in equation (16.23)).


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(ii) Thirty-two sampled points: The period of the signal is same as 8 sec, hence the fundamental

frequency would also be same as0 1/ 1/8 0.125f T= = = Hz. The sampling interval is now

/ 8 / 32 0.25t T N = = = sec. The Nyquist frequency would be ( )1/ 2 1/ 2 0.25 2cf t= = = Hz.

Hence, the maximum harmonics which will be valid is0/ 2 / 0.125 16cn f f= = = . Total number of

harmonics are same as number of sampled points, i.e., 32.

When the sampling interval is narrowed (from 0.25 sec to 0.125 sec) the number of spectra increases

(from 16 to 32) as shown in Figure 16.27, and therefore such a spectra diagram written in the interval

T/2 = (which remains constant since Tis constant) extends to the right (from 2 Hz to 4 Hz).

An envelope is shown by dashed line of the spectra in Figure 16.27. If the sampling frequency is

shortened continuously, the sampled data become substantially equal to the continuous wave, and

therefore its spectra will approach those of the Fourier series shown in right half of Figure 16.23(b).

The magnitude of 0X is 0.313 in Figure 16.26(a) and 0X is 0.281 in Figure 16.27. This value

approaches 0 0.25c = in Figure 16.23(b) as the number of data sampled increases.

Different types of definition of DFT and IDFT are used, depending upon personal preference. Some

use the following definitions, in which the magnitudes of nX coincide with that of Fourier transform

( )X in Figure 16.23(b).

1j2 /

0

Nnk N

n k

k

X t x e

=

= (n= 0, 1, 2, N-1) (16.26)

and1

j2 /

0

1 N nk Nn n

n

x X eT

=

= (k= 0, 1, 2, N-1) (16.27)

Some use following expressions, which have the coefficient 1/N in the counter-part expression:

(MATLAB uses this)

1j2 /

0

Nnk N

n k

k

X x e

=

= (n= 0, 1, 2, N-1) (16.28)

and

1j2 /

0

1 N nk Nn n

n

x X eN

=

= (n= 0, 1, 2, N-1) (16.29)


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Of course, every definition has the same function as mapping from time domain to frequency domain.

Especially, when we are interested in the critical frequencies at which amplitudes have peaks, then all

the definitions can be used equally well. However, we must be careful when we interpret the physical

meaning of the magnitude of the spectra. For example, for ttx sin)( = , it is equation (16.14) that

gives a spectrum with magnitude 1.

16.6 Fast Fourier Transform

The vast computational task necessary for DFT prevented its practical utilization. In 1965, Cooley and

Tukey proposed a computer algorithm that enabled the fast computation of DFT. The algorithm is

called Fast Fourier Transform (FFT), has made real-time spectrum analysis a practical tool. In the

calculation of the DFT given by equation (16.18), as

1j2 /

0

1 N nk Nn k

k

X x eN

=

= ; (n= 0, 1, 2, ,N-1)

If we were working out values ofnX by a direct approach we should have to make Nmultiplications

of the form ( ) ( )j2 /nk Nkx e

for each ofNvalues of nX and so the total work of calculating the full

sequence nX would require2N multiplications. However, the same calculation appears repeatedly

since the function ( ){ } ( ){j2 / cos 2 jsin 2nk Ne nk N nk N = has a periodic characteristic. The

FFT algorithm eliminated such repetition and allowed the DFT to be computed with significantly

fewer multiplications than direct evaluation of DFT. The FFT reduces this work to a number of

operation of the order of22 log ( ).N N The FFT algorithm has the restriction that the number of data

must be ( )2 1, 2, ,n n= . This allows certain symmetries to occur reducing the number of

calculations (especially multiplications) which have to be done. When the number of data N is

nN 2= , DFT needs 2N multiplications and FFT needs 2Nn multiplications, which is only 2 /n N

of the number of operations. For example, when11

2 2048= = , about 4,194,304 multiplications are

necessary in DFT and about 45,056 in FFT, which is only about 1/93rdof number of operations. IfN

increases this difference increases extremely large. The FFT therefore offers the added bonus of an

increase in accuracy. Since fewer operations have to be performed by the computer, round-off errors

due to the truncation of products are reduced, and the accuracy increases. If the length of the discrete

time series is not equal to ( )2 1, 2, ,n n= , then the discrete time series are zero padded (add

zeros in the series as discrete data) till it reaches up to the next 2nvalue. For example, if number of

discrete data are 1000, then we need to zero-pad 24 more data to make it to N= 1024 =10

2 . This will

not affect the accuracy of the FFT at all. For example, if we have 520 discrete data then instead ofzero-padding 1024-520 = 504 data, the original data may be truncated to next lower value of


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92 512= . This may introduce some error due to truncation of the original data, howver, if the discrete

time series is long enough then error introduced will be acceptably small. However, the state of the art

analog-to-digital equipments now digitizes data ( )2 1, 2, ,n n= in number. MATLAB has FFT

function name )( nxfft where 1}{ = nn xx .

Example of zero padding/truncation to be included.

16.7 Leakage Error and Countermeasures

(a) Original signal and measured range

(b) Assumed signal and its spectrum (case A)

(c ) Assumed signal and its spectrum (case B)

Figure 16.28 Leakage error

In FFT or DFT, computations are based on the assumption that the data sampled over a time

period are repeated before and after data measurement. Figure 16.28 shows the assumed signals

and their spectra for two types of measurement of a sinusoidal signal ttx sin)( = . Both cases

have 32 sampled data, but their sampling intervals are different. In case A, the sampling interval is

4 / 32 0.3927t = = and the range measured is exactly twice the fundamental period. The

Measured

Range case A AMeasured

Range case B B

1.0

-1.0

2 4 6 8 10 12 t

T

2 4 6 8 10

26

0 2 4 6 8k

10 12 16

2

4

6

T T

2 4 6 8 10

0

2

4

6

2 4 6 8 10 12 14 16


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computation of FFT or DFT is performed for the wave as shown by the dotted line (Figure

16.28(b). In this case the assumed wave is same as the original signal and therefore we get a

correct signal spectrum. In case B, the sampling interval is 5 / 32 0.4909t = = , and the range

measured is about 2.5 times the period of the original signal. In this case, the assumed wave

shown in Figure 16.28(c) is not smooth at the junction and differs from the original signal in time

domain. As a result, the magnitude of the correct spectrum decreases and spectra that do not exist

in the original signal appear. As seen in this example, if the time duration measured and the

period of the original signal do not coincide, the magnitude of the correct spectrum decreases and

spectra that do not exist in the original signal appear on both sides of the correct spectrum. This

phenomenon is called the leakage error.

Example 16.6 Illustrate the leakage error with the help of a simple sine wave. Take two full-

cycles as sampled signal and then take various sampled signals between two full-cycle and two-

and-half cycles.

Answer: Figure 16.29(a) show a continuous sine wave and its FFT. Figs. 16.29(b-f) shows

sampled signal of the continuous sine wave at various time length and corresponding FFT. It can

be seen that except 16.29(b) all others have leakage error. The leakage error is seen to increase

with the discontinuity at the junction point of two sampled signal for FFT (see 16.29(c-f)).

(a) Original continuous sinusoidal signal and its FFT

0 2pi 4pi 6pi 8pi

-1

-0.5

0

0.5

1

0 10 20 30 40 500

500

1000

1500

2000

Time, s Frequency

x(t)

FFT[x(t)]


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(b) Sampled sinusoidal signal (case 1) and its FFT

(c) Sampled sinusoidal signal (case 2) and its FFT

(d) Sampled sinusoidal signal (case 3) and its FFT

(e) Sampled sinusoidal signal (case 4) and its FFT

0 2pi 4pi 6pi

-1

-0.5

0

0.5

1

0 5 10 15 20 250

500

1000

1500

0 2pi 4pi 6pi 8pi

-1

-0.5

0

0.5

1

0 5 10 15 20 250

500

1000

1500

0 2pi 4pi 6pi 8pi

-1

-0.5

0

0.5

1

0 5 10 15 20 250

500

1000

1500

0 2pi 4pi 6pi 8pi

-1

-0.5

0

0.5

1

0 5 10 15 20 250

500

1000

1500

Time, s Frequency

Time, s

Time, s

Time, s

Frequency

Frequency

Frequency

FFT[x(t)]

x(t)

x(t)

x(t)

x(t)

FFT[x(t)]

FFT[x(t)]

FFT[x(t)]


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(f) Sampled sinusoidal signal (case 5) and its FFT

(g) Sampled sinusoidal signal (case 6) and its FFT

Figure 16.29 A continuous sine wave and its sampled signals with corresponding FFTs

(ii) Countermeasures for leakage error(Window Function):

There is a difficulty if the time signal is not periodic in the time record, especially at the edges of the

record (i.e., window). If the DFT or FFT could be made to ignore the ends and concentrate on the

middle of the time record, it is expected to get much closer to the correct signal spectrum in the

frequency domain. This may be achieved by a multiplication by a function that is zero at the ends of

the time record and large in the middle. This is known as windowing.

It should be realized that, the time record is tempered and perfect results shouldn't be expected. For

example, windowing reduces spectral leakage but does not totally eliminate it. It should also be noted

that, windowing is introduced to force the time record to be zero at the ends; therefore transient

signals which occur (starts and ends) inside this window do not require a window. They are called

self-windowed signals, and examples are impulses, shock responses, noise bursts, sine bursts, etc.

To decrease the leakage error due to discrepancy between the time duration measured and the period

of the original signal, we must connect the repeated wave smoothly. For this purpose we multiply a

weighting function that decrease gradually at both sides. This weighting function is called time

0 2pi 4pi 6pi 8pi

-1

-0.5

0

0.5

1

0 5 10 15 20 250

500

1000

1500

0 2pi 4pi 6pi 8pi

-1

-0.5

0

0.5

1

0 5 10 15 20 250

500

1000

1500

Time, s Frequency

Time s Fre uenc

x(t)

FF

T[x(t)]

FFT[x(t)]

x(t)


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window. Representative time windows: the Hanning window, Hamming window and Blackman-

Harris window are shown in Figure 16.30. These windows are defined in the range: 10 Nn as

Rectangular window: ( ) 1.0w n =

Hanning window: ( )( ) 0.5 0.5cos 2 / w n n N =

Hamming window ( )( ) 0.54 0.46cos 2 / w n n N =

Gaussian: ( ) ( )2

exp 0.5 2 / w n n N =

Blackman-Harris window: ( ) ( )( ) 0.423 0.498cos 2 / 0.0792cos 4 / w n n N n N = +

and outside 10


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Fig 16.31 With the Hanning window

Fig. 16.32 With the Hamming window

Fig. 16.33 With the Blackman-Harris window

0 5 10 15 20 25 30

-1

-0.5

0

0.5

1

0 5 10 15 20 25 30

- 1

-0.5

0

0.5

1

0 5 10 15 20 25 30

-1

-0.5

0

0.5

1

Time, s

Time, s

x(t)

x(t)

x(t)

w(n)

w(n)

w(n)

Time, s


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(a) Original signal

(b) The signal with the Hanning window

(c) The signal with the Hamming window

(d) The signal with the Blackman-Harris window

Fig. 16.34 Illustration of leakage error and application of various window functions on a same signal

0 2pi 4pi 6pi 8pi

-1

-0.5

0

0.5

1

0 5 10 15 20 250

500

1000

1500

0 2pi 4pi 6pi 8pi 10pi

-1

0

1

0 5 10 15 20 250

500

1000

1500

0 2pi 4pi 6pi 8pi 10pi

-1

0

1

0 5 10 15 20 250

500

1000

1500

0 2 i 4 i 6 i 8 i 10 i

-1

0

1

0 5 10 15 20 250

500

1000

1500

Time, s Frequency

x(t)

FF

T[x(t)]

Time, s

Time, s

Time, s

Frequency

Frequency

Frequency

x

(t)

x(t)

x(t)

FFT

[x(t)]

FFT[x(t)]

FFT[x(t)]


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For a discussion of their characteristics and the effects of these window functions, refer to some books

on signal processing (Bendat and Piersol, 2010).

(iii) Prevention of leakage by coinciding periods: As discussed above, we can obtain the correct result

if the time duration measured coincides with the integer multiple of the period of the original signal. If

we can attain this by some means, it is better than the use of window functions, which distorts the

original signal. For example, for numerical calculations that can be repeated in exactly the same way

and whose sampling interval can be adjusted freely, we can determine the measurement duration after

we know the period of the original signal by trial simulation, and then execute the actual numerical

simulation. On the contrary, for experiments, fine adjustment of sampling intervals is generally

impossible using practical measuring instruments. However, if the phenomenon appears within a

speed range, we can change the rotational speed little by little and adopt the best result where the

period, often determined by the rotational speed, and the sampling interval fit.

16.8 Applications of FFT to Rotor Vibrations

In the investigation of rotor vibration, we must know the direction of a whirling motion as well as its

angular velocity. In FFT (or DFT), elements of data sequence { }kx obtained by sampling are

considered as real numbers and those of data sequence { }nX obtained by discrete Fourier transform

are considered as complex numbers. In the following, we introduce a method that can distinguish

between whirling directions utilizing the revised FFT. In this FFT, rotor whirling motion is

represented by a complex number by overlapping the whirling plane on the complex plane and

applying FFT to these complex sampled data. Let us assume that a disc mounted on a elastic shaft is

whirling in the y-z-plane. We get sampled data { }ky and { }kz by measuring the deflections )(ty

and )(tz in the y and z directions respectively. Taking the y-axis as real axis and the z-axis as

imaginary axis, we overlap the whirling plane on the complex plane. Using sampled data ky and kz ,

we define the complex numbers as follows:

kkk jzyS += (k= 0, 1, 2,,N-1) (16.31)

and apply FFT (DFT) to them. We call such a method the complex-FFT method.


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Fig. 16.35(a) shows sub-harmonic vibration component of order in vibration signal. In

Figs.16.35(b) and (c), the same spectrum are shown with windowing and tuning of sampling interval,

respectively, for clarity of the frequency components. Fig. 16.36 shows the complex-FFT with distinct

forward and backward whirl frequency components.

(a)Spectrum without window (b ) Spectrum with window

(c) Spectrum obtained by adjusting sampling interval

Figure 16.35 Spectra of the sub-harmonic resonance of order of a forward whirling mode

Figure 16.36 Spectrum of the combination resonance (complex FFT method)

-2 -1 0 1 2Backward Forward

Subharmonic resonance

of order of a forward2/+ +

-2 -1 0 1 2

2/+ +

-2 -1 0 1 2

f

b

Backward Forward

-2 -1 0 1 2

2/+ +


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16.9 Properties of Random Discrete Signals

16.9.1 Probability, Probability Distribution Function, and Probability Density Function

A probability can be defined as simply the fraction of favourable events out of all possible events.

Probabilities are inherently non-negative; i.e., they can be only be positive or zero. Neither

assumption is strictly true in practice but both provide useful engineering solutions when they are

suitably interpreted. The measure of probability used is based on a scale such that the probability of

the occurrence of an event which cannot possibly occur is taken to be zero; the probability of

occurrence of an event which is absolutely certain to occur is taken to be unity. Any other event

clearly must have a probability between zero and unity. Suppose that we spin a coin: the probability

of the result heads H is equal to that of the result tails T, which will be equal to

Pr[ ] Pr[ ]H T= (16.32)

where Pr[ ] represents the probability. As the probability of either heads or tails, which will be equal

to Pr[ ] Pr[ ]H T+ , must be unity, i.e.,

Pr[ ] Pr[ ] 1H T+ = (16.33)

From equations (16.32) and (16.33), we get

1

2Pr[ ] Pr[ ]H T= = (16.34)

Similarly, the probability of throwing any given number with a symmetrical six-sided die would be

1/ 6 , since all numbers from 1 to 6 are equally probable and their total probability must be unity. The

probability of throwing an odd number with the die is 12

, since Pr[Odd] Pr[Even] 1+ = and

Pr[Odd] Pr[Even]= .

Now define a probability of throwing a given number by the die as

Pr[ ] ( ) 1/ 6N n p n= = = , 1 6n (16.35)

The probability of throwing the die thatNis odd

1 1 1 1Pr[ 1 or 3 or 5] (1) (3) (5)6 6 6 2

N p p p= = + + = + + = (16.36)


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which we obtained earlier also. Hence, it can be generalized as

1Pr[ ] ( ) ( )

n

N n p r P n = =

(16.37)

The quantities ( )p n and ( )P n provide alternative means of describing the distribution of probability

between the various possible values of n. Definition of ( )p n and ( )P n are given in equations (16.35)

and (16.37), respectively.

The expectation, ( )E N , is the expected result in any given trial, assumed to be equal to the mean

result of a very large number of trials. In the case of the die we can assume that in many throws all thenumbers from one to six will recur with equal frequency, so the expectation here is the mean of 1, 2,

3, 4, 5, 6, i.e.,

( ) (1 2 3 4 5 6) / 6 3.5E N = + + + + + = (16.38)

The above expectation can also be obtained as weighted mean of the numbers 1 to 6, as (with

( ) 1/ 6 and 6p n n= = ), i.e.,

6

1

( ) ( )E N np n= (16.39)

The above discussions of the coil and the die involves discrete values, however, the results obtained

can be extended for a continuous values also. Suppose we have a large number of displacement data

of different amplitudes. For this example, Pr[ ] 0X x= = , i.e., the probability that a chosen

displacement value X will be equal to a fixed displacement x will be zero. However, it will be

appropriate to use ( ) Pr[ ]P x X x= , i.e., the probability that the chosen displacement x is below a

certain displacement valueX. This quantity , ( )P x , is known as theprobability distribution function.

Figure 16.37 shows a typical displacement signal and Figure 16.38 shows the corresponding

probability distribution function.


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Figure 16.37 A typical random signal

Figure 16.38 The probability distribution function

Figure 16.39 The probability density


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As the probability can be added, they can also be subtracted. The probability of any randomly selected

displacement having a magnitude between two limits1x and 2x can be written as

1 2 2 1 2 1Pr[ ] Pr[ ] Pr[ ] ( ) ( )x X x X x X x P x P x = = (16.40)

The probability thatXlies betweenxand x dx+ can be written as

Pr[ ] Pr[ ] Pr[ ] ( ) ( ) ( )x X x dx X x dx X x P x dx P x dP x + = + = + = (16.41)

For dx to be infinitesimal, we have

( ) ( )dP x p x dx= or ( )( ) dP xp xdx

= (16.42)

where ( )p x is called the probability density function, which is an alternate way of describing the

probability distribution of a random variable, x. Figure 16.39 shows the probability density of the

random displacement signal shown in Figure 16.37. Equation (16.42) can be expressed as

( ) ( )x

P x p s ds

= (16.43)

Suppose we want to know the steady and fluctuating component of our signal. The steady component

is simply the mean value or expectation [ ( )]E x t of ( )x t . Noting equation (16.39), on similar analogy

we can write

[ ( )] ( )E x t xp x dx

= (16.44)

where square brackets indicate the ensembleaverageof the quantity. It can be seen that [ ( )]E x t is

given by the position of the centroid of the ( )p x diagram, since

( ) 1p x dx

= (16.45)


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16.9.2 Random Process, Ensemble, and Sample Function

The central notion involved in the concept of a random process ( )x t is that not just one time history

is described but the whole family or ensembleof possible time histories which might have been the

outcome of the same experiment are described. Any single individual time history belonging to the

ensemble is called a samplefunction. A random process can have several sample functions( )

( )jx t (

1,2, ,j n= ) defined in the same time interval (see Fig. 16.40). Each( ) ( )jx t is a sample function of

the ensemble. Hence, a random process can be thought of as an infinite ensemble of sample functions.

Figure 16.40 Ensemble of sample functions( )

( )jx t

16.9.3 Stationary and Ergodic Process

Often the random process is simplified and is assumed to be stationary process. This assumption isanalogous to the assumption of steady state forced response in deterministic vibration.

Displacement, ( )x t , at a particular location in test rig measured under identical conditions for n

number of times will have the following form:(1) (2) ( )

( ), ( ), , ( )mx t x t x t { }( )x t , which is called the

random process. These are the same physical quantities, however, they will not themselves be

identical but will have certain statistical properties in common. A random process is said to be

stationaryif its probability distributions are invariant under a shift of the time scale of the signal. For

example, the random process will be stationarywhen displacements(1) (2) ( )( ), ( ), , ( )m

n n nx t x t x t have


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probability distributions independent of time ( )1,2,nt n= . By ergodic we mean the probability

distribution of(1) (2) ( )( ), ( ), , ( )m

n n nx t x t x t at any one time is equal to the probability distribution of

any one displacement signal( )

( )j

x t with respect to time. Hence, an ergodic signal will be stationary,

however, a stationary signal will not necessarily be ergodic. That means ensemble and temporal

averages of the ergodic signal will be same, whereas, these averages are different for the stationary

process. It should be noted that for the ergodic process a signal will be enough to define the statistical

properties of the process completely.

For stationary process, in particular the probability density ( )p x becomes a universal distribution

independent of time. This implies that all the averages based on ( )p x (e.g., the mean [ ]E x and the

variance ( )

22 2

[ ]E x E x =

) are constants independent of time. The autocorrelation function is

defined as

[ ( ) ( )] ( )E x t x t R + = (16.46)

which is also independent of t and function of time lag 2 1t t = (for present case 2t t = + and 1t t=

). It should be noted that (0)R reduces to the mean square 2E x . In case x has zero mean, [ ] 0E x = ,

then the mean square is identical with the variance and 2(0)R = . The stationary assumptions can be

verified for experimental signals by calculating the mean and auto-correlation functions at different

times and checking for its invariance.

There are certain properties of autocorrelation function and its derivatives which are useful in

analysis. The autocorrelation function is an even function, since for stationary random process we can

write

( ) [ ( ) ( )] [ ( ) ( )] ( )R E x t x t E x t x t R = + = = (16.47)

Using a prime to indicate differentiating with respect to the contents of a bracket,

( ) [ ( ) ( )] [ ( ) ( )] ( )R E x t x t E x t x t R = + = = (16.48)

For 0 = , we have [ ( ) ( )] [ ( ) ( )]E x t x t E x t x t = , which is only true when both are zero. Hence,

(0) 0R = (16.49)


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Similarly,

( ) [ ( ) ( )] [ ( ) ( )] [ ( ) ( )]R E x t x t E x t x t E x t x t

= + = =

(16.50)

For 0 = , we have

2(0) [ ( ) ( )] [ ( )]R E x t x t E x t = = (16.51)

16.9.4 Probability Distribution and Probability Distribution Function

Many naturally occurring random vibrations have the Gaussian probability distribution, which is

defined as

2

2

( )

21

( )2

x m

p x e

= (16.52)

where mand represent mean and variance, which are constants for a particular random process.

Fig. 16.41 is a comparison of probability density with Gaussian approximation and the actual

distribution (without smoothening) from signal in Fig. 16.37. Fig. 16.42 shows the comparison of

probability density with Gaussian approximation and the actual distribution (with smoothening).

Figure 16.41 Gaussian probability density versus actual probability density without smoothening


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Figure 16.42 Gaussian probability density versus smoothened actual probability density

16.9.5 Ensemble Average, Temporal Average, Mean, Variance

The variance is defined as the mean-square value of the difference from the mean value. Thus

[ ] [ ] [ ]

[ ] [ ] [ ]

2 22 2

2 2 22 2

( ) ( ) ( ) 2 ( ) ( ) ( )

( ) 2 ( ) ( ) ( ) ( )

E x t E x t E x t x t E x t E x t

E x t E x t E x t E x t E x t

= = +

= + =

(16.53)

where2( )E x t is the square mean and [ ]

2( )E x t is the mean square. It should be noted that as

[ ]( )E x t is given by the first moment of area of the probability density curve about the ( )p x axis, so

2 which by equation (16.53) is the expectation of [ ]2

( ) ( )x t E x t is given by the second moment

of area about [ ( )]x E x t= , i.e.

[ ]22

[ ( )] ( )x E x t p x dx

= (16.54)

is the radius of gyration of the probability density curve about [ ( )]x E x t= . If the mean value is

zero the variance is given by


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2 2( )x p x dx

= (16.55)

To evaluate ensemble averages, it is necessary to have information about the probability distribution

of the samples or at least a large number of individual samples. Given a single sample ( )j

x of duration

Tit is, however, possible to obtain averages by averaging with respect to time along the sample. Such

an average is called a temporal average in contrast to the ensemble or statistical averages described

previously. The temporal mean of ( )x t

/ 2

/ 2

1( ) ( )

T

T

x t x t dtT

< >= (16.56)

and the temporal mean square is

/ 2

2 2

/ 2

1( ) ( )

T

T

x t x t dtT

< >= (16.57)

where the angular brackets represent the temporal average. When ( )x t is defined for all time

averages are evaluated by considering the limits as T . For such a function a temporal

autocorrelation ( ) can be defined

/ 2

/ 2

1( ) ( ) ( ) lim ( ) ( )

T

TT

x t x t x t x t dtT

=< + >= + (16.58)

When ( ) is defined for a finite interval, then similar expression can be used by carefully choosing

the limits. Note that when (0) reduces to the temporal mean square. Within the subclass of

stationary random process a further subclass known as ergodic process. An ergodic process is one for

which ensemble averages are equal to the corresponding temporal averages taken along any

representative sample function. Thus for an ergodic process ( )x t with samples( )

( )jx t we have

( )[ ] jE x x=< > and ( ) ( )R = (16.59)

An ergodic process is necessarily stationary since( )jx< > is a constant while [ ]E x is generally a

function of time1t t= at which the ensemble average is performed except in the case of a stationary


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process. A random process can, however, be stationary without being ergodic. Each sample of an

ergodic process must be completely representative of the entire process.

16.9.6 Auto-correlation Function and Covariance

The autocorrelationis defined as

1 2 1 2 1 2 1 2[ ( ) ( )] ( , )E x t x t x x p x x dx dx

=

The prefix autorefers to the fact that1 2x x represents a product of values on the same sample at two

instants. For fixed 1t and 2t this average is simply a constant; however, in subsequent applications 1t

and2t will be permitted to vary and the autocorrelation will in general be a function of both 1t and 2t .

In an important special case the autocorrelation function is a function only of 2 1t t = .

A related average, the covariance is obtained by averaging the product of the deviation from the

means at two instants. Thus, we have the covariance as

( )( ) ( )( ) ( )( ) ( )( ) ( )

[ ] [ ] [ ]

1 1 2 2 1 1 2 2 1 2 1 2

1 2 1 2

,E x E x x E x x E x x E x p x x dx dx

E x x E x E x

=

=

(16.60)

When 1x and 2x have zero means, the covariance is identical to the auto-correlation. When 1 2t t= , the

covariance becomes identical with the mean square.

A frequency decomposition of the ( )R can be made in the following way

j( ) ( )

tR S e d

= (16.61)

where ( )S is the Fourier transform of ( )R , except for the factor 2 . A physical meaning can be

given to ( )S by considering the limited case of equation (16.61) in which the time shift 0 = is

taken

2(0) ( )R E x S d

= = (16.62)


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The mean square of the process equals the sum over all frequencies of ( )S d so that ( )S can be

interpreted as a mean square spectral density. It should be noted that the dimensions of ( )S are

mean square per unit of circular frequency. Note that according to equation (16.62) both negative and

positive frequencies are considered, which is convenient

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