Wang X. Vehicle Noise and Vibration Refinement
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Random signal processing and spectrum analysis 103
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If x(t) is a periodic signal of period T, from Equation 5.29, this gives
a
T
xt t t
nn
T
T
=
(
)
−
∫
2
2
2
cos
ω
d
(5.30)
b
T
xt t t
nn
T
T
=
(
)
−
∫
2
2
2
sin
ω
d
(5.31)
caib
nnn
=−
(
)
1
2
(5.32)
c
T
xt t
n
nt
T
T
=
(
)
−
∫
−
1
2
2
1
ed
ω
(5.33)
Introducing
n
T
nn
ωω ω ω
11
2
===,
π
Δ
(5.34)
we have
xt
T
Tc Tc
n
n
t
n
t
n
n
nn
()
=
()
=
()
=−∞
∞
=−∞
∞
∑∑
11
2
ee
ii
ZZ
Z
π
Δ
(5.35)
Tc x t t
n
t
T
T
n
=
()
−
∫
−
2
2
ed
i
Z
(5.36)
XTcxtt
T
n
t
n
Z
Z
Z
()
=
()
=
()
→∞
→
−∞
∞
−
∫
lim
Δ 0
ed
i
(5.37)
xt Tc X
T
n
n
t
n
t
n
n
()
=
()
=
()
→∞
→
=−∞
∞
−∞
∞
∑
∫
lim
Δ
Δ
Z
ZZ
ZZZ
0
1
2
1
2ππ
eed
ii
(5.38)
5.3.1 Fourier transforms
The Fourier transform of a real-valued function x(t) extending from
−∞ < t < +∞ is a complex-valued function X(f ) defi ned by:
X f xt xt t xt t X
ft t
()
=
()
[]
=
()
=
()
=
()
−∞
∞
−
−∞
∞
−
∫∫
F ed ed
i.e. i i2π
Z
Z
(5.39)
Assuming this complex-valued X(f ) exists for all f over −∞ < f < +∞, the
inverse Fourier transform brings X(f ) back to x(t) as defi ned by:
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104 Vehicle noise and vibration refi nement
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xt Xf Xf f X
ft t
()
=
()
[]
=
()
=
()
−
−∞
∞
−∞
∞
∫∫
F
12
1
2
ed ed
iiπ
π
ZZ
Z
(5.40)
where
Xf X f X f
(
)
=
(
)
+
(
)
RI
i
(5.41)
The real part X
R
(f ) and the imaginary part X
I
(f ) are:
Xxt ftt
R
d=
(
)
−∞
∞
∫
cos2π
(5.42)
Xxt ftt
I
d=
(
)
−∞
∞
∫
sin2π
(5.43)
The magnitude |X(f )| and phase φ
x
(f ) are given by:
Xf Xf Xf f f
x
f
xx
(
)
=
(
)
=
(
)
(
)
+
(
)
(
)
+
(
)
ei
i
φ
φφ
cos sin
(5.44)
Therefore:
Xf Xf f
xR
(
)
=
(
)
(
)
(
)
cos
φ
(5.45)
Xf Xf f
xI
(
)
=
(
)
(
)
(
)
sin
φ
(5.46)
Xf X f X f
(
)
=
(
)
+
(
)
[]
RI
22
1
2
(5.47)
φ
x
f
Xf
Xf
(
)
=
(
)
(
)
⎡
⎣
⎢
⎤
⎦
⎥
−
tan
1
I
R
(5.48)
Fourier transforms of several conventional time domain functions are
shown in Table 5.1. A number of properties of Fourier transforms are given
Table 5.1 Fourier transforms
x(t) x(f )
cos 2πf
0
t
1
2
00
[( ) ( )]
δδ
ff ff−+ +
sin 2πf
0
t
1
2
00
i
[( ) ( )]
δδ
ff ff−− +
sint
t
π
ππ
0
−≤≤
(
)
⎧
⎨
⎪
⎩
⎪
1
2
1
2
f
otherwise
1
1
2
+t
π
π
π
π
e
e
−
>
<
{
2
2
0
0
f
f
f
f
e
−c|t|
cos 2πf
0
t
c
cff
c
cff
22
0
22 2
0
2
44+−
+
++ππ() ()
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Random signal processing and spectrum analysis 105
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in Table 5.2, where X(f ) =
F
[x(t)], Y(f ) =
F
[y(t)] are Fourier transforms of
x(t) and y(t).
5.3.2 Digital FFT analysis
In order to implement the above Fourier transform, a quality data acquisi-
tion system is required for signal time recording. Ahead of FFT analysis,
the time signal is subdivided into n time segments of length Δt = n/f
S
(f
S
= sampling frequency) as shown in Fig. 5.9. In order to avoid signal alias,
fi lters and windows will have to be applied to the sampled data. An FFT
analysis is conducted from the digital integration of Equations 5.39 and 5.40
using fast computing algorithms for each time signal segment. The number
of samples per time segment can then be entered as spectrum size input. If
there is more than one time segment (as is normally the case), the individual
partial analyses are averaged ahead of representation. The calculated FFT
spectrum for side mirror power fold actuator noise of a passenger car is
shown in Plate III (between pages 114 and 115).
Table 5.2 Fourier transform properties
Operation Property
Linear
F
[ax(t) + by(t)] = aX(f ) + bY(f ) for any functions x(t), y(t)
and constants a and b
Shift
F
[x(t − a)] = e
−2πifat
X(f )
Fourier transform
of Fourier
transform
F
[X(f )] = x(−t)
Inverse Fourier
transform of X(f )
xt Xf Xf f
f
() [ ()] ()==
−
−∞
∞
∫
F
12
ed
itπ
Odd and even
functions
X(f ) = X
R
(f ) + iX
I
(f ); x(t) is even ⇔ X(f ) = X
R
(f ) is even;
x(t) is odd ⇔ X(f ) = jX
I
(f ) is odd
Similarity
F[( )]xat
a
X
f
a
=
(
)
1
Energy
xt t Xf f xtyt t XYf f
2
2
() () , () () ()dd d d
−∞
∞
−∞
∞
−∞
∞
−∞
∞
∫∫ ∫ ∫
==∗
Autocorrelation
If d thenRt xuxu t u Rt Xf() ( ) ( ) [ ()] ()=+ =
−∞
∞
∫
F
2
Product
F x t y t X f Y f Xf Yf XuYf u u
() ()
[]
=
()
∗
()
∗= −
−∞
∞
∫
where d() () ( ) ( )
Convolution
xt yt xuyt u u xt yt XfYf() () ( ) ( ) [ () ()] () (
)
∗= − ∗ =
−∞
∞
∫
dwhereF
Parseval’s theorem
xtxt t XfXf f x fXf f
12 1 2 1 2
() () () () () ()d*d
*
d
−∞
∞
−∞
∞
−∞
∞
∫∫ ∫
==
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106 Vehicle noise and vibration refi nement
© Woodhead Publishing Limited, 2010
From the averaged FFT spectrum results, the constant percentage band-
width fi lters defi ned in Table 4.5 and Equation 4.13 in Chapter 4 can be
used to calculate the constant percentage bandwidth spectrum as shown in
Plate IV (between pages 114 and 115).
If a signal process is a non-stationary or transition process, a 3-D spec-
trum map of FFT versus time will have to be used to present the measured
spectrum in both time and frequency domains. In ‘standard’ FFT analysis,
individual partial analyses (spectrum size) are averaged over the entire
signal curve. In the analysis versus time, this averaging is not carried out;
the analysis result for each time window is graphically displayed. This
requires an additional axis. Since the fl at monitor screen or paper page
allows for only two axes, the third axis (the Z-axis) is represented with the
aid of various colours. The three axes are (a) the X-axis (previously the
frequency axis), now the time axis; (b) the Y-axis (previously the level axis),
now the frequency axis; and (c) the Z-axis, which represents level in the
time and frequency domains via different colours as shown in Plate V
(between pages 114 and 115).
5.4 Spectral density analysis
For stationary random data, the following expected values of Fourier trans-
form quantities can defi ne the two-sided auto-spectral density functions
S
xx
(f ) and S
yy
(f ) and the two-sided cross-spectral density function S
xy
(f ):
Sf
T
EXf
xx
(
)
=
(
)
⎡
⎣
⎤
⎦
1
2
(5.49)
Sf
T
EYf
yy
(
)
=
(
)
⎡
⎣
⎤
⎦
1
2
(5.50)
Sf
T
EX fY f
xy
(
)
=
(
)
(
)
[]
1
*
(5.51)
where E[ ] denotes an expected value ensemble average over the quantities
inside the brackets. The X(f ) and Y(f ) are infi nite Fourier transforms of
ΔtΔtΔt
t
L
5.9 Time records for digital FFT analysis.
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Random signal processing and spectrum analysis 107
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length T for n
d
distinct sample records and the frequency f can theoretically
be any value in −∞ < f < ∞. For digitized data f is restricted to discrete values
spaced 1/T apart in a bounded frequency range, and Fourier transform
formulas become Fourier series formulas.
The spectral density functions S
xx
(f ) and S
yy
(f ) are positive, real-valued
even functions of f; the cross-spectral is a complex-valued function of f. If
x(t) and y(t) are measured in volts, S
xx
(f ) and S
yy
(f ) will have units of volts
2
per hertz, while t has units of seconds. S
xx
(f ), S
yy
(f ) and S
xy
(f ) always exist
for fi nite-length records.
For stationary random data, the one-sided measurable auto-spectral
density functions and cross-spectral density function are zero for f < 0 and
for f ≥ 0 defi ned by:
Gf
T
EXf
xx
(
)
=
(
)
⎡
⎣
⎤
⎦
2
2
(5.52)
Gf
T
EYf
yy
(
)
=
(
)
⎡
⎣
⎤
⎦
2
2
(5.53)
Gf
T
EX fY f
xy
(
)
=
(
)
(
)
[]
2
*
(5.54)
Note that:
Gf Sf f
Gf f
xy xy
xy
(
)
=
(
)
≥
(
)
=<
⎧
⎨
⎩
20
00
(5.55)
which includes G
xx
(f ) and G
yy
(f ) as special cases.
For f ≥ 0 the following relationships hold:
Hf
Sf
Sf
Gf
Gf
xy
xy
xx
xy
xx
(
)
=
(
)
(
)
=
(
)
(
)
(5.56)
Hf
Sf
Sf
Gf
Gf
xy
yy
xx
yy
xx
(
)
=
(
)
(
)
=
(
)
(
)
2
(5.57)
γ
xy
xy
xx yy
xy
xx yy
Sf
SfSf
Gf
GfGf
2
22
==
()
() ()
()
() ()
(5.58)
where H
xy
is the frequency response function between variables x and y,
and γ
2
xy
is the coherence function between two stationary random processes
x(t) and y(t). As proved in the works in the bibliography (Section 5.10), this
ordinary coherence function for all f is a dimensionless constant that is
bounded between zero and one, namely:
01
2
≤
(
)
≤
γ
xy
f
(5.59)
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108 Vehicle noise and vibration refi nement
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The coherence function of Equation 5.59 is a causal relationship unless one
has other physical grounds for knowing that an input x(t) produces an
output y(t). For example, x(t) could be excitation force and y(t) could be
vibration response. If a perfect linear relationship exists between x(t) and
y(t) at some frequency f
0
, then the coherence function γ
2
xy
(f
0
) will be unity
at the frequency. If x(t) and y(t) are such that S
xy
(f
0
) = 0 at frequency f
0
,
then the coherence function will be zero at that frequency, and x(t) is not
correlated to y(t).
There are four main reasons in practice why a computed coherence func-
tion between a measured input x(t) and a measured output y(t) will differ
from unity. These are:
• Extraneous noise in the input and output measurements
• Bias and random errors in the spectral density function estimates
• Output y(t) being due in part to other inputs as well as the measured
input x(t)
• Non-linear system operations existing between x(t) and y(t).
If the fi rst three possibilities are ruled out by good data acquisition, proper
data processing and physical understanding, it is reasonable to conclude
that low coherence at particular frequencies is due to non-linear system
effects at these frequencies. Thus the ordinary coherence function can
provide a simple practical way to detect non-linearity without identifying
its precise nature.
Table 5.3 shows the basic formulas of two- and one-sided spectral density
functions for stationary random data. Three frequently encountered signals
such as sine wave, wideband random signal and narrowband random signal
and their conversions for time traces, the cumulative probability distribu-
tion, the probability density distribution, autocorrelation and the auto-
spectrum are compared and plotted in Fig. 5.6 where the relationship and
trend of the signals changing with bandwidth can be seen.
Table 5.3 Basic spectral density functions, stationary random data
Two-sided spectra
Sf
T
EX fYf f
xy
( ) [ ( ) ( )],=−∞<<∞
1
*
Sf
T
EX f Xf
T
EXf
xx
() [ () ()] [ () ]==
11
2
*
One-sided spectra
Gf
T
EX fYf f
xy
() () (),=
[]
≥
2
0* only
Gf
T
EX f Xf
T
EXf
xx
() [ () ()] [ () ]==
22
2
*
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5.5 Relationship between correlation functions and
spectral density functions
Comparing Equations 5.49 – 5.51 with Equations 5.24 and 5.26 and applying
Fourier transform relationships give the conclusion that the relationship
between the correlation function and the spectral density function is a
Fourier transform pair. That is:
RS
xy xy
τωω
ωτ
(
)
=
(
)
−∞
∞
∫
1
2π
ed
i
(5.60)
SR
xy xy
ωττ
ωτ
(
)
=
(
)
−∞
∞
−
∫
ed
(5.61)
F RSSf
xy xy xy
τω
(
)
[]
=
(
)
=
(
)
(5.62)
FF
−−
(
)
[]
=
(
)
[]
=
(
)
11
Sf S R
xy xy xy
ωτ
(5.63)
When τ = 0, x = y and
RSSff
xx xx xx
0
1
2
2
(
)
==
(
)
=
(
)
−∞
∞
−∞
∞
∫∫
σωω
π
dd
(5.64)
5.6 Linear systems
Ideal physical systems should be linear, physically realizable, have constant
parameters and be stable. Properties of ideal physical systems are shown
in Table 5.4. A system H is a linear system if, for any inputs X
1
= X
1
(t) and
X
2
= X
2
(t) and for any constants c
1
and c
2
:
HcX cX cHX cHX
11 22 1 1 2 2
+
[]
=
[]
+
[]
(5.65)
The additive property is
HX X HX HX
12 1 2
+
[]
=
[]
+
[]
(5.66)
Table 5.4 Properties of ideal physical systems
System property Requirement Benefi t
Linear Additive and homogeneous Output stays Gaussian
Physically realizable h(τ) = 0 for τ < 0 Output follows input
Constant-parameter h(t, τ) = h(τ) Output stays stationary
Stable
h()
ττ
d
−∞
∞
∫
<∞
Output stays bonded
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110 Vehicle noise and vibration refi nement
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The homogeneous property is
HcX cHX
00
[]
=
[]
(5.67)
where c
0
is a constant.
For a constant-parameter linear system:
y
tHxt+
(
)
=+
(
)
[]
ττ
(5.68)
for any τ.
For a second-order dynamic system:
Hm
t
c
t
k=++
d
d
d
d
2
2
(5.69)
Hx Ft
[]
=
(
)
(5.70)
where m is mass, c is damping, k is stiffness and F(t) is excitation force.
If two excitations F
1
(t) and F
2
(t) are considered, the responses x
1
(t) and
x
2
(t) are given by
Hx F t Hx F t
11 22
[]
=
(
)
[]
=
(
)
,
(5.71)
If the excitation F
3
(t) is a linear combination of F
1
(t) and F
2
(t), namely
F
CF t CF t
311 22
=
(
)
+
(
)
(5.72)
and if the response x
3
(t) to the excitation F
3
(t) satisfi es the relation
xt Cxt Cxt
31122
(
)
=+() ()
(5.73)
Hxt HCxtCxt Ft CFtCFt
CH x
3112231122
11
(
)
[]
=
(
)
+
(
)
[]
=
(
)
=
(
)
+
(
)
=
[]
+CCH x
22
[]
(5.74)
and the system satisfi es the superposition principle, and is a linear system.
Classifi cation of systems is shown in Fig. 5.10.
System
Linear
Constant-
parameter
Time-varying
Special types of non-linear
systems
Non-linear
5.10 Classifi cations of systems.
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5.7 Weighting functions
The unit impulse delta function is defi ned by
δ
δ
()
()
ta t a
ta
−= ≠
=∞ =
{
0
0
(5.75)
δ
tat−
(
)
=
−∞
∞
∫
d1
(5.76)
If δ(t − a) is multiplied by any time function f(t), as shown in Fig. 5.11, the
product will be zero everywhere except at t = a, and its time integral will
be:
ft t a t fa a
(
)
−
(
)
=
(
)
<<∞
∞
∫
δ
0
0d
(5.77)
Since
ˆ
F
= Fdt = mdv, the impulse
ˆ
F
acting on the mass m will result in a
sudden change in its velocity equal to
ˆ
F
/m without an appreciable change
in its displacement.
According to the free non-damped spring–mass system equation with
initial conditions x(0) and
x
(0),
x
x
tx t
n
nn
=
(
)
+
(
)
0
0
ω
ωω
sin cos
(5.78)
where ω
n
is a natural frequency of the system. The response of a mass–
spring system initially at rest and excited by an impulse
ˆ
F
is:
x
F
m
tFht
n
n
==
ˆ
sin
ˆ
()
ω
ω
(5.79)
a
t
F
F
ε
ε
5.11 Unit impulse delta function.
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where
ht
m
t
n
n
(
)
=
1
ω
ω
sin
(5.80)
which is the response to a unit impulse.
When damping is present:
x
F
m
t
n
t
n
n
=
−
−
−
ˆ
sin
e
ωξ
ωξ
ξω
1
1
2
2
(5.81)
where ξ is the relative damping ratio.
xFht=
(
)
ˆ
(5.82)
where
ht
m
t
n
t
n
n
(
)
=
−
−
−
e
ωξ
ωξ
ξω
1
1
2
2
sin
(5.83)
which is the impulse response function.
For arbitrary force to be considered as a series of impulses as shown in
Fig. 5.12, the strength of one of the impulses (shown crosshatched) at time
t = a is:
ˆ
Ffaa=
(
)
Δ
(5.84)
a = t
a = t
Δaa
a
a
a
t
p
f(a)
f(a)Δa
f(a)Δa h(t – a)
t – a
5.12 Excitation force and response.
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