AP01_6.vp
1 Envelope analysis
Envelope analysis is deeply connected to the Hilbert
transform [1], [5]. The Hilbert transform of signal x(t) is
defined by the following equation
� � � �� �
� �~x t H x t
x
t
� �
�
� �
�
1
�
�
�
�d (1)
where � �~x t is Hilbert image of signal x(t), also refered to
as the quadrature part to signal x(t).
The inverse Hilbert transform is defined by
� � � �� � � �x t H x t x
t
� � �
�
�
� �
�
1
1~
~
�
�
�
�d . (2)
Using the definition of a convolution we can define the
Hilbert transform by
� � � �~x t
t
x t�
1
�
(3)
� � � �x t
t
x t� �
1
�
~ . (4)
The complex signal �(t) whose imaginary part � �~x t is
the Hilbert transform of the real part x(t) is called the analytic
signal.
� � � � � �� t x t j x t� � ~ (5)
where � � � �� �~x t H x t� .
The analytical signal as a complex function in the time
domain can be expressed as a complex function in Euler form
by
� � � � � �� �t E t e j t� � , (6)
where is E(t) is the amplitude of a complex function in the
time domain,
�(t) is the phase shift of a complex function in the
time domain.
The following equations are valid for E(t) and �(t):
� � � � � �E t x t x t� �2 2~ (7)
� �
� �
� �
� t
x t
x t
� atan
~
. (8)
Function E(t) is called a signal x(t) envelope.
The following equations are valid for the Hilbert trans-
form of a harmonic signal:
� �� � � �H t tcos sin� � � �� � � (9)
� �� � � �H t tsin cos� � � �� � � � . (10)
24 © Czech Technical University Publishing House http://ctn.cvut.cz/ap/
Acta Polytechnica Vol. 41 No. 6/2001
Parasitic Events in Envelope Analysis
J. Doubek, M. Kreidl
Envelope analysis allows fast fault location of individual gearboxes and parts of bearings by repetition frequency determination of
the mechanical catch of an amplitude-modulated signal. Systematic faults arise when using envelope analysis on a signal with strong
changes. The source of these events is the range of function definition of 1 �t used in convolution integral definition. This integral is
used for Hilbert image calculation of analyzed signal. Overshoots (almost similar to Gibbs events on a synthetic signal using the Fourier
series) are result from these faults. Overshoots are caused by parasitic spectral lines in the frequency domain, which can produce faulty
diagnostic analysis.
This paper describes systematic arising during faults rising by signal numerical calculation using envelope analysis with Hilbert transform.
It goes on to offer a mathematical analysis of these systematic faults.
Keywords: gearbox, bearing, envelope analysis, Hilbert transform, parasitic spectral lines.
)t̂xt � )(t�
)()}(Re{ txt ��
0
0.5
1
txt ~Im ��
txt ��Re
0.5
1
1.5
U [V]
)(~ tx
)(tx
a)
b)
Envelope ( ) = 1 VE t
Fig. 1: Analytical signal and envelope of � � � �x t t� �sin 2 10� � ,
� � � �~ cos ,x t t� � �2 10 0� � �
These equations show that the envelope E(t) of the har-
monic signal x(t) = sin (�t) is equal to:
� � � � � � � � � �E t x t x t t t� � � � �2 2 2 2 1~ sin cos� � . (11)
The analytical signal and envelope is shown in Fig. 1.
2 Envelope analysis faults
Fig. 1 shows a part of harmonic signal � �x t , its analytically
calculated quadrature part � �~x t and envelope E(t). Figure 1a
shows analytical signal �( t) in complex space. Figure 1b
shows the same signal in two dimensions.
2.1 The edge measurement interval effect on
a harmonic signal envelope
The envelope of a harmonic signal is shown in Figure 2a.
The quadrature part was obtained by a numerical calculation
of the analytical signal in comparison to an analytical cal-
culation (see Figure 1b). The envelope distortion at the edge
of the measurement interval is shown in Figure 2a. Figure 2b
shows the same signal but with a phase shift of � �� 2. Figure
2a shows a 50 % deflection of the envelope in comparison to
the ideal. Figure 2b shows a 70 % deflection of the envelope in
comparison to the ideal.
2.2 The step change effect on a harmonic
signal envelope
The envelope of harmonic signal modulated by a square
pulse is shown in Figure 3a. The step is in the time when the
harmonic signal is crossing zero (phase �sk = 0), and the
quadrature part is numerically calculated. Figure 3b shows the
same signal but the step is at the time when the harmonic
signal reaches the maximum (phase � �sk � 2).
Figure 3 shows almost the same form of envelope as
Figure 2. The overshoots produced by the step changes are
smaller than the overshoot produced by the edge of the
measurement interval.
2.3 Envelope analysis errors recapitulation
Figure 1 to Figure 3 show the different form of the
envelope if it is calculated analytically or numerically at points
of step change and at points of the edge of the measurement
interval. The overshoot is theoretically infinite at the point of
the step change (see Chapter 3.3). Calculation in discrete time
only approximates this one. The signal in Figure 2b has an
overshoot of 70 % of the original envelope.
© Czech Technical University Publishing House http://ctn.cvut.cz/ap/ 25
Acta Polytechnica Vol. 41 No. 6/2001
)(~ tx
)(tx
)(tx
)(~ tx
a)
b)
0.5 V – 50%�
1.7 V –
Fig. 2: Envelope distortion on the edge of the measurement in-
terval, � � � �x t t� �sin 2 10� � , sampling frequency 5 kHz
a) for � = 0, b) for � �� 2
)(tx
)(~ tx
)(tx
)(~ tx
a)
b)
Fig. 3: Envelope distortion on step change of harmonic sign-
al, sampling frequency 5 kHz, � = 0
a) � � � � � � � �� ��
x t t sign t sign t� � � � � �sin . . .2 10 1 0 5 0 3 0 7� �
b) � � � � � � � �� ��
x t t sign t sign t� � � � � �sin . . .2 10 1 0 5 0 35 0 75� �
Figure 4 is a recapitulation of the systematic errors and
a recapitulation of the theoretical values of the overshoots
during the step change. The recapitulation is done for the
harmonic signal � � � � � �x t E t t� �sin 2 10� � for the phase shift
� � 0 and for phase shift � �� 2. The step change from
� �E t � 0 to � �E t � 1 is in time t � 0.
The following lines show recapitulate the errors from
Fig. 4:
a) for � � 0 and step change at time t � 0 the envelope of sig-
nal x(t) has:
�the first maximum overshoot (point p1) 50 %, at time
t = 0 ms
�the second maximum overshoot (point m1) 9.06 %, in
time t = 0.045 ms, i.e., 0.45 of the signal,
�stabilization (ripple < 1 %) (point u1), at time
t = 0.405 ms, i.e., 4.05 of the harmonic signal period.
b) for � �� 2 and step change at time t � 0 the envelope of
signal x(t) has:
�the first maximum overshoot (point p2) �, at time
t = 0 ms,
�the second maximum overshoot (point m2) 9.44 %, at
time t = 0.012 ms, i.e., 0.12 of the signal period,
�stabilization (ripple < 1 %) (point u2), at time
t = 0.080 ms, i.e., 0.80 of the harmonic signal period.
An analytical calculation of the theoretical overshoot
values of points p1 and p2 is given in Section 3.3.
3 Analytical calculation of envelope
overshoots
This section gives a mathematical analysis of envelope
behavior (parasitic overshoots) based on discovering the “non
standard” behavior of the envelope (described in the previous
chapter) calculated by the Hilbert transform in areas of step
changes.
3.1. Reason for envelope overshoots
As was said in the introduction, the reason for these over-
shoots is the convolution function of the Hilbert transform
1 �t. This function for t � 0 reaches as very high values, and
for t = 0 it is not defined. Thus it is necessary for the integral
of the Hilbert transform
� �
� �~x t
x
t
�
�
� �
�
1
�
�
�
�d (12)
to be calculated as the values of the intrinsic integral accord-
ing to the equation
� �
� � � �~ lim limx t
x
t
x
t
t
t
�
�
�
���
� �
�
��
�
�
�
�
�
�
�
�
�
�
�
�
�
�
0 0
1 1
d d . (13)
The value of the integral is given by summation of limits.
The limits calculated from the left and right side have a high
difference when the calculation is made at the point of the
step change. This is the source of the overshoots in the areas
with step changes. The areas of step changes are also the
beginning and the end of measurement signal.
3.2 Overshoot amplitude calculation of
a trapezoidal pulse
Envelope overshoot behavior is mathematically analyzed
on the basis of the example of the impulse combined from
a trapezoidal and square pulse (further trapezoidal). This
signal can simulate more types of signals by parameter
changes. The trapezoidal signal is shown in Figure 5.
The function shown in Figure 5 can be expressed as:
� � � �x t
t t t t
e t t t t t t
g t t t
g t t
t t
t
�
�
� � � �
�
�
0 1 8
1 3 6 8
4 5
2
4 2
3
a
a
� �
t t
g t t
t t
t t t
�
�
�
�
�
�
�
�
��
�
�
�
�
�
4
7
7 5
5 6
(14)
26 © Czech Technical University Publishing House http://ctn.cvut.cz/ap/
Acta Polytechnica Vol. 41 No. 6/2001
0
50
m1
u2
envelope for
envelope for
[%]
2
�
� �
0��
2
Fig. 4: Envelope error evaluation of harmonic signal for a step
change in time t � 0
e
g
x t( )
Fig. 5: The pulse combined from a trapezoidal and square pulse
Using equation (13) for the quadrature part of trapezoidal pulse calculation
� �~ limx t
e
t
g
t t
t
t
t
t
t
t
� �
�
�
�
�
��
�
�
�
�
�
�
�
� �
�
�
�
�
0 4 2
21
1
3
3
4
d d
�
�
�
�
�
� � � �
�
�
�
�
�
�
�
�
�
�
�
�
�
�
�
g
t
g
t t
t
t
e
t
t
t
t
t
d d d
4
5
5
7
7 5
7 �
�
�
t
t
6
8
�
�
�
�
�
�
�
�
�
�
�
�
(15)
Figure 6 illustrates the quadrature part � �~x t and envelope
� � � � � �E t x t x t� �2 2~ of signal � �x t calculated by Equation
(17), defined by Equation (14). Parameter e
0 was chosen to
provide an easier view of the quadrature part and envelope.
The dependency of the overshoot size (Figure 6) on the
rising edge gradient is shown in Figure 7. This gradient
is defined by parameter s. The dependency is determined
numerically based, on any analytical expression of the quad-
rature part (17). The dependency cannot be determined ana-
lytically due to the complexity of Equation (17). Finding the
local extreme by using the first derivative and then getting
the overshoot dependency is very difficult.
It is evident from the graph that there is an overshoot am-
plitude of 50 % for gradient s = 46,2° of the rising edge. It is
derived in [2] that the quadrature part of a square pulse
(a trapezoidal pulse with a 90° gradient) reaches an infinite
value at this point. The difference between the theoretical am-
plitude (�) and the amplitude given numerically (250 %, see
Figure 7) is due to numerical calculation. Numerical
calculation is not able to simulate a trapezoidal pulse with
a 90° gradient according to the finite distance of two following
time samples. The maximum gradient that can be found by
numerical calculation is given by
� ��
s g e fmax � �
180
�
atan vz (18)
where fvz is the sampling frequency.
It is possible to reach the gradient smax = 89.942° for
fvz = 1 kHz and g � e = 1, smax = 89.999° for fvz = 1 MHz.
Further increasing of the sampling frequency only very
slowly reaches the theoretical infinite amplitude.
3.3 Overshoot amplitude calculation of
a harmonic signal modulated by
a trapezoidal pulse
The following section describes the same analytical calcu-
lation as in the previous chapter, but for a harmonic signal
modulated by the trapezoidal pulse shown in Figure 8.
The function in Figure 8 can be expressed as:
� �
� �
� �
x t
t t t t
e t t t t t t t
g t t t t
�
� �
� � � � �
� � �
0 1 8
1 3 6 8
4
a
acos
cos
� �
� �
� �
� �
� �
� �
5
2
4 2
3 4
7
7 5
5 6
g t t
t t
t t t t
g t t
t t
t t t t
�
�
� � �
�
�
�
� � �
cos
cos
� �
� �
�
�
�
�
��
�
�
�
�
�
(19)
27
Acta Polytechnica Vol. 41 No. 6/2001
� �~ limx t
e
t
g
t t
t
t
t
t
t
t
� �
�
�
� ��
�
�
�
�
� �
�
�
�
0 4 2
21
1
3
3
4
d d
�
� �
�
�
�
�
�
�
�
�
�
�
�
�
�
�
g
t t t
g
t
g
t
t
t
t
t
4 2
4
4
7
3
5
�
�
�
�
�
d
d
t t
g
t t
t
t
e
t
t
t
t
t
t
5 7 5
7
5
6
5
6
6
�
�
�
�
�
�
�
�
�
� �
�
�
�
�
�
�
�
d d d
t 8 �
�
�
�
�
�
.
(16)
Substituting for the integral yields (16)
� �~ ln ln lnx t e
t t
t t
e
t t
t t
gt
t t
t t
t
� �
�
�
�
�
�
�
�
�
�
�
�1 3
1
8
6
2
4 2
4
3� �
�
�
�
�
�
�
�
�
�
�
� �
t
gt
t t
t t
t t
g
t t
t t
g
t t
t t t
7
7 5
6
5
5
4
4 2
4 3
ln ln
ln
t t
t t
g
t t
t t t
t t
t t
4
3 7 5
7 5
6
5
�
�
�
�
�
�
�
�
�
� � �
� �
�
�
�
�
�
�
�
�
�
�
�
�ln
�
.
(17)
)(tx
)(
~
tx
)(tE
t
Fig. 6: Trapezoidal pulse given by (14), quadrature part � �~x t
and envelope E(t) with parameters e = 0; g = 1; t3 = �6.5 s;
t4 = �5.5 s; t5 = 5.5 s; t6 = 6.5 s; fvz = 1 kHz
150
200
250
%
10 20 30 40 50 60 70 80 90
Fig. 7: Normalized graph of overshoot dependency on the gradi-
ent s of a trapezoidal pulse rising edge.
Parameters � � � ��
s g e t t� � �180 6 5atan �, where e � 0,
g � 1, t6 6� s, � �t t6 5 0 5� � s s,
e
g
x (t)
Fig. 8: Harmonic signal modulated by a trapezoidal pulse
3.3.1 Invariant part of a harmonic signal
First we make a calculation of the invariant part of the
harmonic signal, because it is easier to describe the calculat-
ion and because the Hilbert transform is linear according to
addition. The invariant part of the harmonic signal modul-
ated by the trapezoidal pulse of the signal in Fig. 8 is between
points t1 and t2. The invariant part of the signal is shown in
Fig. 9.
The function shown in Fig. 9 can be expressed as:
� �
� �
x t
t t t t
t t t t
�
� � � �
� �
�
cos ,
,
� � �0 1 2 0
1 2
0
0
pro
pro
(20)
The use of Equation (13) for calculating the quadrature
part of function (20) yields
� �
� �~ cosx t
t
t
t
� �
�
��
1 0
1
2
�
� � �
�
�d . (21)
Substituting
z t
z
z t t
z t t
� �
�
� �
� �
�
�d d
1 1
2 2
(22)
Equation (21) can be written as:
� �
� �~ cosx t
z t
z
z
z
z
� �
�
�
1 0 0
1
2
�
� � � �
d . (23)
28
Acta Polytechnica Vol. 41 No. 6/2001
1
x t( )
0
�1
Fig. 9: Invariant part of the harmonic signal
Equation (23) can be decomposed by equation (24) to equation (25)
� � � � � � � � � �cos cos cos sin sin� � � � � � (24)
and thus
� � � �
� �
� �
� �~ cos
cos
sin
sin
x t t
z
z
z t
z
z
z
z
z
z
z
� � � � ��
1
0
0
0
0
1
2
1
�
� �
�
� �
�
d d
2
�
�
�
�
�
�
�
�
�
�
. (25)
The integral calculation inside Equation (25) has to be done separately according to a valid condition of integration [3]. The
properties of odd and even functions, their integrals and integral sine and cosine are (27) in [2].
� for t t z1 1 0� �, and for t t z2 2 0� �,
� � � �
� � � �~ cos
cos cos
x t t
z
z
z
z
z
z
z z
� � � �
�
�
�
�
�
�
�
�
� �
� �
1
0
0 0
1 2
�
� �
� �
d d � �
� � � �
�
�
� � �
�
�
�
�
�
�
�
�
�
�
�
� �sin
sin sin
� �
� �
0
0
0
0
0
2 1
t
z
z
z
z
z
z
z z
d d
�
�
�
�
�
�
�
�
. (26)
E;quation (26) changes by using inversion substitution (22) and substitution integral sine and cosine (27) to equation (30).
� �
� �
� �
� �
Ci x
t
t
t Si x
t
t
t
x
x
� � �
�
� �
cos
,
sin
d d
0
(27)
� � � � � � � �� � � � � �~ cos sinx t t Ci z Ci z t Si z Si z� � � � � � � �
1
0 0 1 0 2 0 0 2 0
�
� � � � � � � �� �� �� �1 (28)
� � � � � � � �� � � � � � � �� �� �~ cos sinx t t Ci z Ci z t Si z Si z� � � � � �1 0 0 1 0 2 0 0 2 0 1
�
� � � � � � � � (29)
� � � � � �� � � �� �� � � � � �� �~ cos sinx t t Ci t t Ci t t t Si t t S� � � � � � � � �
1
0 0 1 0 2 0 0 2
�
� � � � � � � � �� �� �� �i t t�0 1 � . (30)
� for t t z1 1 0� �, and for t t z2 2 0� �,
The extrinsic value of the integral has to be calculated on this condition, because the integral is not defined at zero. The
function � �lim
�0
f x will be written as � �f x for easier notation. Simplification (31) will also be used further
� �
� �
C z
z
z
�
cos �0 , � �
� �
S z
z
z
�
sin �0 (31)
� � � � � � � � � � � �~ cosx t t C z C z C z C z
z
� � � � � �
��
�
�� �
�
� � �
1
0
0
0
1
�
� �
z d z d z d z d � � � � � �
z z
z
t S z S z
2 1
2
0
0
0
� �
�
� � �
�
�
�
�
��
�
�
�
�
��
� � �
�
�
�
�sin � �
z d z d
�
�
�
�
�
�
�
�
�
�
�
�
�
(32)
3.3.2 Semifinite part of a harmonic signal
A special case of the previous section is the semifinite part
of the harmonic signal shown in Figure 10.
The function shown in the figure above can be expressed
as:
� �
� �
x t
t t
t
�
�
�
�
�
�
cos ,� � �0 00 0
0 0
pro
pro
(44)
© Czech Technical University Publishing House http://ctn.cvut.cz/ap/ 29
Acta Polytechnica Vol. 41 No. 6/2001
The Equation (33) and subsequently Equation (34) will be given by using the properties of even function S(z) and odd
function C(z).
� � � � � � � �~ lim cosx t t C z C z
z
z
� � � � �
�
�
�
�
��
�
�
�
�
�
��
�
� � �0 0
1
1
2
z d z d � � � � � �
��
� � �
�
�
�
�
�
�
�
�
�
�
!
"
#
#
#
$
%
&
�
�
sin � �
0
0
01
2
t S z S z
z
z
z d z d &
&
�
�
�
�
�
�
�
�
�
�
(33)
� � � � � � � �~ lim cos sx t t C z C z
z z
� � � �
�
�
�
�
�
�
�
�
�
�
�
�
� �
� � �0 0
1
1 2
z d z d � � � � � �in � �
0
0 0
1 2
t S z S z
z z
� �
�
�
�
�
�
�
�
�
�
�
!
"
#
#
#
$
%
&
&
&
�
�
�
z d z d (34)
Equation (36) is given by conversion and by inversion substitution (22)
� � � � � � � �� � � � � �~ cos sinx t t Ci z Ci z t Si z Si� � � � � � � � �
1
0 0 1 0 2 0 0 2 0
�
� � � � � � � �� �� ��
z1 (35)
� � � � � �� � � �� �� � � � � �� �~ cos sinx t t Ci t t Ci t t t Si t t S� � � � � � � � �
1
0 0 1 0 2 0 0 2
�
� � � � � � � � �� �� ��
i t t�0 1� (36)
� for t t z1 1 0� �, and for t t z2 2 0� �,
� � � � � � � �~ cos sinx t t C z C z
z z
� � � �
�
�
�
�
��
�
�
�
�
��
�
�� ��
1
0
2 1
�
� � z d z d � � � � � �� �0
0 0
1 2
t S z S z
z z
� �
�
�
�
�
�
�
�
�
�
�
!
"
#
#
#
$
%
&
&
&
z d z d (37)
Equation (38) will be given by using the properties of even function S(z) and odd function C(z).
� � � � � � � �~ cos sinx t t C z C z
z z
� � � �
�
�
�
�
��
�
�
�
�
��
�
�
�
�
�
1
0
1 2
�
� � z d z d � � � � � �� �0
0 0
1 2
t S z S z
z z
� �
�
�
�
�
�
�
�
�
�
�
!
"
#
#
#
$
%
&
&
&
� �
z d z d (38)
Equation (40) is given by conversion and inversion substitution.
� � � � � � � �� � � � � �~ cos sinx t t Ci z Ci z t Si z Si� � � � � � � � � � �
1
0 0 1 0 2 0 0 1
�
� � � � � � � � �� ��
��0 2z (39)
� � � � � �� � � �� �� � � � � �� �~ cos sinx t t Ci t t Ci t t t Si t t S� � � � � � � � �
1
0 0 1 0 2 0 0 1
�
� � � � � � � � �� �� ��
i t t�0 2� (40)
Let us rewrite Equations (30), (36) and (40) as a function with parameters.
From Equation (30) it holds for t t t� �1 2
� � � � � �� � � �� �� � � �Y t t t t Ci t t Ci t t t Si1 1 2 0 0 1 0 2 0 0
1
, , cos sin� � � � � � �
�
� � � � � � � � �� � � �� �� ��
t t Si t t2 0 1� � �� (41)
From Equation (36) it holds for t t t1 2� �
� � � � � �� � � �� �� � � �Y t t t t Ci t t Ci t t t Si2 1 2 0 0 1 0 2 0 0
1
, , cos sin� � � � � � �
�
� � � � � � � � �� � � �� �� ��
t t Si t t2 0 1� � �� (42)
From equation (40) holds for t t t1 2� �
� � � � � �� � � �� �� � � �Y t t t t Ci t t Ci t t t Si3 1 2 0 0 1 0 2 0 0
1
, , cos sin� � � � � � �
�
� � � � � � � � �� � � �� �� ��
t t Si t t� � �1 0 2� (43)
1
x (t)
Fig. 10: Semifinite part of a harmonic signal
The same result we obtained if we use the function defined
by Equation (20), when t1 0� , t2 � �. Thus for Equation (30)
and (36) it holds:
� for t < 0
� � � � � ��
� � � �
~ cos
sin
x t t Ci t
t Si t
� � � �
� � � �!
"#
$
%&
1
1
2
0 0
0 0
�
� � �
�
� �
�
�
(45)
� for t > 0
� � � � � ��
� � � �
~ cos
sin
x t t Ci t
t Si t
� � �
� � � �!
"#
$
%&
1
1
2
0 0
0 0
�
� � �
�
� �
�
�
(46)
� for t = 0 and � ��
�
� �
2
2 1k , where k = 1, 2, … is a positive
integer, the value of � �~x t according to Equation (47) is
equal to �0.5.
� � � �~x k0
1
2
1 1� � � (47)
For t = 0 and � ��
�
' �
2
2 1k , and therefore for other
phases except � 2, 3 2� , 5 2� , … the quadrature signal is
discontinuous, and it has an extrinsic limit of a logarithmic
character [3] – see Equation (50).
� �
� �
� �
� �
Ci x
t
t
t x
n
x
n
x
n
n
n
� � � � � �
�
�
�
(
cos
ln
!
d � 1
1
2 2
2
1
(48)
where � �� � � � � � ��
�
�
�
�
� �
��
lim ln .
n n
n1
1
2
1
3
1
0 58� (49)
� �
� �
Ci
t
t
t0
0
� � � �
�
cos
d (50)
3.3.3 Linearly rising part of a harmonic signal
Let us make a calculation of the linearly rising part of
harmonic signal modulated by a trapezoidal pulse between
points t3 and t4, see Figure 8. This part is shown in Figure 11,
but points t3 and t4 are replaced by points t1 and t2.
The function shown in Figure 11 can be expressed as:
� �
� � � �
x t
at b t t t t
t t t t
�
� � � �
�
�
�
�
cos ,
,
� � �0 1 2 0
1 2
0
0
pro
pro
(51)
Using Equation (13) for the quadrature part, the calcula-
tion of function (52) yields
� �
� � � �~ cosx t
a b
t
t
t
� �
� �
�
1 0
�
� � � �
�
�d
1
2
(52)
� �
� � � �~ cos cosx t a
t
b
t
t
t
t
t
� �
�
�
�
�
�
�
�
1 0 0
�
� � � �
�
� �
� � �
�
�d d
1
2
1
2
�
�
�
�
�
�
�
(53)
Substituting
z t
z
z t t
z t t
� �
�
� �
� �
�
�d d
1 1
2 2
(54)
30
Acta Polytechnica Vol. 41 No. 6/2001
b
x(t)
Fig. 11: Linearly rising part of a harmonic signal
we can write Equation (53) as Equation (57)
� �
� � � �� � � �� �~ cos cosx t a
z t z t
z
z b
z t
z
z
z
z
z
z
� �
� � � � �
!
1 0 0
�
� �
�
� �
d d
1
2
1
2
"
#
#
#
$
%
&
&
&
(55)
� � � �� �
� �� �~ cos
cos co
x t a z t z at
z t
z
z b
z
z
z
z
� � � � �
� �
1
0
0
�
� �
� �
�d d
1
2
1
2
� �� �s � �0 z t
z
z
z
z
� �
!
"
#
#
#
$
%
&
&
&
d
1
2
(56)
� � � �� � � �
� �� �~ cos
cos
x t a z t z at b
z t
z
z
z
z
z
z
� � � � � �
� �
!
"
1
0
0
�
� �
� �
d d
1
2
1
2
#
#
#
$
%
&
&
&
(57)
Let us make a separate calculation of the integral using
(58) in Equation (57)
� � � � � ��
~x t I t I t� � �
1
1 2
�
(58)
where there is � � � �� �I t a z t z
z
z
1 0
1
2
� � � cos � � d (59)
Let us rewrite the Equations (62), (63), (64) and (65) as
a function with parameters.
For Equation (62), (63) and for t t t� �1 2 it holds:
� � � � � ��
Y t t t a b I t t t a I t t t a ba4 1 2 1 1 2 2 1 2
1
, , , , , , , , , , ,� � �
�
(66)
For Equation (62), (64) and for t t t1 2� � it holds
� � � � � ��
Y t t t a b I t t t a I t t t a bb5 1 2 1 1 2 2 1 2
1
, , , , , , , , , , ,� � �
�
(67)
For equation (62), (65) and for t t t1 2� �
� � � � � ��
Y t t t a b I t t t a I t t t a bc6 1 2 1 1 2 2 1 2
1
, , , , , , , , , , ,� � �
�
(68)
3.3.4 Total analytical calculation of the quadrature part of
a harmonic signal modulated by trapezoidal pulse
As was mentioned above, it was necessary to make
the calculation of the quadrature part of a harmonic signal
modulated by a trapezoidal pulse in separate parts, due to
the complexity of the analytical calculation.
Now we will make a summary of all the parts, see Figure 8.
We obtain the entire quadrature signal � �~x t by the sum of all
parts with respect to important parameters in the individual
part.
© Czech Technical University Publishing House http://ctn.cvut.cz/ap/ 31
Acta Polytechnica Vol. 41 No. 6/2001
� � � �
� �� �
I t at b
z t
z
z
z
z
2
0� �
� �
cos � �
d
1
2
(60)
For calculation of � �I t1 it holds:
� � � �� � � �� ��
I t
a
z t z t1
0
0 2 0 1� � � � � �
�
� � � �sin sin (61)
by using inverse substitution (54) we have
� � � � � ��
I t
a
t t1
0
0 2 0 1� � � �
�
� � � �sin sin (62)
The result of Equations (30), (36) and (40) will be used for calculating of � �I t2 .
� for t t t� �1 2
� � � � � � � �� � � �� ��
� �I t at b t Ci t t Ci t t t Si ta2 0 0 1 0 2 0 0 2� � � � � � � � �cos sin� � � � � � � � �� � � �� ��
� �t Si t t� ��0 1 (63)
� for t t t1 2� �
� � � � � � � �� � � �� ��
� �I t at b t Ci t t Ci t t t Si tb2 0 0 1 0 2 0 0 2� � � � � � � � �cos sin� � � � � � � � �� � � �� ��
� �t Si t t� ��0 1 (64)
� for t t t1 2� �
� � � � � � � �� � � �� ��
� �I t at b t Ci t t Ci t t t Si t tc2 0 0 1 0 2 0 0� � � � � � � � �cos sin� � � � � � � � �� � � �� ��
� �1 0 2� �Si t t� (65)
� for t t� 1
� � � �Z t Y t t t Y t t t
g e
t t
et gt
t t
1 1 1 3 4 3 4
4 3
4 3
4 3
� �
�
�
�
�
�
�
�
�
�
, , , , , , � �� � �
�
�
�
�
�
�
�
�
�
� �Y t t t Y t t t
e g
t t
gt et
t t
1 4 5 4 5 6
6 5
6 5
6 5
, , , , , , � �Y t t t1 6 8, , (69)
� for t t t1 3� �
� � � �Z t Y t t t Y t t t
g e
t t
et gt
t t
2 2 1 3 4 3 4
4 3
4 3
4 3
� �
�
�
�
�
�
�
�
�
�
, , , , , , � �� � �
�
�
�
�
�
�
�
�
�
� �Y t t t Y t t t
e g
t t
gt et
t t
1 4 5 4 5 6
6 5
6 5
6 5
, , , , , , � �Y t t t1 6 8, , (70)
� for t t t3 4� �
� � � �Z t Y t t t Y t t t
g e
t t
et gt
t t
3 3 1 3 5 3 4
4 3
4 3
4 3
� �
�
�
�
�
�
�
�
�
�
, , , , , , � �� � �
�
�
�
�
�
�
�
�
�
� �Y t t t Y t t t
e g
t t
gt et
t t
1 4 5 4 5 6
6 5
6 5
6 5
, , , , , , � �Y t t t1 6 8, , (71)
� for t t t4 5� �
� � � �Z t Y t t t Y t t t
g e
t t
et gt
t t
4 3 1 3 6 3 4
4 3
4 3
4 3
� �
�
�
�
�
�
�
�
�
�
, , , , , , � �� � �
�
�
�
�
�
�
�
�
�
� �Y t t t Y t t t
e g
t t
gt et
t t
2 4 5 4 5 6
6 5
6 5
6 5
, , , , , , � �Y t t t1 6 8, , (72)
� for t t t5 6� �
� � � �Z t Y t t t Y t t t
g e
t t
et gt
t t
5 3 1 3 6 3 4
4 3
4 3
4 3
� �
�
�
�
�
�
�
�
�
�
, , , , , , � �� � �
�
�
�
�
�
�
�
�
�
� �Y t t t Y t t t
e g
t t
gt et
t t
3 4 5 5 5 6
6 5
6 5
6 5
, , , , , , � �Y t t t1 6 8, , (73)
� for t t t6 8� �
� � � �Z t Y t t t Y t t t
g e
t t
et gt
t t
6 3 1 3 6 3 4
4 3
4 3
4 3
� �
�
�
�
�
�
�
�
�
�
, , , , , , � �� � �
�
�
�
�
�
�
�
�
�
� �Y t t t Y t t t
e g
t t
gt et
t t
3 4 5 6 5 6
6 5
6 5
6 5
, , , , , , � �Y t t t2 6 8, , (74)
� for t t8 �
� � � �Z t Y t t t Y t t t
g e
t t
et gt
t t7
3 1 3 6 3 4
4 3
4 3
4 3
� �
�
�
�
�
�
�
�
�
�
, , , , , , � �� � �
�
�
�
�
�
�
�
�
�
� �Y t t t Y t t t
e g
t t
gt et
t t
3 4 5 6 5 6
6 5
6 5
6 5
, , , , , , � �Y t t t3 6 8, , (75)
The entire equation for the quadrature part is done by the
sum of Equations (69)–(75):
� � � � � � � � � � � � � � � �~x t Z t Z t Z t Z t Z t Z t Z t� � � � � � �1 2 3 4 5 6 7 (76)
The quadrature part � �~x t calculated by (76) and the enve-
lope � � � � � �E t x t x t� �2 2~ of signal x ( t) defined by Equation
(19) is shown in Figure 12.
The dependency of the overshoot size (Fig. 12) on the
rising edge gradient is shown in Figure 13. This gradient
is defined by parameter s. The dependency is determined
numerically based on an analytical expression of the
quadrature part (76). The dependency cannot be determined
analytically due to the complexity of Equation (76). Finding
the local extreme by using the first derivative and then getting
the overshoot dependency is very difficult.
It is evident from the graph that there is an overshoot
amplitude of 50 % for gradient s = 89.5° of the rising edge.
It is derived in [2], that the quadrature part of a harmonic
signal modulated by a square pulse (a trapezoidal pulse with
a 90° gradient) reaches an infinite value at this point. It is
also evident that there is a high amplitude rise for gradient
s > 86°. It was derived in Equations (44) – (50) that the
quadrature part of the harmonic signal � �cos � �0t � reaches
an infinite amplitude at any point with a step change and
when the relation � �cos � �0 0t � ' is valid.
The difference between the theoretical amplitude (�) and
the amplitude given numerically (90 %, see Figure 13) is due
to numerical calculation. Numerical calculation is not able to
simulate a trapezoidal pulse with a 90° gradient according to
the finite distance of two following time samples. The maxi-
mum gradient that can be reached by numerical calculation is
given by
� �� �s g e fmax � �
180
�
atan vz
where fvz is the sampling frequency.
It is possible to reach the gradient smax = 89.936° for
fvz = 1 kHz and g – e = 1, smax = 89.999° for fvz = 1 MHz.
A further increase in sampling frequency only very slowly
approaches to the theoretical amplitude.
4 Conclusion
The dependency of the size overshoot on the gradient
edge of a pulse was found. This was done by an analytical
calculation of the quadrature part of a trapezoidal pulse
and of a harmonic signal modulated by a trapezoidal pulse.
The same source of the envelope size overshoot is the
gradient edge of a pulse and also that beginning and end
of the measurement interval. The size of the overshoot of
a harmonic signal modulated by a trapezoidal pulse is 80 %
with a gradient of edge almost 90°. The size of the overshoot
of the trapezoidal pulse is 250 %. The analytical dependency
of the size of the overshoot as not found, due to mathematical
complexity. Only numerical calculation was done, based on
previous analytical calculation of the quadrature part.
Acknowledgement
This research work has received support from research
program No J04/98:210000015 “Research of New Methods
for Physical Quantities Measurement and Their Application
in Instrumentation” of the Czech Technical University in
Prague (sponsored by the Ministry of Education, Youth and
Sports of the Czech Republic).
References
[1] Hahn, S. L.: Hilbert Transforms in Signal Processing.
Artech House, Boston, 1966
[2] Doubek, J.: Integrální transformace ve zpracování vibro-
diagnostického signálu. Dizertační práce, ČVUT Fakulta
elektrotechnická, 2001, (In Czech: Integral Transform
in a Vibrodiagnostic Signal), Ph.D. thesis, FEE, CTU,
Prague, 2001
[3] Čížek, V.: Discrete Hilbert Transform. IEEE Transaction on
Audio and Electronic, Volume AU-18, No. 4, December
1970, pp. 340–342
[4] Doubek, J., Kreidl, M.: New Algorithms of Gearbox Faults
Detection Using Hilbert Transform and Cross Corelation.
Workshop, CTU Publishing House, Prague, 2001,
pp. 320–321
[5] Kreidl, M., et al: Diagnostické systémy. Monografie, Ediční
středisko ČVUT, Praha, (In Czech: Diagnostic Systems).
Monograph, FEE CUT, Prague, 2001
Doc. Ing. Marcel Kreidl, CSc.
e-mail: kreidl@fel.cvut.cz
Ing. Jan Doubek, Ph.D.
e-mail: doubek@fel.cvut.cz
Department of Measurement
Czech Technical University in Prague
Faculty of Electrical Engineering
Technická 2, 166 27 Praha 6, Czech Republic
32 © Czech Technical University Publishing House http://ctn.cvut.cz/ap/
Acta Polytechnica Vol. 41 No. 6/2001
-10 -8 -6 -4 -2 0 2 4 6 8 10
-1
-0.5
0
0.5
1
t [s]
)(tx
)(tEovershoot
)(~tx
Fig. 12: Harmonic signal modulated by trapezoidal pulse x(t)
given (19), quadrature part � �~x t and envelope E(t)
with parameters e = 0; g = 1; t3 = �6.25 s; t4 = �5.75 s;
t5 = 5.75 s; t6 = 6.25 s; �0 = 2�; � = 0, fvz = 1 kHz
Notice: Parameters t1 and t8 lose their relevance if the parameter
e = 0. For this reason these parameters were not men-
tioned. Parameter e = 0 was chosen to provide an easier
view of the quadrature part and envelope.
60
80
100
%
Fig. 13: Normalized graph of overshoot dependency on the gra-
dient s of the signal (19) rising edge, parameters
� � � ��
s g e t t� � �180 6 5atan � , where e = 0, g = 1,
t6 = 6 s, � �t t6 5 0 5� � s s, , �0 = 2�, � = 0