Application Note Practical use of the “Hilbert transform” The envelope

Application Note Practical use of the “Hilbert transform” The envelope
Application Note
Practical use of the “Hilbert transform”
by N.Thrane, J.Wismer, H.Konstantin-Hansen & S.Gade, Brüel&Kjær, Denmark
The Brüel&Kjær Signal Analyzer
Type 3550 and 2140 families implement the Hilbert transform to open up
new analysis possibilities in the time
domain. By means of the Hilbert transform, the envelope of a time signal can
be calculated, and displayed using a
logarithmic amplitude scale enabling a
large display range. Two examples
which use the Hilbert transform are
presented here:
❍ The determination of the damping
or decay rate at resonances, from
the impulse response function.
❍ The estimation of propagation time,
from the cross correlation function.
h(t) = Ae– σ t sinωdt
a)
c)
∇
h(t) = Ae– σ t
∇
∼
h(t) = h(t) + ih(t)
∇
∼
h(t) = √h2(t) + h2(t)
b)
∼
h(t) = Ae– σ t cosωdt
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Fig.1 The Hilbert transform enables computation of the envelope of the impulse-response
function
The envelope
Many application measurements result in a time signal containing a rapidly oscillating component. The
amplitude of the oscillation varies
slowly with time, and the shape of
the slow time variation is called the
“envelope”. The envelope often contains important information about
the signal. By using the Hilbert
transform, the rapid oscillations can
be removed from the signal to produce a direct representation of the
envelope alone.
For example, the impulse response
of a single degree of freedom system
is an exponentially damped sinusoid,
h ( t ) . This is shown as (a) in Fig. 1.
The envelope of the signal is determined by the decay rate. See Fig. 1.
The Hilbert transform,
is used
to calculate a new time signal h̃ ( t )
from the original time signal h ( t ) .
The time signal h̃ ( t ) is a cosine function whereas h ( t ) is a sine: both are
shown in Fig.1.
The magnitude of the analytic sig∇
nal h ( t ) can be directly calculated
∇
from h and h̃ . The magnitude of h ( t )
is the envelope of the original time
signal and is shown above as (c). It
has the following advantages over
h(t) :
Brüel & Kjær
B
K
1. Removal of the oscillations allows
detailed study of the envelope.
∇
2. Since h ( t ) is a positive function,
it can be graphically represented
using a logarithmic amplitude
scale to enable a display range of
1:10,000 (80 dB), or more. The
original signal, h ( t ) , includes both
positive and negative values and
is traditionally displayed using a
linear amplitude scale. This limits
the display range to about 1:100
(40 dB).
30
20
10
0
– 10
– 20
– 30
– 40
– 50
0
0.5 k
1.0 k
1.5 k
2.0 k
2.5 k
3.0 k
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Fig.2
Decay rate estimation
Determining the frequency and corresponding damping at resonances is
often the first step in solving a vibration problem for a structure. Fig. 2
shows the log. magnitude of a mechanical
mobility
measurement.
Within the excitation frequency
range of 0 Hz to 3.2 kHz, five resonances are clearly seen. The resonance frequencies can be read
directly with an accuracy determined
by the resolution of the analysis, i.e.
4 Hz. The decay rate at the resonances is often determined by the halfpower (or 3 dB) bandwidth, B 3 dB, of
4k
3k
2k
1k
0
–1k
–2k
–3k
–4k
0
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10 m 30 m 50 m 70 m 90 m 110 m
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Fig.3
the resonance peak. B3 dB = 2σ. In
this case B 3 dB is of the order of the
resolution; consequently a determination of the B 3 dB (and hence σ) will be
very inaccurate. Two methods can be
used to obtain a more accurate estimate of the damping:
1. A (time consuming) zoom analysis
using a much smaller ∆ f. This involves a new analysis for each resonance,
making
five
new
measurements in total.
2. The damping at each resonance can
be determined from the envelope of
the associated impulse response
function. This method is illustrated
in Figs. 2 to 7, from which σ (decay
constant) for each resonance can be
easily found from the original
measurement.
Fig. 2 shows the frequency response function, and Fig. 3 shows the
corresponding impulse response function. However, this cannot be used to
calculate σ, as it contains five exponentially damped sinusoids (one for
each resonance) superimposed.
Fig. 4 shows a single resonance
which has been isolated using the frequency weighting facility of Type
3550. The corresponding impulse response function, shown in Fig. 5,
clearly shows the exponential decaying sinusoid.
Fig. 6 shows the magnitude of the
analytic signal of the impulse response function on a linear amplitude
scale. By using a log. amplitude axis,
the envelope is a straight line, see
Fig. 7. The analyzer’s reference cursor
is used to measure the time constant
τ corresponding to an amplitude decay of 8.7 dB. From τ, the decay constant and hence the damping of the
resonance can be calculated directly
(σ = 1/τ).
By using the Hilbert transform, it
is possible to determine the decay
constant for the five individual resonances, without having to make new,
more narrow banded measurements.
This method applies to the 3550 family. The 2140 family does not support
frequency weighting.
Brüel & Kjær
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2.0 k
30
20
1.5 k
10
1.0 k
0
0.5 k
– 10
0
– 20
– 0.5 k
– 30
– 1.0 k
– 40
– 1.5 k
– 50
0
0.5 k 1.0 k
1.5 k
2.0 k
2.5 k
3.0 k
– 2.0 k
0
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Fig.4
Fig.5
2.0 k
70
1.8 k
65
1.6 k
60
1.4 k
55
1.2 k
50
1.0 k
45
– 8.7 dB
τ
40
0.6 k
35
0.4 k
30
0.2 k
0
0
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Fig.6
Fig.7
Propagation time
estimation
1.0
The propagation time (from point A
to B) of a signal is usually estimated
by measuring the signal at A and B,
and calculating the cross correlation
function RAB(t).
By using the Hilbert transform, the
correct propagation time can easily
be found from the envelope of the cross
correlation function, see Fig. 8, whether
or not the peak of RAB(t) corresponds
to the envelope maximum.
0.6
0.5
0.4
References
A short discussion of the Hilbert
transform can be found in ref. [1],
while ref. [2] discusses the properties
and applications of the Hilbert transform. Ref. [3] gives additional information about damping measurement
in general.
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0.8
0.7
0.3
0.2
0.1
0
– 10 m – 5 m
0
5m
10 m
15 m
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Fig.8
[1]. N. Thrane: “The Hilbert Transform”, Technical Review No. 3 1984,
Brüel&Kjær, BV 0015
[2]. J.S. Bendat: “The Hilbert Transform and Applications to Correlation
Measurements”, Brüel&Kjær, 1985,
BT 0008
[3]. S.Gade, H.Herlufsen: “Digital
Filter Techniques vs. FFT Techniques
for Damping Measurements”, Technical Review No. 1 1994, Brüel&Kjær,
BV 0044
K
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