Application Note

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h(t) = Ae t

c)

h(t) = h(t) + ih(t)

h(t) = h2(t) + h2(t)

the signal. By using the Hilberttransform, the rapid oscillations canbe removed from the signal to pro-duce a direct representation of theenvelope alone.

For example, the impulse responseof a single degree of freedom systemis an exponentially damped sinusoid,

. This is shown as (a) in Fig. 1.The envelope of the signal is deter-mined by the decay rate. See Fig. 1.

The Hilbert transform, is usedto calculate a new time signal from the original time signal .The time signal is a cosine func-tion whereas is a sine: both areshown in Fig.1.

h t( )

h t( )h t( )

h t( )h t( )

1:10,000 (80 dB), or more. Theoriginal signal, , includes bothpositive and negative values andis traditionally displayed using alinear amplitude scale. This limitsthe display range to about 1:100(40 dB).

Decay rate estimation

Determining the frequency and cor-responding damping at resonances isoften the first step in solving a vibra-tion problem for a structure. Fig. 2shows the log. magnitude of a me-chanical mobility measurement.Within the excitation frequency

h t( )

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tion of the envelope of the impulse-responseThe magnitude of the analytic sig-nal can be directly calculatedfrom and . The magnitude of is the envelope of the original timesignal and is shown above as (c). Ithas the following advantages over

h

t( )h h h

t( )

range of 0 Hz to 3.2 kHz, five reso-nances are clearly seen. The reso-nance frequencies can be readdirectly with an accuracy determinedby the resolution of the analysis, i.e.4 Hz. The decay rate at the resonanc-es is often determined by the half-

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1 kPractical use of the Hby N.Thrane, J.Wismer, H.Konstantin-H

The envelope

Many application measurements re-sult in a time signal containing a rap-idly oscillating component. Theamplitude of the oscillation variesslowly with time, and the shape ofthe slow time variation is called theenvelope. The envelope often con-tains important information about

The Brel&Kjr Signal AnalyzerType 3550 and 2140 families imple-ment the Hilbert transform to open upnew analysis possibilities in the timedomain. By means of the Hilbert trans-form, the envelope of a time signal canbe calculated, and displayed using alogarithmic amplitude scale enabling alarge display range. Two exampleswhich use the Hilbert transform arepresented here:m The determination of the damping

or decay rate at resonances, fromthe impulse response function.

m The estimation of propagation time,from the cross correlation function.Brel & Kjr B K

:h t( )ilbert transformansen & S.Gade, Brel&Kjr, Denmark

a)

b)

h(t) = Ae t cosdt

h(t) = Ae t sindt

1. Removal of the oscillations allowsdetailed study of the envelope.

2. Since is a positive function,it can be graphically representedusing a logarithmic amplitudescale to enable a display range of

h

t( )

Fig.1 The Hilbert transform enables computafunctionpower (or 3 dB) bandwidth, B 3 dB, of Fig.3

K 45 80 05 00 Fax: +45 45 80 14 05 Internet: http://www.bk.dk e-mail: info@bk.dkelgium 016/44 92 25 Brazil (011) 246-8166 Canada: (514) 695-8225 China 10 6841 9625 / 10 6843 7426 nce (01) 69 90 69 00 Germany 0610 3/908-5 Holland (0)30 6039994 Hong Kong 254 8 7486 779-8671 Republic of Korea (02) 3473-0605 Norway 66 90 4410 Poland (0-22) 40 93 92 Portugal (1) 47114 53

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the resonance peak. B3 dB = 2. Inthis case B3dB is of the order of theresolution; consequently a determina-tion of the B 3dB (and hence ) will bevery inaccurate. Two methods can beused to obtain a more accurate esti-mate of the damping:1. A (time consuming) zoom analysis

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Fig.5

to B) of a signal is usually estimatedby measuring the signal at A and B,and calculating the cross correlationfunction RAB(t).

By using the Hilbert transform, thecorrect propagation time can easilybe found from the envelope of the crosscorrelation function, see Fig.8, whetheror not the peak of RAB(t) correspondsto the envelope maximum.

References

A short discussion of the Hilberttransform can be found in ref. [1],while ref. [2] discusses the propertiesand applications of the Hilbert trans-form. Ref. [3] gives additional infor-mation about damping measurementin general.

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[1]. N. Thrane: The Hilbert Trans-form, Technical Review No. 3 1984,Brel&Kjr, BV 0015

[2]. J.S. Bendat: The Hilbert Trans-form and Applications to CorrelationMeasurements, Brel&Kjr, 1985,BT0008

[3]. S.Gade, H.Herlufsen: DigitalFilter Techniques vs. FFT Techniquesfor Damping Measurements, Techni-cal Review No. 1 1994, Brel&Kjr,BV 0044Brel & Kjr BWORLD HEADQUARTERS:DK-2850 Naerum Denmark Telephone: +45Australia (02 ) 9450-2066 Austria 00 43-1-865 74 00 BCzech Republic 02-67 021100 Finland 90-229 3021 FraHungary (1) 215 83 05 Italy (02) 57 60 4141 Japan 03-3Singapore (65) 275-8816 Slovak Republic 07-37 6181 United Kingdom and Ireland (0181) 954-236 6 USA 1 - Local representatives and service organisations worldwide

using a much smaller f. This in-volves a new analysis for each res-onance, making five newmeasurements in total.

2. The damping at each resonance canbe determined from the envelope ofthe associated impulse responsefunction. This method is illustratedin Figs.2 to 7, from which (decayconstant) for each resonance can beeasily found from the originalmeasurement.Fig. 2 shows the frequency re-

sponse function, and Fig. 3 shows thecorresponding impulse response func-tion. However, this cannot be used tocalculate , as it contains five expo-nentially damped sinusoids (one foreach resonance) superimposed.

Fig. 4 shows a single resonancewhich has been isolated using the fre-quency weighting facility of Type3550. The corresponding impulse re-sponse function, shown in Fig. 5,clearly shows the exponential decay-ing sinusoid.

Fig. 6 shows the magnitude of theanalytic signal of the impulse re-sponse function on a linear amplitudescale. By using a log. amplitude axis,the envelope is a straight line, seeFig. 7. The analyzers reference cursoris used to measure the time constant corresponding to an amplitude de-cay of 8.7 dB. From , the decay con-stant and hence the damping of theresonance can be calculated directly( = 1/).

By using the Hilbert transform, itis possible to determine the decayconstant for the five individual reso-nances, without having to make new,more narrow banded measurements.This method applies to the 3550 fam-ily. The 2140 family does not supportfrequency weighting.BO 0437 11 500.5 k0 1.0 k 1.5 k 2.0 k 2.5 k 3.0 k

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Propagation time estimation

The propagation time (from point A

Practical use of the Hilbert transformThe envelopeDecay rate estimationPropagation time estimationReferences

FiguresFig.1 The Hilbert transform enables computation ...