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 exampl

Practical use of the “Hilbert transform”

by N.Thrane, J.Wismer, H.Konstantin-Hansen & S.Gade, Brüel&Kjær, Denmark

The Brüel&Kjær 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:

rThe determination of the dampingor decay rate at resonances, fromthe impulse response function.

rThe estimation of propagation time,from the cross correlation function.

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 the“envelope”. The envelope often con-tains important information aboutthe 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,h(t). 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 used

(t)to calculate a new time signal h

from the original time signal h(t).

(t) is a cosine func-The time signal h

tion whereas h(t) is a sine: both areshown in Fig.1.

The magnitude of the analytic sig-

nal h(t) can be directly calculated

. The magnitude of h(t)from h and h

is the envelope of the original timesignal and is shown above as (c). Ithas the following advantages overh(t):

function

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

2.Since h(t) is a positive function,it can be graphically representedusing a logarithmic amplitudescale to enable a display range of1:10,000 (80dB), or more. Theoriginal signal, h(t), includes bothpositive and negative values andis traditionally displayed using alinear amplitude scale. This limitsthe display range to about 1:100(40dB).

Fig.2

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 frequencyrange of 0Hz to 3.2kHz, five reso-nances are clearly seen. The reso-nance frequencies can be readdirectly with an accuracy determinedby the resolution of the analysis, i.e.4Hz. The decay rate at the resonanc-es is often determined by the half-power (or 3dB) bandwidth, B3dB, of

Fig.3

Brüel & Kjær

B