8. AUTOMATIC ANALYSIS OF AN ACCELERATION SIGNAL
8.1 Feature extraction
8.1.3 Envelope detectors for the envelope spectrum method
In this chapter envelope detectors are tested in conjunction with the envelope spectrum method.
The envelope detectors tested are the Fourier method with a Gabor- filter implementation, the Hilbert transformer and the band pass filter. Also the peak detector using a median filter is tested.
The schemas of the actual envelope detectors used in the tests are illustrated in Figure 8-5. En-velopes are detected with a band pass demodulation (Figure a), with a Gabor-filter (Figure b) and with a Hilbert transformer (analytic signal) (Figure c). The band pass demodulation proce-dure filters the natural frame frequency with two 30. order Butterworth filters forming a pass band of 2-5 kHz. The filtered signal is rectified then decimated with a 2-phase decimator from sampling frequency of 20 kHz to a sampling frequency of 1.67 kHz. 2048- point FFT is calcu-lated for the power spectral estimate (periodogram method). The peak detector is otherwise similar to the band pass filter implementation but the band pass filter is replaced with a length ten median filter and peak selector. In Figure b, the Gabor filter uses the central frequency of 2.5 kHz. The magnitudes of the coefficients form a new time series that is low pass filtered and the power spectral estimate is formed as in the previous implementation. In the procedure of Figure c, the analytic signal is formed as a sum of the Hilbert transformer and the original signal according to Equation 4.19. The magnitude of the obtained complex signal is the signal enve-lope that is decimated and the PSD is formed as in other procedures.
s(n) D
4 D
3
|h(n)| FFT,
PSD
s(n) FFT,
G{s(n)} PSD
s(n) D
4 D
3
|h(n)| FFT,
H{s(n)} + PSD a)
b)
c)
Fig. 8-5. Envelope detection with a) a band pass demodulation, b) with a Gabor-filter and with c) a Hil-bert transformer (analytic signal). The procedure a) filters the natural frame frequency with two Butterworth filters, then rectifies the signal and performs a 2-phase decimation (Fs/(3x4)). FFT is used for the calculation of the power spectral estimate (PSD). The procedure b) consists of a Ga-bor-filter, a low pass filter and an FFT-PSD block. In the procedure c) the analytic signal is formed as a sum of the Hilbert transformed and the original signal. The magnitude of the obtained complex signal is the signal envelope that is decimated and the PSD is formed as in procedure a.
In Figure 8-6 the envelope detection methods are compared. In the left side figures the enve-lopes using a) a band pass demodulation technique, c) a Gabor filter and e) an analytic signal are presented. The light curve is the original accelerometer signal, the dark line is the envelope.
In the right side figures the power spectrum estimates of envelopes are presented. Although all of the methods reveal a bearing pass frequency of 99.8 Hz, the signal to noise ratio in the spec-trum from the Gabor filter based detection (Figure d) is lower that in the spectra from the band pass detection (Figure b) or from Hilbert transform based detection.
In Figure 8-7, the signal and its envelope is formed using a rectifier with several pass bands and an analytic signal. With narrow bands (band width from 2 to 3 kHz) the results are highly de-pendent on selected frequency band. The use of a wider band attenuates the spectrum components slightly compared to correctly selected narrow band or to Hilbert transformer based envelope spectrum. In this test, the Hilbert transformer produces the best ratio between the noise and the bearing pass frequency components. However, all of the previous tests with a Hilbert transformer (and analytic signal) were made with a quite complex algorithm that trans-forms a signal into the frequency domain, multiplies the obtained coefficients with the corresponding Hilbert filter coefficients and performs the inverse FFT. Fortunately, the Hilbert-transformers can be approximated with relatively short FIR or IIR filters when the unity magni-tude amplification is required only for a restricted band. For example, a 31 order Hilbert transformer FIR filter (Remez) for the pass band 1-9 kHz (Fs=20 kHz) can detect the envelope as well as a well tuned band pass filter and better than a wide band pass band filter. This is presented in Figure 8-8 where envelope spectra are formed using a) wide band pass demodula-tion, 31. order FIR Hilbert filter demodulation and c) also the peak detection demodulation. The peak detection method can also find the bearing frequency but more noise components are pre-sent in the spectrum than in the spectra of the band pass filter based methods.
30 40 50 60 70 -5
0 5 a)
t [ms]
a [g]
0 100 200 300
-30 -20 -10 0
f [Hz]
(dB/Hz)
b)
30 40 50 60 70
-5 0 5 c)
t [ms]
a [g]
0 100 200 300
-30 -20 -10 0
f [Hz]
(dB/Hz)
d)
30 40 50 60 70
-5 0 5 e)
t [ms]
a [g]
0 100 200 300
-30 -20 -10 0
f [Hz]
(dB/Hz)
f)
Fig. 8-6. Envelope detection using a) a band pass demodulation technique, c) a Gabor filter and e) an analytic signal. The light curve is the original acceleration signal, the dark line is its envelope.
On the right side figures the power spectrum estimates of the envelopes are presented. All of the spectra reveal the bearing pass frequency of 99.8 Hz.
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0
5 a) band pass demod. envelope
t [ms]
a [g]
0 100 200 300
-30 -20 -10 0
f [Hz]
(dB/Hz)
b) pass band 2-4 kHz
0 100 200 300
-30 -20 -10 0
f [Hz]
(dB/Hz)
c) pass band 5-8 kHz
0 100 200 300
-30 -20 -10 0
f [Hz]
(dB/Hz)
d) pass band 2-8 kHz
30 40 50 60 70
-5 0 5
e) Hilbert demod. envelope
t [ms]
a [g]
0 100 200 300
-30 -20 -10 0
f [Hz]
(dB/Hz)
f) Hilbert demodulation
Fig. 8-7. Envelope spectra of band pass demodulation with different pass bands and envelope spectrum with Hilbert demodulation in the case of a small bearing fault. With narrow bands (2 or 3 kHz) the results are highly dependent on the selected frequency band. The use of a wider band attenuates the spectrum components slightly compared to correctly selected narrow band or to Hilbert trans-former based envelope spectrum.
30 40 50 60 70 -5
0
5 a) band pass demodulation
t [ms]
5 c) length 31 Hilbert (FIR)
t [ms]
d) Hilbert FIR 1-9 kHz
30 40 50 60 70
-5 0
5 e) peak detection demodulation
t [ms]
Fig. 8-8. Envelope spectra of band pass demodulation with different pass bands and envelope spectrum with Hilbert demodulation in the case of a small bearing fault. With narrow bands (2 or 3 kHz) the results are highly dependent on the selected frequency band. The use of a wider band attenu-ates the spectrum components slightly compared to correctly selected narrow band or to Hilbert transformer based envelope spectrum.