A Comparison Between Hilbert Transformand a New Method for Signal Enveloping

Abdullah Al-Ahmari, Jyoti K. Sinha and Erfan Asnaashari

Abstract Envelope analysis of vibration signals is a well known tool for amplitudedemodulation and diagnosis of a number of vibration problems in machines andstructures. The typical application is the fault diagnosis in the anti-friction bearingsand gearboxes. Hilbert transformation (HT) is often used to extract the envelopesignals (upper and lower) from a time domain signal. However it is observed thatthe envelope signals obtained by the HT are not always without any error. In thispaper, 4 different signals, 3 simulated; sine, amplitude modulated and random, andmeasured vibration data on anti-friction bearings are analyzed using the HT. Thepaper compares the accuracy of the envelope data obtained for these differentsignals using the HT. A new method called three-point moving window (TPMW)method is also developed to generate envelope signals and applied to the 4 differentsignals that gives better results.

Keywords Envelop analysis � Bearings � Condition monitoring

1 Introduction

Envelope analysis is a well-known technique used for detecting incipient failure ofrolling element bearing. This method is also known by other names such as,amplitude demodulation [1], demodulated resonance analysis, narrow band enve-lope analysis [2, 3] or high frequency resonance technique [4–7]. Envelope analysis

A. Al-Ahmari (&) � J.K. Sinha � E. AsnaashariSchool of Mechanical, Aerospace and Civil Engineering, The University of Manchester,Oxford Road, Manchester M13 9PL, UKe-mail: Abdullah.al-ahmari@postgrad.manchester.ac.uk

J.K. Sinhae-mail: jyoti.sinha@manchester.ac.uk

E. Asnaasharie-mail: Erfan.asnaashari@postgrad.ac.uk; Erfan.asnaashari@postgrad.manchester.ac.uk

© Springer International Publishing Switzerland 2015J.K. Sinha (ed.), Vibration Engineering and Technology of Machinery,Mechanisms and Machine Science 23, DOI 10.1007/978-3-319-09918-7_14

163

or amplitude demodulation is the technique that can be used to extract the periodicimpacts from the modulated random noise generated within a faulty or deterioratingrolling element bearing. This is a possible process even when the signal from therolling element bearing is relatively low in energy and buried within the othervibrations produced from the machine.. The effectiveness of the envelope analysishas been assessed by a number of researchers [2–9]. Hilbert transform is animportant tool in envelope analysis which can be used for machine fault detectionand diagnostics [10–12]. There are also some literatures which used the HT com-bined with other techniques to make the fault diagnosis easier [7, 13, 14].

It has been observed that the envelope signals obtained by the HT are not alwayswithout any error. In this paper, 4 different signals, 3 simulated; sine, amplitudemodulated, random, and the measured vibration data for anti-friction bearings areexamined by the HT. The paper compares the accuracy of the envelope dataobtained for these signals using the HT. A new method called three-point movingwindow (TPMW) method is then developed to generate the envelope data for thesame signals. The new method is shown to be much more accurate compared to theHT.

2 Hilbert Transformation

The HT is used for solving integral equations. The mathematical definition of HT ofthe signal xðtÞ is defined as:

H x tð Þ½ � ¼ ~x tð Þ ¼ 1p

Zþ1

�1

x sð Þt � s

ds ð1Þ

The integral is considered as a Cauchy principal value due to possible singularityat t ¼ s. The HT can be expressed physically as a special type of a linear filterwhich keeps all the amplitudes of spectral components but shifts their phases by �p

2 .Therefore, the HT of the original signal is equal to the convolution integral of xðtÞwith function ðptÞ�1 which can be written as ~x tð Þ ¼ x tð Þ � ðptÞ�1.

The HT has attracted great attention in vibration analysis over the past twodecades [15]. It can be applied to vibration signals to present useful informationregarding the amplitude, instantaneous phase and frequency of vibrations. Simonet al. [16] and Tomlinson [17] utilised HT for identification of structural non-linearities using sinusoidal input excitation in frequency domain. They identifiedcommon types of nonlinearity from the frequency response function (FRF) dis-tortions. Feldman [18] employed HT of input and output time domain measure-ments for identification of non-linearities in stiffness and damping characteristics ofa vibration system.

Signal demodulation is another area which the HT can be used. An amplitudemodulated wave can be represented as:

164 A. Al-Ahmari et al.

x tð Þ ¼ u tð Þcosw tð Þ ð2Þ

where u tð Þ is the modulating envelope function or message waveform and coswðtÞis the oscillations or carrier waveform. To demodulate waveform x tð Þ, the carrierwaveform should be removed while the message waveform is preserved. The HTshifts the phase of every frequency component of the modulated wave by 90° toobtain a quadrature signal:

H x tð Þ½ � ¼ ~x tð Þ ¼ u tð Þsinw tð Þ: ð3Þ

The message waveform can then be acquired by adding the squares of wave-forms 2 and 3:

u2 tð Þ ¼ u2 tð Þcos2w tð Þ þ u2 tð Þsin2w tð Þ ð5Þ

3 Application of HT

In order to visualize the envelope using the HT, this section demonstrates the upperand lower envelopes for the simulated and measured vibration data extractedexperimentally on an anti-friction bearing.

3.1 Sine and Multi-sine Signals

A sine wave;

x tð Þ ¼ A sin 2pftð Þ ð5Þ

was generated with a sampling frequency of 20 kHz. Amplitude A and frequency fwere chosen to be 1.5 and 60 Hz respectively. Figure 1 illustrates xðtÞ and its upperand lower envelopes using the Hilbert transform.

To demonstrate more elaborations, a combination of two sine waves wasinvestigated:

y tð Þ ¼ A sin 2pf1tð Þ þ B sin 2pf2tð Þ ð6Þ

where amplitude B and frequency f2 were chosen to be 1.5 and 80 Hz respectively.The upper and lower envelopes of this combined signal are shown in Fig. 2. As itcan be seen, at some locations, the upper envelope is tangent to one side of thepeaks instead of the peak tips.

A Comparison Between Hilbert Transform… 165

3.2 Amplitude Modulated Signals

Three different amplitude modulated signals were tested separately. The first signalis a sine wave considered as the original signal (Fig. 3). The second and thirdsignals are superposition of two and three sinusoidal wave components respectivelywith different frequencies (Figs. 4 and 5). In all the cases, the modulation index is50 % while the sampling frequency and the frequency of the carrier signal are 10and 2 kHz respectively.

3.3 Random Signal

The HT was also performed on a random signal to generate upper and lowerenvelopes as can be seen in Fig. 6. Samples were taken at the frequency of 20 kHz.Figure 6 shows that the envelope does not precisely cross the local maxima andlocal minima for upper and lower envelopes respectively.

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08-2

-1.5-1

-0.50

0.51

1.52

Time, s

Am

plit

ud

esine signal x(t)top envelopbottom envelop

Fig. 1 Envelope using HT for the sine wave xðtÞ

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2-4-3-2-101234

Time, s

Am

plit

ud

e

sine signal y(t)top envelopbottom envelop

Fig. 2 Envelope using HT for yðtÞ

166 A. Al-Ahmari et al.

3.4 Measured Signal

The HT was applied to measured vibration data of an anti-friction bearing toproduce upper and lower envelopes as seen in Fig. 7. Similarly, the same error as inthe random signal appears in the measured vibration signal.

0 0.005 0.01 0.015 0.02 0.025 0.03-6

-4

-2

0

2

4

6

Time, s

Am

plit

ud

eamplitude modulated signaltop envelopbottom envelop

Fig. 3 Envelope using HT for an amplitude modulated signal with 1 sine wave

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05-6

-4

-2

0

2

4

6

Time, s

Am

plit

ud

e

amplitude modulated signaltop envelopbottom envelop

Fig. 4 Envelope using HT for an amplitude modulated signal with 2 sine waves

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035-10

-5

0

5

10

Time, s

amplitude modulated signaltop envelopbottom envelop

Am

plit

ud

e

Fig. 5 Envelope using HT for an amplitude modulated signal with 3 sine waves

A Comparison Between Hilbert Transform… 167

4 Proposed Method

In order to avoid the error associated with the HT in random and measured vibrationdata, a three-point moving window method (TPMW) was developed. To find theupper envelope, the TPMW method scans the whole signal and search for the localmaximum points. When the value of the middle point is greater than its twoadjacent points, both the index and value of the middle point are stored. By movingthe scanning window over the whole signal, all maximum points and their indicescan be stored. After finding the local maximum points, they are connected linearlyby interpolating the points in between. To produce the lower envelope, the TPMWsearches for local minimum points and then connect those points linearly as it isdone with the upper envelope.

In other words, Let’s assume xi�1, xi and xiþ1, are three consecutive data pointsthat form a three-point moving window in the time domain, where i ¼ 2; 3; . . .; nand n is the number of samples collected. To consider a point as the local minimumor local maximum the following conditions should be satisfied simultaneously:

xi ¼ local minimum; xi\; 0 and xi\xi�1 and xi\xiþ1

xi ¼ local maximum; xi [ 0 and xi [ xi�1 and xi [ xiþ1

�

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

x 10-3

-3

-2

-1

0

1

2

3

Time, s

Am

plit

ud

e

random signaltop envelopbottom envelop

Fig. 6 Envelope using HT for a random signal

0 1 2 3 4 5 6 7 8

x 10-3

-10

-5

0

5

10

Time, s

measured signaltop envelopbottom envelop

Am

plit

ud

e

Fig. 7 Envelope using HT for a measured vibration signal on anti-friction bearing

168 A. Al-Ahmari et al.

If the conditions for either local minimum or maximum are satisfied, the valueand index of the middle point (xi) are stored and the window is shifted by twoindices. Otherwise, the window will move one index forward and test the newpoints.

5 Application of the TPMW Method

The TPMW was applied to the same signals which were previously tested by theHT in Sect. 3. Figure 8 shows the upper and lower envelopes for the signal xðtÞ andFig. 9 displays it for yðtÞ where the envelopes cross the local maxima and localminima of the signal.

5.1 Amplitude Modulated Signal

The amplitude modulated signals generated in 3.2 is again utilized here, but theenvelope was made using the TPMW method (Figs. 10, 11, 12, 13 and 14). In boththe cases of random and measured vibration signals (Figs. 12–14), the TPMW

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08-2

-1.5-1

-0.50

0.51

1.52

Time, s

sine signal x(t)top envelopbottom envelop

Am

plit

ud

e

Fig. 8 Envelope using HT for the sine wave xðtÞ

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2-4-3-2-101234

Time, s

sine wave y(t)top envelopbottom envelop

Am

plit

ud

e

Fig. 9 Envelope using HT for the sine wave yðtÞ

A Comparison Between Hilbert Transform… 169

0 0.005 0.01 0.015 0.02 0.025 0.03-6

-4

-2

0

2

4

6

Time, s

Am

plit

ud

eamplitude modulated signaltop envelopbottom envelop

Fig. 10 Envelope using TPMW for an amplitude modulated signal with 1 sine wave

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035-10

-5

0

5

10

Time, s

amplitude modulated signaltop envelopbottom envelop

Am

plit

ud

e

Fig. 12 Envelope using TPMW for an amplitude modulated signal with 3 sine waves

0 1 2 3 4 5 6 7 8x 10-3

-3

-2

-1

0

1

2

3

Time, s

random signaltop envelopbottom envelop

Am

plit

ud

e

Fig. 13 Envelope using TPMW for a random signal

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05-6

-4

-2

0

2

4

6

Time, s

amplitude modulated signaltop envelopbottom envelop

Am

plit

ud

e

Fig. 11 Envelope using TPMW for an amplitude modulated signal with 2 sine waves

170 A. Al-Ahmari et al.

clearly connects the local maxima and local minima to generate upper and lowerenvelopes respectively.

6 Conclusion

In conclusion, the HT was found to produce good envelope for sine and modulatedsignals; however, it was unable to accurately connect the local maxima and localminima of the random and measured vibration signals. The developed method,TPMW, proposed in this paper, generated better envelopes that can be used inenvelope spectrum analysis.

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