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Fig. 13. (a) Vibration signal and (b) magnitude of the complex envelope c}°(n) which locally maximises the kurtogram, together with its 0.1% signification threshold. (c) Envelope spectrum as provided by the Fourier transform of |c, 10()/?. J. Antoni / Mechanical Systems and Signal Processing 4 (1111) 114i

Figure 13 (a) Vibration signal and (b) magnitude of the complex envelope c}°(n) which locally maximises the kurtogram, together with its 0.1% signification threshold. (c) Envelope spectrum as provided by the Fourier transform of |c, 10()/?. J. Antoni / Mechanical Systems and Signal Processing 4 (1111) 114i

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Fig. 1. (a) Elementary low-pass and high-pass decomposition. (b) Fast computation of the kurtogram by means of an arborescent filter- bank structure. J. Antoni / Mechanical Systems and Signal Processing © (W400) 1i—8H) sampling by factor 2, this sequence offsprings wo new sequences cj’, (n) and cj'+!(n) at level k + 1. This process is iterated from level k = 0, where co(n) = x(n) down to level K — 1. heir respective length is also decreased by the same factor so filtered sequences is increased by a factor 2, but that overall the total amount of data remains the branches of the tree after filtration by /; (but be same. Noteworthy is the multi At each level the number of plication by (—/)” in the lower fore down-sampling) in order to turn the high-pass sequences into low-pass sequences and hence to respect the frequency ordering. As a consequence, coefficients ci.(m) can be interpreted as the complex envelope of signa x(n) positioned on the centra frequency
Fig. 1. (a) Elementary low-pass and high-pass decomposition. (b) Fast computation of the kurtogram by means of an arborescent filter- bank structure. J. Antoni / Mechanical Systems and Signal Processing © (W400) 1i—8H) sampling by factor 2, this sequence offsprings wo new sequences cj’, (n) and cj'+!(n) at level k + 1. This process is iterated from level k = 0, where co(n) = x(n) down to level K — 1. heir respective length is also decreased by the same factor so filtered sequences is increased by a factor 2, but that overall the total amount of data remains the branches of the tree after filtration by /; (but be same. Noteworthy is the multi At each level the number of plication by (—/)” in the lower fore down-sampling) in order to turn the high-pass sequences into low-pass sequences and hence to respect the frequency ordering. As a consequence, coefficients ci.(m) can be interpreted as the complex envelope of signa x(n) positioned on the centra frequency
Fig. 2. Fast kurtogram paving of the (frequency/frequency resolution) plane. J. Antoni {| Mechanical Systems and Signal Processing © (1HHN) 110K 2.4. Prototype filter specifications
Fig. 2. Fast kurtogram paving of the (frequency/frequency resolution) plane. J. Antoni {| Mechanical Systems and Signal Processing © (1HHN) 110K 2.4. Prototype filter specifications
Fig. 3. Filter-bank transfer functions obtained from a 8 coefficient FIR prototype filter with cut-off frequency f, = 0.3 (all transfer functions have been normalised to a unit maximum). The relationship between prototype filter A(m) and the filter-bank transfer functions is illustrated in Figs. and 4. Fig. 3 displays the filter-bank transfer functions in the first five decomposition levels obtained from a coefficient FIR prototype filter with cut-off frequency f, = 0.3. Note that this design produces a certat amount of overlap between adjacent filters, which is sometimes a desirable property. It also results in a estimator very similar to the STFT-estimator used in Ref. [3], but with a much faster implementation! Fo comparison, Fig. 4 displays the filter-bank transfer functions obtained from a 32 coefficient FIR prototyp filter with same cut-off frequency f,. This design produces filters with a higher-frequency selectivity.
Fig. 3. Filter-bank transfer functions obtained from a 8 coefficient FIR prototype filter with cut-off frequency f, = 0.3 (all transfer functions have been normalised to a unit maximum). The relationship between prototype filter A(m) and the filter-bank transfer functions is illustrated in Figs. and 4. Fig. 3 displays the filter-bank transfer functions in the first five decomposition levels obtained from a coefficient FIR prototype filter with cut-off frequency f, = 0.3. Note that this design produces a certat amount of overlap between adjacent filters, which is sometimes a desirable property. It also results in a estimator very similar to the STFT-estimator used in Ref. [3], but with a much faster implementation! Fo comparison, Fig. 4 displays the filter-bank transfer functions obtained from a 32 coefficient FIR prototyp filter with same cut-off frequency f,. This design produces filters with a higher-frequency selectivity.
Fig. 4. Filter-bank transfer functions obtained from a 32 coefficient FIR prototype filter with cut-off frequency f,, = 0.3 (all transfer functions have been normalised to a unit maximum).
Fig. 4. Filter-bank transfer functions obtained from a 32 coefficient FIR prototype filter with cut-off frequency f,, = 0.3 (all transfer functions have been normalised to a unit maximum).
Fig. 5. Paving of the (frequency/frequency resolution) plane in the case of the 1/3-binary tree kurtogram estimator. J. Antoni / Mechanical Systems and Signal Processing © (400K) 11-01 2.5. Extension to a 1/3-binary tree
Fig. 5. Paving of the (frequency/frequency resolution) plane in the case of the 1/3-binary tree kurtogram estimator. J. Antoni / Mechanical Systems and Signal Processing © (400K) 11-01 2.5. Extension to a 1/3-binary tree
Fig. 6. The (frequency/frequency resolution) region spanned by (a) the wavelet transform (light grey area for an octave decomposition and dark grey area for a quint decomposition), and (b) the STFT (light grey area for a short analysis-window and dark grey area for a long analysis-window).
Fig. 6. The (frequency/frequency resolution) region spanned by (a) the wavelet transform (light grey area for an octave decomposition and dark grey area for a quint decomposition), and (b) the STFT (light grey area for a short analysis-window and dark grey area for a long analysis-window).
Fig. 7. Transfer functions in the first five decomposition levels of a wavelet packet transform obtained from a Daubechies10 mother wavelet. J. Antoni / Mechanical Systems and Signal Processing © (400K) 1-41
Fig. 7. Transfer functions in the first five decomposition levels of a wavelet packet transform obtained from a Daubechies10 mother wavelet. J. Antoni / Mechanical Systems and Signal Processing © (400K) 1-41
Fig. 8. (a) Vibration signal and (b) magnitude of the complex envelope c}(n) which maximises the kurtogram, together with its 0.1% signification threshold.
Fig. 8. (a) Vibration signal and (b) magnitude of the complex envelope c}(n) which maximises the kurtogram, together with its 0.1% signification threshold.
Fig. 9. (a) Fast kurtogram of the signal of Fig. 8(a) computed over K = 7 levels with a 1/3 binary-tree and a 8 coefficient prototype filter. The maximum is reached at dyad {f'5;(Af)4} = {6750 Hz; 1500 Hz}. (b) Full kurtogram computed from the STFT with analysis-window lengths ranging “continuously” from N,, = 4 to 400. The maximum is reached at dyad {f; Af} = {6750 Hz; 1460 Hz} (Ay ~ 1.5F;/Ny fora Hanning analysis-window).
Fig. 9. (a) Fast kurtogram of the signal of Fig. 8(a) computed over K = 7 levels with a 1/3 binary-tree and a 8 coefficient prototype filter. The maximum is reached at dyad {f'5;(Af)4} = {6750 Hz; 1500 Hz}. (b) Full kurtogram computed from the STFT with analysis-window lengths ranging “continuously” from N,, = 4 to 400. The maximum is reached at dyad {f; Af} = {6750 Hz; 1460 Hz} (Ay ~ 1.5F;/Ny fora Hanning analysis-window).
Ball bearing characteristics I a a 0, SU frequency band where to filter and demodulate the signal. The first analysed signal is displayed in Fig. 11(a). It was measured on a test-rig at the Vibrations and Acoustics Laboratory of the University of New South Wales (Sydney). The system is a one-stage gearbox with primary and secondary shafts supported by ball bearings, with characteristics outlined in the following table. The corresponding fast kurtogram is displayed in Fig. 12. It clearly reveals the presence of sharp abnorma shocks in the signal, with a maximum kurtosis of about 40 at dyad {f3;(Af))} = {13 500 Hz; 3000 Hz}. The corresponding envelope magnitude |c3(n)| is shown in Fig. 11(b) and its envelope spectrum in Fig. 11(c). Note that the envelope spectrum immediately indicates a inner race fault (BPFI° = 71 Hz) with typical modulations at the shaft rotation speed (10 Hz).
Ball bearing characteristics I a a 0, SU frequency band where to filter and demodulate the signal. The first analysed signal is displayed in Fig. 11(a). It was measured on a test-rig at the Vibrations and Acoustics Laboratory of the University of New South Wales (Sydney). The system is a one-stage gearbox with primary and secondary shafts supported by ball bearings, with characteristics outlined in the following table. The corresponding fast kurtogram is displayed in Fig. 12. It clearly reveals the presence of sharp abnorma shocks in the signal, with a maximum kurtosis of about 40 at dyad {f3;(Af))} = {13 500 Hz; 3000 Hz}. The corresponding envelope magnitude |c3(n)| is shown in Fig. 11(b) and its envelope spectrum in Fig. 11(c). Note that the envelope spectrum immediately indicates a inner race fault (BPFI° = 71 Hz) with typical modulations at the shaft rotation speed (10 Hz).
Fig. 10. Spectrogram of the signal of the signal of Fig. 8(a) computed with a N,, = 256 Hanning window (Af ~ 300 Hz)
Fig. 10. Spectrogram of the signal of the signal of Fig. 8(a) computed with a N,, = 256 Hanning window (Af ~ 300 Hz)
Fig. 11. (a) Vibration signal and (b) magnitude of the complex envelope c3(n) which maximises the kurtogram, together with its 0.1% signification threshold. (c) Envelope spectrum as provided by the Fourier transform of I(n)/’.
Fig. 11. (a) Vibration signal and (b) magnitude of the complex envelope c3(n) which maximises the kurtogram, together with its 0.1% signification threshold. (c) Envelope spectrum as provided by the Fourier transform of I(n)/’.
Fig. 12. Fast kurtogram of the signal of Fig. 11(a) computed over K = 7 levels with a 1/3 binary-tree and a 8 coefficient prototype filter. The maximum is reached at dyad {f3; (Af))} = {13 500 Hz; 3000 Hz}.
Fig. 12. Fast kurtogram of the signal of Fig. 11(a) computed over K = 7 levels with a 1/3 binary-tree and a 8 coefficient prototype filter. The maximum is reached at dyad {f3; (Af))} = {13 500 Hz; 3000 Hz}.
Fig. 14. Fast kurtogram of the signal of Fig. 13(a) computed over K = 7 levels with a 1/3 binary-tree and a 8 coefficient prototype filter. Several local maxima are observed. One relevant maximum is reached at dyad {f4; (Af),9} = {14250 Hz; 1500 Hz}.
Fig. 14. Fast kurtogram of the signal of Fig. 13(a) computed over K = 7 levels with a 1/3 binary-tree and a 8 coefficient prototype filter. Several local maxima are observed. One relevant maximum is reached at dyad {f4; (Af),9} = {14250 Hz; 1500 Hz}.
Fig. 15. (a) Vibration signal and (b) magnitude of the complex envelope c}*(n) which locally maximises the kurtogram, together with its 0.1% signification threshold. (c) Envelope spectrum as provided by the Fourier transform of |cl4()/*. J. Antoni {| Mechanical Systems and Signal Processing © (440K) 1-H
Fig. 15. (a) Vibration signal and (b) magnitude of the complex envelope c}*(n) which locally maximises the kurtogram, together with its 0.1% signification threshold. (c) Envelope spectrum as provided by the Fourier transform of |cl4()/*. J. Antoni {| Mechanical Systems and Signal Processing © (440K) 1-H
Fig. 16. Fast kurtogram of the signal of Fig. 15(a) computed over K = 7 levels with a 1/3 binary-tree and a 8 coefficient prototype filter Several local maxima are observed. One relevant maximum is reached at dyad {f4; (Af),4} = {20 250 Hz; 1500 Hz}.
Fig. 16. Fast kurtogram of the signal of Fig. 15(a) computed over K = 7 levels with a 1/3 binary-tree and a 8 coefficient prototype filter Several local maxima are observed. One relevant maximum is reached at dyad {f4; (Af),4} = {20 250 Hz; 1500 Hz}.
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