Abstract: The response of Integrated Kurtosis-Based Algorithm for Z-Notch Filter (I-kazTM) Multilevel coefficients toward the simultaneous changes in amplitude and frequency of signals was unknown. This study presents the coefficients responses toward the synthetic signals which vary in amplitude and frequency simultaneously. The I-kazTM Multilevel coefficients, 3rd order (3Z∞), 4th order (4Z∞), 5th order (5Z∞), 6th order (6Z∞), 7th order (7Z∞) and 8th order (8Z∞) were investigated by analyzing their response using three types of synthetic signals, FIAI, FIAD and FDAI. The responses of the I-kazTM Multilevel coefficients were analyzed by either increasing or decreasing the amplitude and frequency in the synthetic signals simultaneously. This study indicated that the response of all orders of I-kazTM Multilevel coefficients showed an increasing trend with respect to the increment in amplitude of FIAI and FDAI. All the coefficients however showed a decreasing trend in FIAD synthetic signal study regardless the increase in the frequency. The study also indicated that the change in amplitude has more influence than the change in frequency in the I-kazTM Multilevel coefficients responses. The influence ratio of the amplitude to frequency change was estimated to be 86.96-13.06%, respectively. Recognizing the behaviour of I-kazTM Multilevel coefficients toward the change in amplitude and frequency is important especially when analysing dynamic signals.
INTRODUCTION
Statistical signal analysis is actually a mathematical science that involves data collection, analysis, interpretation and presentation (Nuawi et al., 2008a). The main objective for the statistical analysis is to provide an easy and a simple analysis of a complex random signal.
Data at several levels could be revealed by the classification and interpretation of the signals (Chatfield and Collins, 1980). Common parameters such as the mean value, standard deviation value, the variance, the skewness, the kurtosis and the root mean square (rms) are used in statistical analysis (Pontuale et al., 2003; Abdullah, 2005).
Average value, standard deviation, variance, skewness, kurtosis and root mean square (rms) are the common Signal Features (SFs) that can be used for extraction from any time domain signal (Sick, 2002; Ghosh et al., 2007; Dong et al., 2006). In order a signal to be able to adequately described and maintained the relevant information, SF from the captured signal need to be properly derived (Teti et al., 2010). Previous works related to I-kazTM method were mainly on the application of this method on analyzing dynamic signals. The main objective of this study on the other hand, is to investigate the I-kazTM Multilevel coefficients responses toward the simultaneous variation of amplitude and frequency in synthetic signals.
The mean value
| (1) |
The standard deviation value is given by:
| (2) |
where, xi is the value of the data point and
| (3) |
Signal classification on the real-life signals base on mean and variance was not compatible mainly due to the signals contain outliers that can bring a noticeable shift in the actual value of both mean and variance (Pontuale et al., 2003).
Skewness is the measurement of the asymmetry from the normal distribution in a set of statistical data. The skewness S, of a set of data is calculated base on the Eq. 4:
| (4) |
where, xi is the value of the data point and
The signal 4th statistical moment Kurtosis K, is an important global signal statistic that is very sensitive to the spikiness of the data. The value of Kurtosis K, for discrete data sets is defined in Eq. 5:
| (5) |
The normal or Gaussian distribution, the Kurtosis value is approximately 3.0. The presence of more extreme value or amplitude than should be found in a Gaussian distribution can be detected when the kurtosis value is higher than 3.0.
In industries, statistical analysis using Kurtosis value were used frequently in detecting defect symptoms due to its sensitivity towards the existence of high amplitude (Pontuale et al., 2003). A proper maintenance can be conducted systematically and accurately base on the measurement of Kurtosis value.
Development of integrated kurtosis-based algorithm (I-kaz™): I-kazTM was formulated base on the concept of data distribution or scattering about its center points. It was developed with the purpose of giving descriptive and inferential statistics which is an advantage in comparison with other statistical methods that only rely on numerical value. I-kazTM coefficient, Z8 and the value was supported by a three dimensional graphical summarisations of frequency distribution (Nuawi et al., 2008a).
In the original I-kazTM coefficient calculation, a dynamic signal in time domain will be decomposed into three frequency band by following the 3rd order of the Daubechies concept in signal decomposition process (Daubechies, 1992). To calculate the sampling frequency of any signal in time domain, the Nyquist number must be 2 or greater than the maximum frequency in order to avoid the content of the sampling signal to be misinterpreted. Nyquist number in this calculation was chosen to be equal to 2 for the purpose of calculation simplification (Figliola and Beasley, 2000).
The sensitivity of the I-kazTM coefficient toward amplitude and frequency change in a signal was proven far better than the current statistical parameters (Nuawi et al., 2008a; Karim et al., 2011). With this advantage, the I-kazTM coefficient is very suitable in Signal Features (SF) translation. In previous researches, the structure-borne sound signal analyzed using I-kazTM which correlated with the internal pipe surface condition showed a high ability to differentiate between the smooth and rough pipe surface (Nuawi et al., 2009). Different study showed that the I-kazTM method was capable of improving the Taylor curve which was unable to exhibit the three typical wear curve for the cutting that use certain cutting speed (Nuawi et al., 2007). Ghani et al. (2011) in used the I-kazTM coefficient to analyze the flank wear during turning process for tool wear prediction purpose.
Development of I-kaz™ multilevel coefficient (LZ∞): The development of I-kaz™ multilevel coefficient (LZ∞) was inspired by the original I-kazTM (Z∞) which was pioneered by Nuawi et al. (2008b). The new symbol for I-kazTM Multilevel coefficient is defined as LZ∞ in which L is referring to the number of order of signal decomposition. The decomposition of signals in time domain into more frequency bands is to get a better coefficient response especially in the lower part of the frequency spectrum. The new developed coefficient (LZ∞) is expected to have more sensitivity towards amplitude and frequency change in a signal. In I-kazTM Multilevel method, signal decomposition using Lth order of Daubechies theorem will result in L number of frequency bands. This algorithm was summarized as presented in Fig. 1.
The frequency ranges of F1, F2, F3 to FL in Fig. 1 are depending on the value of n and fmax. For I-kaz™Multilevel with Lth order of signal decomposition and for i = 1, 2, 3...L, the frequency ranges are shown below (Karim et al., 2011):
| Fig. 1: | Flowchart of the I-kazTM Multilevel method |
| (6) |
| (7) |
| (8) |
The related I-kazTM Multilevel coefficient can be calculated as (Karim et al., 2011):
| (9) |
where, L indicates the order of signal decomposition. Karim et al. (2012) in their study used the I-kazTM Multilevel coefficient at level 7 of signal decomposition to correlate the wear rate of connecting rod bearing.
MATERIALS AND METHODS
Creating synthetic signals FIAI, FIAD and FDAI: Three different synthetic signals, Frequency Increase and Amplitude Increase (FIAI), Frequency Increase and Amplitude Decrease (FIAD) and Frequency Decrease and Amplitude Increase (FDAI) were created with the same initial specifications. All signals were created by using MATLAB® and were defined with 512 data points and sampled at 1000 Hz (Fs = 1000 Hz). The synthetic signals originally consist of 10, 25, 40, 55, 70, 100, 140, 200, 300 and 350 Hz sinusoidal waves. The plots of the signals in time and frequency domain are shown in Fig. 2a and b. The unit used in time domain for y axis is volt (V).
| Fig. 2(a-b): | (a) Time domain of the test signal and (b) Frequency domain of the test signal |
The Fast Fourier Transform (FFT) method was used to transform the signal in time domain to frequency domain (Nuawi et al., 2008b).
The amplitude and frequency of the synthetic signals were increased by 10, 20, 30, 40% and 10, 20, 30 and 10 Hz, respectively. For each pair of incremental value, the higher order of I-kazTM Multilevel coefficients were calculated and compared.
Creating synthetic signals FRAI and FIAR: Another two types of synthetic signals were created in order to identify the influence ratio of the amplitude and frequency in I-kazTM Multilevel coefficient response. Frequency Remain and Amplitude Increase (FRAI) signal was used to investigate how much the I-kazTM Multilevel coefficient response towards amplitude change. Whereas, Frequency Increase and Amplitude Remain (FIAR) signal was used to investigate how much the I-kazTM Multilevel coefficient response towards frequency change. The initial specification of both FRAI and FIAR signals were the same as the specification in signal FIAI, FIAD and FDAI.
The amplitude of signal FRAI was increased by 40% while the frequencies were kept constant. For the amplitude increment, the higher order of I-kazTM coefficients, 4Z∞, 5Z∞, 6Z∞, 7Z∞ and 8Z∞ were calculated and compared.
In FIAR signal, the frequency was increased by the incremental of 40 Hz while the amplitude was kept constant. Similarly, for the frequency increment, the higher order of I-kazTM coefficients, 4Z∞, 5Z∞, 6Z∞, 7Z∞ and 8Z∞ were calculated and compared.
RESULTS AND DISCUSSION
The I-kazTM method was applied in various field of study, such as automotive engine performance monitoring (Nuawi et al., 2008b), fatigue analysis (Abdullah et al., 2007; Putra et al., 2010) and machining condition monitoring (Nuawi et al., 2007; Jaharah et al., 2009). Previous works related to I-kazTM method were mainly on the application of this method on analyzing dynamic signals. There is no specific work on the investigation of the I-kazTM Multilevel coefficients responses toward the simultaneous variation of amplitude and frequency in synthetic signals. The most related works related to this study is the study on the I-kazTM Multilevel coefficient response toward the change in amplitude and frequency when one of them was increased and one of them was kept constant (Karim et al., 2011; Nuawi et al., 2008b). The results from these studies showed that the I-kazTM Multilevel coefficient and the normal I-kazTM coefficient response were mainly due the change in amplitude of signals. These results was in accordance with the result in this study.
I-kazTM multilevel coefficient response towards the FIAI signal: Four types of signals from original FIAI signal were created by increasing its amplitude and frequency from 10-40% by 10% incremental. At the same time, the signal frequency is changed by 10-40 Hz by 10 Hz incremental. Figure 3a-b and 4a-b show the sample plot of FIAI signals in time and frequency domain after 20 and 40% amplitude increment and 20 and 40 Hz frequency increment, respectively.
The results of I-kazTM (Z∞) and I-kazTM Multilevel coefficients (LZ∞ ) toward different level of amplitude and frequency for the FIAI signal are presented in Fig. 5 and Table 1.
The I-kazTM Multilevel coefficient values increase linearly with the increase in amplitude and frequency of the FIAI signal. The superscript number on the top left of letter Z represents the level of signal decomposition. The sensitivity of all coefficients toward the amplitude change can be seen clearly from Table 1.
| Fig. 3(a-b): | The 20% Amplitude and 20 Hz frequency increase of FIAI signal (a) Time-domain (b) Frequency domain |
| Table 1: | The respond of each coefficient in deviation percentage with respect to FIAI signal |
| The coefficient symbol 3Z∞, 4Z∞, 5Z∞, 6Z∞, 7Z∞ and 8Z∞ represent the value of the coefficient calculated when the FIAI signal decomposed to 3, 4, 5, 6, 7 and 8 levels, respectively | |
The higher the order of the I-kazTM Multilevel coefficients, the more sensitive it responded to the amplitude and frequency change. For this type of particular synthetic signal, the sensitivity of the I-kazTM Multilevel coefficient saturated at the 7th order (7Z∞).
I-kazTM multilevel coefficient response toward the FIAD signal: Four types of signals from original FIAD signal were created by increasing its sinusoidal frequencies by 10 Hz incremental and decreasing its amplitude by 10% decremental as shown in Fig. 6a and b.
| Fig. 4(a-b): | The 40% Amplitude and 40 Hz frequency increase of FIAI signal (a) Time-domain (b) Frequency domain |
Figure 7a and b show the sample plot of FIAD signal in time and frequency domain after 40 Hz frequency increment and 40% amplitude decrement of the FIAD.
The result of I-kazTM (Z∞) and I-kazTM Multilevel coefficients (LZ∞ ) toward different degree of the FIAD signal are presented in Fig. 8 and Table 2.
The I-kazTM Multilevel coefficient values decrease linearly when frequency is increased and amplitude is decreased. At this stage, the change in amplitude has more influence on the response of I-kazTM Multilevel values. From Table 2, it was found that the most sensitive I-kazTM Multilevel value was found at level 3.
I-kazTM multilevel coefficient response toward the FDAI signal: Four types of signals from original FDAI signal were created by deceasing its sinusoidal frequencies by 10 Hz decremental and increasing its amplitude by 10% incremental.
| Fig. 5: | The response of I-kazTM Multilevel coefficient in FIAI signal |
| Fig. 6(a-b): | The 10% amplitude decrease and 10 Hz frequency increase of FIAD signal (a) Time-domain (b) Frequency domain |
Figure 9a and b show the sample plot of FDAI signal in time and frequency domain after 10 Hz frequency increment and 10% amplitude reduction of FDAI signal.
| Fig. 7(a-b): | The 40% amplitude decrease and 40 Hz frequency increase of FIAD signal (a) Time-domain (b) Frequency domain |
| Fig. 8: | The response of I-kazTM Multilevel coefficient in FIAD signal |
Figure 10a and b show the sample plot of FDAI signal in time and frequency domain after 40 Hz frequency increment and 40% amplitude reduction of FDAI signal.
| Fig. 9(a-b): | The 10% amplitude increase and 10 Hz frequency decrease of FDAI signal (a) Time-domain (b) Frequency domain |
| Table 2: | The response of each coefficient in deviation percentage for FIAD signal |
| 3Z∞, 4Z∞, 5Z∞, 6Z∞, 7Z∞ and 8Z∞: Value of the coefficient calculated when the FIAD signal decomposed to 3, 4, 5, 6, 7 and 8 levels, respectively | |
The result of I-kazTM (Z∞) and I-kazTM Multilevel coefficients (LZ∞ ) toward different degree of the FDAI signal are presented in Fig. 11 and Table 3.
The I-kazTM Multilevel value increase linearly when the amplitude is increased and the frequency is decreased in FDAI signal.
| Fig. 10(a-b): | The 40% amplitude increase and 40 Hz frequency decrease of FDAI signal (a) Time-domain (b) Frequency domain |
| Table 3: | The response of each coefficient in deviation percentage for FDAI signal |
| 3Z∞, 4Z∞, 5Z∞, 6Z∞, 7Z∞ and 8Z∞: Value of the coefficient calculated when the FDAI signal decomposed to 3, 4, 5, 6, 7 and 8 levels, respectively | |
The change of amplitude in FDAI signal has more influence in the I-kazTM Multilevel value. The I-kazTM Multilevel values are increasing following the increasing trend in amplitude even though the frequency is decreased. From Table 3, the most sensitive I-kazTM Multilevel coefficient occur at level 3.
| Fig. 11: | The response of I-kazTM multilevel coefficient in FDAI signal |
| Table 4: | The Response of I-kazTM Multilevel coefficients toward FIAR signal |
| 3Z∞, 4Z∞, 5Z∞, 6Z∞, 7Z∞ and 8Z∞: Value of the coefficient calculated when the FIAR signal decomposed to 3, 4, 5, 6, 7 and 8 levels, respectively | |
| Table 5: | The Response of I-kazTM Multilevel coefficients toward FRAI signal |
| 3Z∞, 4Z∞, 5Z∞, 6Z∞, 7Z∞ and 8Z∞: Value of the coefficient calculated when the FRAI signal decomposed to 3, 4, 5, 6, 7 and 8 levels, respectively | |
I-kazTM multilevel coefficient response toward the FRAI and FIAR signal: Synthetic signals FRAI and FIAR were created to investigate the percentage contribution of frequency and amplitude toward the change in I-kazTM Multilevel coefficient value. Table 4 shows the value of I-kazTM Multilevel coefficients calculated on FIAR signal when the frequency was increased to 50 Hz while the amplitude was kept constant. In Table 5, FRAI synthetic signals were used to calculate the value of I-kazTM Multilevel coefficients in which the amplitude of this was increased to 50% while the frequency was kept constant.
The above results also clearly indicate that the amplitude change has more influence than the frequency change in FRAI and FIAR signals, respectively for the I-kazTM Multilevel coefficient value.
| Fig. 12: | FIAI original with calculated I-kaz multilevel coefficient (7Z∞) = 0.0099 |
| Fig. 13: | FIAI after 40% and 40 Hz increased with calculated I-kaz multilevel coefficient (7Z∞) = 0.0023 |
A very similar results were reported in the previous study by Karim et al. (2011) and Nuawi et al. (2008b) in their study to investigate the sensitivity of I-kazTM coefficient versus the current statistical coefficient such as standard deviation, kurtosis, skewness and root mean square (rms).
The influence ratio of amplitude and frequency toward I-kazTM Multilevel coefficient can be estimated by below equation:
I-kazTM multilevel representation for FIAI signals: The original I-kazTM technique provide not only the coefficient value, but also a 3-D graphical illustration. The higher value of LZ∞ refers to the bigger space scattering of I-kazTM Multilevel representation.
By using Eq. 6-8, for the Nyquist number equal to 2, frequency span equal to 1000 Hz, fmax equal to 500 Hz and L equal to 3, the frequency ranges of the I-kazTM Multilevel representation can be summarized as follows:
| • | x-axis: Low Frequency (LF) range of 0-0.25 fmax |
| • | y-axis: High Frequency (HF) range of 0.25 fmax-0.5 fmax |
| • | z-axis: Very high frequency (VF) range of 0.5 fmax-fmax |
The I-kazTM Multilevel 3D representation and their coefficients values are shown in Fig. 12 and 13 for the original FIAI signal and the FIAI signal after 40% amplitude and 40 Hz 000011 frequency increase, respectively. The data distribution of the I-kazTM Multilevel 3D representation for the Z-notch filtered signal in Fig. 13 was spread compared to the I-kazTM Multilevel 3D representation in Fig. 12. Thus, the judgment of the existence of different signals can be based on the I-kazTM Multilevel coefficient and also the I-kazTM Multilevel 3D representation. A similar single I-kazTM 3D scattering pattern was reported by Nuawi et al. (2008b) when studying for the filtered and the unfiltered signal in machining.
CONCLUSION
This study discussed the response of I-kazTM Multilevel coefficients toward the simultaneous change in amplitude and frequency of signals. This new I-kazTM Multilevel method was proven to be very sensitive and detects very well in amplitude and frequency changes of measured signals.
In FIAI and FDAI signal study, all level of I-kazTM Multilevel coefficients were noted to be increasing with the increase in amplitude regardless of the frequency condition, either increasing or decreasing. For both types of signals, the optimized value of I-kazTM was found to be at the highest order of signal decomposition which is 8Z∞. The increasing rate of I-kazTM Multilevel coefficients in response with FIAI signal is greater compare to the response in FDAI signal.
The study of FIAD showed that the I-kazTM Multilevel coefficients were decreasing with the reduction in amplitude regardless of the increasing of the frequency in the signal. The optimized value of I-kazTM Multilevel coefficient was found to be at the lowest order of signal decomposition which is 3Z∞.
In FIAI, FIAD and FDAI signal study, the response of I-kazTM Multilevel coefficients are greatly depending on the amplitude change in the signals. The increasing or decreasing of the I-kazTM Multilevel coefficients would follow the trend in the amplitude of the measured signals. The influence of amplitude and frequency in the I-kazTM Multilevel coefficients response can be estimated in the FRAI and FIAR signals study. From this study, the influence ratio of amplitude and frequency is 86.96 and 13.04%, respectively. The study of this five different synthetic signals showed that the I-kazTM Multilevel coefficients saturated at the highest level of signal decomposition except in the response of FIAD signal. Recognizing and understanding the behaviour of I-kazTM Multilevel coefficients toward the change in amplitude and frequency is important especially when analysing dynamic signals.
ACKNOWLEDGMENTS
The authors wish to express their gratitude to Universiti Kebangsaan Malaysia and the Government of Malaysia, through the fund of 03-01-02-SF0647, for supporting this research.