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 Ikaz^{TM} 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 Ikaz^{TM}
Multilevel coefficients responses toward the simultaneous variation of amplitude
and frequency in synthetic signals.
The mean value
for a signal with nnumber of data points is mathematically defined trough Eq.
1, where x_{i} is the value of the data point. The mean value is
one of the most important and often used parameters in indicating the tendency
of the data toward the center:
The standard deviation value is given by:
where, x_{i} is the value of the data point and
is the mean of the data. Base on Eq. 2, standard deviation
value measures the spread of the data about the mean value. Variance is the
square of the standard deviation as shown in Eq. 3:
Signal classification on the reallife 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:
where, x_{i} is the value of the data point and
is the mean of the data and s is the standard deviation value. Azrulhisham
et al. (2012) in their study used skewness value to estimate the
accelerated test model in fatigue life reliability evaluation of stub axle.
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:
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 kurtosisbased algorithm (Ikaz™): Ikaz^{TM}
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. Ikaz^{TM} coefficient, Z^{8
}and the value was supported by a three dimensional graphical summarisations
of frequency distribution (Nuawi et al., 2008a).
In the original Ikaz^{TM} 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 Ikaz^{TM} 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 Ikaz^{TM} coefficient
is very suitable in Signal Features (SF) translation. In previous researches,
the structureborne sound signal analyzed using Ikaz^{TM} 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 Ikaz^{TM} 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 Ikaz^{TM} coefficient to analyze the flank wear during
turning process for tool wear prediction purpose.
Development of Ikaz™ multilevel coefficient (^{L}Z^{∞}):
The development of Ikaz™ multilevel coefficient (^{L}Z^{∞})
was inspired by the original Ikaz^{TM} (Z^{∞}) which
was pioneered by Nuawi et al. (2008b). The new
symbol for Ikaz^{TM} Multilevel coefficient is defined as ^{L}Z^{∞}
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 (^{L}Z^{∞}) is
expected to have more sensitivity towards amplitude and frequency change in
a signal. In Ikaz^{TM} 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 F_{1}, F_{2}, F_{3} to F_{L
}in Fig. 1 are depending on the value of n and f_{max}.
For Ikaz™Multilevel with L^{th }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 Ikaz^{TM} Multilevel method 
The related Ikaz^{TM} Multilevel coefficient can be calculated as
(Karim et al., 2011):
where, L indicates the order of signal decomposition. Karim
et al. (2012) in their study used the Ikaz^{TM} 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 (F_{s} = 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(ab): 
(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 Ikaz^{TM} 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 Ikaz^{TM} Multilevel coefficient response. Frequency Remain and Amplitude Increase (FRAI) signal was used to investigate how much the Ikaz^{TM} Multilevel coefficient response towards amplitude change. Whereas, Frequency Increase and Amplitude Remain (FIAR) signal was used to investigate how much the Ikaz^{TM} 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 Ikaz^{TM} coefficients,^{ 4}Z^{∞}, ^{5}Z^{∞}, ^{6}Z^{∞}, ^{7}Z^{∞} and ^{8}Z^{∞} 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 Ikaz^{TM} coefficients, ^{4}Z^{∞}, ^{5}Z^{∞}, ^{6}Z^{∞}, ^{7}Z^{∞} and ^{8}Z^{∞} were calculated and compared.
RESULTS AND DISCUSSION
The Ikaz^{TM} 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 Ikaz^{TM} method
were mainly on the application of this method on analyzing dynamic signals.
There is no specific work on the investigation of the Ikaz^{TM} 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 Ikaz^{TM} 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 Ikaz^{TM}
Multilevel coefficient and the normal Ikaz^{TM} coefficient response
were mainly due the change in amplitude of signals. These results was in accordance
with the result in this study.
Ikaz^{TM} multilevel coefficient response towards the FIAI signal:
Four types of signals from original FIAI signal were created by increasing its
amplitude and frequency from 1040% by 10% incremental. At the same time, the
signal frequency is changed by 1040 Hz by 10 Hz incremental. Figure
3ab and 4ab 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 Ikaz^{TM} (Z^{∞}) and Ikaz^{TM} Multilevel coefficients (^{L}Z^{∞} ) toward different level of amplitude and frequency for the FIAI signal are presented in Fig. 5 and Table 1.
The Ikaz^{TM} 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(ab): 
The 20% Amplitude and 20 Hz frequency increase of FIAI signal
(a) Timedomain (b) Frequency domain 
Table 1: 
The respond of each coefficient in deviation percentage with
respect to FIAI signal 

The coefficient symbol ^{3}Z^{∞}, ^{4}Z^{∞},
^{5}Z^{∞}, ^{6}Z^{∞}, ^{7}Z^{∞}
and ^{8}Z^{∞} 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 Ikaz^{TM} Multilevel coefficients, the
more sensitive it responded to the amplitude and frequency change. For this
type of particular synthetic signal, the sensitivity of the Ikaz^{TM}
Multilevel coefficient saturated at the 7th order (^{7}Z^{∞}).
Ikaz^{TM} 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(ab): 
The 40% Amplitude and 40 Hz frequency increase of FIAI signal
(a) Timedomain (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 Ikaz^{TM} (Z^{∞}) and Ikaz^{TM} Multilevel coefficients (^{L}Z^{∞} ) toward different degree of the FIAD signal are presented in Fig. 8 and Table 2.
The Ikaz^{TM} 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 Ikaz^{TM} Multilevel values. From Table 2, it was found that the most sensitive Ikaz^{TM} Multilevel value was found at level 3.
Ikaz^{TM} 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 Ikaz^{TM} Multilevel coefficient
in FIAI signal 

Fig. 6(ab): 
The 10% amplitude decrease and 10 Hz frequency increase of
FIAD signal (a) Timedomain (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(ab): 
The 40% amplitude decrease and 40 Hz frequency increase of
FIAD signal (a) Timedomain (b) Frequency domain 

Fig. 8: 
The response of Ikaz^{TM} 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(ab): 
The 10% amplitude increase and 10 Hz frequency decrease of
FDAI signal (a) Timedomain (b) Frequency domain 
Table 2: 
The response of each coefficient in deviation percentage for
FIAD signal 

^{3}Z^{∞}, ^{4}Z^{∞},
^{5}Z^{∞}, ^{6}Z^{∞}, ^{7}Z^{∞}
and ^{8}Z^{∞}: Value of the coefficient calculated
when the FIAD signal decomposed to 3, 4, 5, 6, 7 and 8 levels, respectively 
The result of Ikaz^{TM} (Z^{∞}) and Ikaz^{TM} Multilevel coefficients (^{L}Z^{∞} ) toward different degree of the FDAI signal are presented in Fig. 11 and Table 3.
The Ikaz^{TM} Multilevel value increase linearly when the amplitude
is increased and the frequency is decreased in FDAI signal.

Fig. 10(ab): 
The 40% amplitude increase and 40 Hz frequency decrease of
FDAI signal (a) Timedomain (b) Frequency domain 
Table 3: 
The response of each coefficient in deviation percentage for
FDAI signal 

^{3}Z^{∞}, ^{4}Z^{∞},
^{5}Z^{∞}, ^{6}Z^{∞}, ^{7}Z^{∞}
and ^{8}Z^{∞}: 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 Ikaz^{TM}
Multilevel value. The Ikaz^{TM} Multilevel values are increasing following
the increasing trend in amplitude even though the frequency is decreased. From
Table 3, the most sensitive Ikaz^{TM} Multilevel
coefficient occur at level 3.

Fig. 11: 
The response of Ikaz^{TM} multilevel coefficient
in FDAI signal 
Table 4: 
The Response of Ikaz^{TM} Multilevel coefficients
toward FIAR signal 

^{3}Z^{∞}, ^{4}Z^{∞},
^{5}Z^{∞}, ^{6}Z^{∞}, ^{7}Z^{∞}
and ^{8}Z^{∞}: 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 Ikaz^{TM} Multilevel coefficients
toward FRAI signal 

^{3}Z^{∞}, ^{4}Z^{∞},
^{5}Z^{∞}, ^{6}Z^{∞}, ^{7}Z^{∞}
and ^{8}Z^{∞}: Value of the coefficient calculated
when the FRAI signal decomposed to 3, 4, 5, 6, 7 and 8 levels, respectively 
Ikaz^{TM} 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 Ikaz^{TM} Multilevel coefficient value. Table 4 shows the value of Ikaz^{TM} 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 Ikaz^{TM} 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 Ikaz^{TM} Multilevel coefficient value.

Fig. 12: 
FIAI original with calculated Ikaz multilevel coefficient
(^{7}Z^{∞}) = 0.0099 

Fig. 13: 
FIAI after 40% and 40 Hz increased with calculated Ikaz multilevel
coefficient (^{7}Z^{∞}) = 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 Ikaz^{TM} 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 Ikaz^{TM} Multilevel coefficient can be estimated by below equation:
Ikaz^{TM} multilevel representation for FIAI signals: The original Ikaz^{TM} technique provide not only the coefficient value, but also a 3D graphical illustration. The higher value of ^{L}Z^{∞} refers to the bigger space scattering of Ikaz^{TM} Multilevel representation.
By using Eq. 68, for the Nyquist number
equal to 2, frequency span equal to 1000 Hz, f_{max }equal to 500 Hz
and L equal to 3, the frequency ranges of the Ikaz^{TM} Multilevel
representation can be summarized as follows:
• 
xaxis: Low Frequency (LF) range of 00.25 f_{max} 
• 
yaxis: High Frequency (HF) range of 0.25 f_{max}0.5 f_{max} 
• 
zaxis: Very high frequency (VF) range of 0.5 f_{max}f_{max} 
The Ikaz^{TM} 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 Ikaz^{TM}
Multilevel 3D representation for the Znotch filtered signal in Fig.
13 was spread compared to the Ikaz^{TM} Multilevel 3D representation
in Fig. 12. Thus, the judgment of the existence of different
signals can be based on the Ikaz^{TM} Multilevel coefficient and also
the Ikaz^{TM} Multilevel 3D representation. A similar single Ikaz^{TM}
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 Ikaz^{TM} Multilevel coefficients toward the simultaneous change in amplitude and frequency of signals. This new Ikaz^{TM} 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 Ikaz^{TM} 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 Ikaz^{TM} was found to be at the highest order of signal decomposition which is ^{8}Z^{∞}. The increasing rate of Ikaz^{TM} Multilevel coefficients in response with FIAI signal is greater compare to the response in FDAI signal.
The study of FIAD showed that the Ikaz^{TM} Multilevel coefficients were decreasing with the reduction in amplitude regardless of the increasing of the frequency in the signal. The optimized value of Ikaz^{TM} Multilevel coefficient was found to be at the lowest order of signal decomposition which is ^{3}Z^{∞}.
In FIAI, FIAD and FDAI signal study, the response of Ikaz^{TM} Multilevel coefficients are greatly depending on the amplitude change in the signals. The increasing or decreasing of the Ikaz^{TM} Multilevel coefficients would follow the trend in the amplitude of the measured signals. The influence of amplitude and frequency in the Ikaz^{TM} 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 Ikaz^{TM} Multilevel coefficients saturated at the highest level of signal decomposition except in the response of FIAD signal. Recognizing and understanding the behaviour of Ikaz^{TM} 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 030102SF0647, for supporting this research.