INTRODUCTION
The time series and the spectrum of signals are the most important and fundamental
indexes in machinery condition monitoring. For nonstationary signals or signal
with transients, the time domain or frequency domain analysis loses faults detection
in many rotating machines. The Shorttime Fourier transform (STFT) is the most
widely tool for the display nonstationary signals in timefrequency domain.
However, the STFT cannot obtain a good resolution image simultaneously in both
time and frequency domains. There exists a fundamental resolution tradeoff:
improving the frequency resolution (by using a large window size) results in
a loss of time resolution or improving the time resolution (by using a small
window size) results in a loss of frequency resolution (Cohen,
1989; Hammond and White, 1996; Sejdic
et al., 2009). Thus, looking for a variable window size scheme for
STFT is of great worth.
Recent study have been developed the signal dependent adaptive timefrequency
analysis, in which the algorithm is allowed to adapt based on optimized performance
(Jones and Baraniuk, 1994; Czerwinski
and Jones, 1997; Liu et al., 2007; Zhong
and Huang, 2010). But these methods often require either computationally
expensive or perform only offline applications. For the purpose of online
applications, an efficient Variable Shorttime Fourier Transform (VSTFT) is
proposed in this study. The VSTFT is a data dependent algorithm, where the local
window size is allowed to change based on a single parameter of the kurtosis
(Wiggins, 1978; Lee and Nandi,
2000). It is a low computing cost scheme.
VARIABLE STFT SCHEME
The basic ideal of shorttime Fourier transform is that if one wants to know what frequencies exist at a particular time. Then take a small part of the signal around that time and Fourier analyzes it, neglecting the rest of signal. Since the time interval is short compared to the whole signal, this process is called taking the shorttime Fourier transform. For a given signal x(t), the STFT is:
where, w(t) is a Gaussian window function (Lee, 2010),
* is a complex conjugate, t_{s} is sampling time, N_{fix} is
a fixed window size, t_{fix} is a fixed time delay and f is frequency.
The proposed scheme is data dependent and requires a measurement of the local
kurtosis that is similar to the high order statistics used for deconvolution
in signal processing (Lee and Nandi, 2000). The local
shorttime timefrequency kurtosis is defined as:
Compared to the STFT of Eq. 1, the proposed variable STFT scheme measured the local kurtosis K_{local} for windowed data over time and the local window size is determined by:
where, K_{local} is a maximum kurtosis for all K_{local} and N_{max} is a preset maximum variable window size. Thus, the local window size N_{local} is proportional to the value of the local kurtosis K_{local}. For a given signal x(t), the variable STFT is:
where, t_{local} is a local time delay. With this procedure, the time and frequency resolutions are linked to the kurtosis. In regions where the kurtosis measures relatively high, a large window size is applied to improve the resolution of timefrequency image. Conversely, a small window size is applied to regions where the kurtosis measures relatively low. The kurtosis based variable STFT algorithm is as follows.
Algorithm: Kurtosis based variable STFT 

Finding an optimal window size solution for an adaptive scheme could require excessive computation power, yet still obtain an insignificant result. By using the proposed VSTFT scheme, the local variable window size is obtained using a simple and timeefficient algorithm without any optimization procedures. Thus, the computing cost of the VSTFT algorithm is only slightly greater than that of the standard STFT.
SIMULATION RESULT AND DISCUSSION
The simulation signals shown in Fig. 1a contain three types of signals: three impulses appear at 0.1, 0.2 and 0.9 sec; mixed sinusoids (0.5 sin (2πf_{1}t+0.5 sin (2πf_{2}t)), where f_{1} = 100 and f_{2} = Hz) appear between 0.4 and 0.5 sec; mixed chirp signals (sin [2π(f_{3}t+f_{3}t)]+sin (sin [2π(f_{4}t+f_{4}t)] here f_{3} = 50 and f_{3} = 100 Hz) appear between 0.7 and 0.8 sec. The signal is sampling at 1 kHz and the Gaussian noise SNR 10 dB is added to the data. The corresponding frequency spectrum is shown in Fig. 1b.
Figure 2a shows the timefrequency contour plot using the
STFT scheme, where the fixed Gaussian window size is N_{fix} = 128.
The STFT scheme smears the impulse components in the time domain, although it
detects the sinusoidal components in the frequency domain. Figure
2b shows the timefrequency contour plot using window size N_{fix}
= 64, where the impulse components match in the time domain, although the sinusoidal
components smear in the frequency domain. These results demonstrate the discussed
tradeoff of the STFT scheme. It is impossible to obtain good time and good frequency
resolutions simultaneously when using a fixed window size only.

Fig. 1(ab): 
Simulation signals with additive SNR 10 dB Gaussian noise
(a) Time series and (b) Frequency spectrum 

Fig. 2(ab): 
Timefrequency contour plot using Fig. 1
data and STFT with fixed window size (a) N_{fix} = 128 and (b) N_{fix}
= 64 

Fig. 3(ab):

Using VSTFT with N_{max} = 128 and Fig.
1 data (a) Timefrequency contour plot and (b) Adaptive window size
along the time axis 
Using the proposed VSTFT scheme, Fig. 3 shows the timefrequency contour plot where a dynamic Gaussian window size is based on the variations of the local kurtosis. Figure 3a shows distinguishable stationary components with transients in time and frequency domains. The variable window size is a function of time, as shown in Fig. 3b. Compared to the timefrequency contour plot using the STFT scheme in Fig. 2, the VSTFT scheme can properly select window sizes for distinguishing between harmonics and transients. This also demonstrates the VSTFT scheme is robust for Gaussian noise.
RESULTS AND DISCUSSION
In this section, the experimental vibration signals from a rotating generator are measured, as shown in Fig. 4. The National Instruments NI 9234 dynamic signal acquisition module and integrated circuit piezoelectric accelerometer (100 mV g^{1}) are applied for signal measurement. Figure 5a shows the time series of the vibration signals at sampling rate 20 kHz. The corresponding frequency spectrum is shown in Fig. 5b. The signals contain the main steadystate sinusoids signal at 6.5 k Hz and the unknown transients caused by coupling faults between two motors.

Fig. 4: 
Experimental setup 

Fig. 5(ab): 
Rotating vibration signals (a) Time series and (b) Frequency
spectrum 
Figure 6a shows the timefrequency contour plot using the STFT scheme, where the fixed Gaussian window size is N_{fix}= 128. The result shows the transients signal, appear around 0.01, 0.03, 0.05, 0.07 and 0.09 sec. It is not clear enough for impacting detection. Figure 6b shows the results of setting N_{fix} = 64 to improve the time resolution while the frequency resolution is reduced because of the short window size. When the proposed VSTFT scheme is applied, Fig. 6c shows a timefrequency contour plot with maximum window size N_{fix} = 128. The stationary components and transient signals are retrieved successfully. This confirms that the dynamic kurtosis measurement is suitably designed for the adaptive scheme, where the window size changes dynamically in conjunction variations in the vibration signals.
Figure 7 shows the timefrequency plots using the measured
signals of Fig. 5 with additive SNR 10 dB Gaussian noise.
Figure 7a shows the timefrequency contour plot using the
STFT scheme with N_{fix} = 64, where the transient signals are smeared
by noise. Using the VSTFT scheme the result shown in Fig. 7b,
the harmonic signals with transients are retrieved successfully.

Fig. 6(ac): 
Timefrequency contour plot using Fig. 5 data (a) STFT with
K_{max} = 128, (b) STFT with _{klocla} = 64 and (c) VSTFT
with N_{max} = 128 

Fig. 7(ab): 
Timefrequency contour plots of Fig. 5 data with 10 dB Gaussian
noise (a) STFT with N_{fix} = 64 and (b) VSTFT with N_{max}
= 128 
This confirms the VSTFT scheme is robust for measurement noise.
CONCLUSION
In this study, an efficient variable timefrequency analysis scheme is proposed. The algorithm varies the window size along the time axis based on the measurement of the local kurtosis. The method is tested using simulation signals and vibration signals. The results show that it is suitable for dealing with vibration signals with transients. Under the noise condition SNR 10 dB, the results show a better performance than the standard STFT and the computational cost is only slightly greater than that of the standard STFT scheme.