of X(f,T) is
（2.5）
At these frequencies, the transformed values give the Fourier components defined by
（2.6）
Where t has been included with X(f ) to have a scale kfactor of unity before the summation. Note that results are unique only out to k=N/2 since the Nyquist frequency occurs at this point. The function defined in Eq. (6) is often referred to as the Discrete Fourier Transform (DFT). A drawback in utilizing the DFT Eq. (6) for discrete signal processing is the time it takes to compute the transform values at all frequencies for large arrays of discrete data. More efficient numerical algorithms were developed to improve computational speed, and this has resulted in numerous the FFT algorithms.
2.3. Short-time Fourier Transform
The STFT is a Fourier Transform performed in successive time frames :
（2.7）
Where t is time, w is frequency, and h(u) is a temporal window function such as Rectangular, Gaussian, Blackman, Hanning, Hamming, etc. The STFT
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characterizes the temporal signal f(t) in both time and frequency domains. The main limitation of this TFAM is caused by the fixed temporal window applied, whose size (number of sampling points) defines the time and frequency resolutions; a large frame size improves the frequency resolution, while decreasing the time resolution and vice versa. Since fixed window is unable to adjust time and frequency resolution by signals, time-frequency resolution about non-stationary signals analysis needs to make mutual balance.
2.4. Wigner-ville Distribution
The WVD provides a relationship between time and frequency during the period of the time data window that is not present in standard Fourier Spectral Analysis. It is capable of displaying any phase and magnitude changes presented. The WVD can be defined in both the time and frequency domains. Due to computational efficiency and good time resolution, the time domain definition is used. The definition of the WVD is
Where x(t) is the time signal, conjugate, t is the time domain variable, and frequency. Any real signal x(t) is not only contaminated by the noise, but also by the interference terms. Suppression of both of these requires a combination of time and frequency windowing with the WVD. This approach is called the Smoothed Pseudo-Wigner-Ville Distribution (SPWVD), and expressed as follows:
（2.9）
In this Eq. (9), two independent windows exist. The function W() is
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（2.8）
a result of the application of a truncating window to the original time data, and determines the frequency resolution of the WVD. The g(t) is a smoothing window, which determine the times resolution. The independence of the two window functions enables them to be applied individually, or in combination, so that the desired degree of interference suppression can be achieved. Discrete smoothed version of the SPWVD can be expressed as
（2.10）
Where the n and m denote the time and frequency indices, respectively, and S is analytic signal obtained form Hilbert transform of the original windowed signal. N is the half-size of the FFT window w(k) and Q is the half-size of the post smoothing window g(p). The SPWVD is achieved by convolving the WVD with smoothing window.
2.5. Wavelet Transform
The WT, a popular tool for studying intermittent and localized phenomena in various signals, is efficient signal processing method to obtain time information lost in the FT, partly irregular fluctuation, time change changing, and discontinuous point. Due to the multi-resolution ability of the WT, the noised signals can be separated into several approximation signals and detail signals. The WT of a signal f(t) is defined as the sum of all of the time of the signal f(t) multiplied by a scaled and shifted version of the wavelet function (t). The coefficients C(a,b) of the WT of the signal f(t) can be expressed as follows
（2.11）
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Where a and b are the scaling and shifting parameters in the WT. Basically, a small scaling parameter corresponds to a compressed wavelet function. As a result, the rapidly changing features in the signal f(t), i.e. high frequency components, can be obtained from the WT by using a small scaling parameter. On the other hand, the low frequency features in the signal f(t) can be extracted by using a large scaling parameter with a stretched wavelet function. For the sampled signal, a digital signal, f(k), k=0, 1, 2, !-, the Discrete Wavelet Transform (DWT) should be considered. The most commonly used DWT is the scaling and shifting of parameters with powers of two. That is:
（2.12）
（2.13）
Where j is the number of levels in the DWT. The coefficients C(a,b) of the DWT can be divided into two parts: that is, one is the approximation coefficients and the other is the detailed coefficients. The approximation coefficients are the high scale and the low frequency components of the signal f(t), while the detail coefficients are the low scale and the high frequency components of the signal f(t). The approximation of the DWT for the sampled signal f(t) at Coefficients level j can be expressed as Eq. (14).
（2.14）
Where (n) is the scaling function associated with j,k (n). Similarly,
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the detail the wavelet function coefficients (D) of the DWT for the sampled signal f(t) at level j can be expressed as follows.
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提取磨削主轴型转子轴承系统在加速度期间振动信号的特征研究外文文献翻译、中英文翻译
