小波功率谱相关译文.doc

小波功率谱的应用译文THE CONTINUOUS WAVELET TRANSFORMThe continuous wavelet transform was developed as an alternative approach to the short time Fourier transform to overcome the resolution problem. The wavelet analysis is done in a similar way to the STFT analysis, in the sense that the signal is multiplied with a function, it the wavelet, similar to the window function in the STFT, and the transform is computed separately for different segments of the time-domain signal. However, there are two main differences between the STFT and the CWT: 1. The Fourier transforms of the windowed signals are not taken, and therefore single peak will be seen corresponding to a sinusoid, i.e., negative frequencies are not computed. 2. The width of the window is changed as the transform is computed for every single spectral component, which is probably the most significant characteristic of the wavelet transform. The continuous wavelet transform is defined as followsEquation 3.1As seen in the above equation , the transformed signal is a function of two variables, tau and s , the translation and scale parameters, respectively. psi(t) is the transforming function, and it is called the mother wavelet . The term mother wavelet gets its name due to two important properties of the wavelet analysis as explained below: The term wavelet means a small wave . The smallness refers to the condition that this (window) function is of finite length ( compactly supported). The wave refers to the condition that this function is oscillatory . The term mother implies that the functions with different region of support that are used in the transformation process are derived from one main function, or the mother wavelet. In other words, the mother wavelet is a prototype for generating the other window functions. The term translation is used in the same sense as it was used in the STFT; it is related to the location of the window, as the window is shifted through the signal. This term, obviously, corresponds to time information in the transform domain. However, we do not have a frequency parameter, as we had before for the STFT. Instead, we have scale parameter which is defined as $1/frequency$. The term frequency is reserved for the STFT. Scale is described in more detail in the next section.The ScaleThe parameter scale in the wavelet analysis is similar to the scale used in maps. As in the case of maps, high scales correspond to a non-detailed global view (of the signal), and low scales correspond to a detailed view. Similarly, in terms of frequency, low frequencies (high scales) correspond to a global information of a signal (that usually spans the entire signal), whereas high frequencies (low scales) correspond to a detailed information of a hidden pattern in the signal (that usually lasts a relatively short time). Cosine signals corresponding to various scales are given as examples in the following figure .Figure 3.2Fortunately in practical applications, low scales (high frequencies) do not last for the entire duration of the signal, unlike those shown in the figure, but they usually appear from time to time as short bursts, or spikes. High scales (low frequencies) usually last for the entire duration of the signal.Scaling, as a mathematical operation, either dilates or compresses a signal. Larger scales correspond to dilated (or stretched out) signals and small scales correspond to compressed signals. All of the signals given in the figure are derived from the same cosine signal, i.e., they are dilated or compressed versions of the same function. In the above figure, s=0.05 is the smallest scale, and s=1 is the largest scale.In terms of mathematical functions, if f(t) is a given function f(st) corresponds to a contracted (compressed) version of f(t) if s 1 and to an expanded (dilated) version of f(t) if s 1 dilates the signals whereas scales s 1则f(t)对应压缩版本；若s 1，规模扩张的信号，s1，压缩信号。在本文中都将采用这种解释。连续小波变换计算消费物价指数在这一节将解释上述方程。设x（t）是要分析信号。母小波选择作为进程中的所有窗口的原型。所有被使用的扩张（或压缩）和移出母小波版本的Windows。有许多是用于此目的的职能。有两个候选函数： Morlet小波和墨西哥帽函数，他们是为小波分析的例子是在本章稍后使用。 一旦选择了母小波开始计算S = 1和连续小波变换为S，体积更小，大于“ 1”所有值计算。然而，在信号的不同，一个完整的转换通常没有必要。对于所有的实际目的，信号是带限的，因此，在变换的尺度有限区间计算通常就足够了。在这项研究中，使用了一些为有限区间的价值观，如将在本章后面介绍。 为方便起见，该过程将开始从规模S = 1和将继续为S，即增加值，分析将开始着手从高向低频率。这第一个值将对应到最压缩的小波。而S值增加，小波会扩张。 小波被放置在一开始即时间对应为0，在小波函数尺度“1”乘以信号，然后对所有次积分。该整合的结果再乘以数量不变1/sqrtS的。乘法是为了让转换后的信号将在每一个规模相同的能量能源正常化的目的。最后的结果是转换，即价值，对连续小波变换在时间价值为零，规模为S=1。换句话说，它是值对应的tau =0，在时间尺度平面s=1。在S=1的小波规模为，然后向右转向tau，在本地令t=tau，上面的公式计算得到在t=tau，在时频平面S=1。 此过程反复进行，直到小波到达信号结束。对于一个上述规模的时间尺度平面上的点s=1现在完成。然后，S是增加了一个较小的值。请注意，这是一个持续变换，因此，无论是tau和S必须不断递增。但是，如果这种转换需要由计算机计算，那么这两个参数是由一个足够小的步长增加。这对应于采样时间尺度的模型。重复上述过程的每一个价值秒每一个给定值计算的S填补了当时规模平面对应一行。当过程是为完成一切所需的值，信号的连续小波变换已计算完毕。下面的数字说明了整个过程的步骤：图3.3在图3.3中，信号和小波函数列的头有四个不同的值。该信号是在图3.1所示的信号被截断的版本。规模值是1，对应的最低规模，或最高频率。注意它是如何紧凑（蓝色窗口）。它应为最高频率分量的信号存在，在狭窄。小波函数的四个不同的位置都显示在图中分别为s=2,s=40,s=90,s=140。在每一个位置，它是乘以信号。显然，该产品是非零只有在信号的下降，对小波支持区域，它是零别处。通过将及时小波，信号是局部的时间，通过改变s的值，信号在尺度（频率）的本地化。 如果信号的频谱组成部分，对应于当前值（这是在这种情况下，s=1），此次与在此位置存在频谱分量信号小波产品给出了一个比较大的值。如果频谱分量对应到目前的价值不在于信号目前，产品的价值会比较少，或零。图3.3信号在s=1，t= 100ms的窗口的宽度频谱分量。 连续小波变换在图3.3信号将产生约时间尺度大值低100毫秒，和其他地方的小值。另一方面，对于高频率，连续小波变换将给予较大的值几乎信号的整个持续时间，因为低频率在任何时候都存在。图3.4图3.5图3.4和3.5说明了他们对尺度值分别为S = 5和S = 20处理的过程相同。注意窗口宽度是如何随规模越来越大（降低频率）的变化而变化的。作为窗口宽度的增加，转换从低频率的部分开始。因此，每个比例和每次转换时间（间隔），一个时间尺度平面点都要被计算。在一个尺度计算中构建时间尺度平面的行，并在不同尺度的计算中构的时间尺度平面的列。现在，让我们来看一个例子，看看小波变换到底是怎样进行的。注意图3.6所示的是一个非平稳信号，这和STFT时所举的例子类似，但在不同的频率。如图所示，信号包含四个频率分量分别是30赫兹，20赫兹，10赫兹和5赫兹。图3.6图3.7是连续小波变换这个信号（简称CWT）。请注意，轴线是平移和尺度，而不是时间和频率。然而，平移是和时间严格相关的，因为它表示母小波的位置。母小波变换可以被看作是时间，在t = 0时结束。但是，尺度完全