小波功率谱相关译文.docx

1、小波功率谱相关译文译文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

2、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 take

3、n, 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.

4、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

5、 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 fun

6、ction 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 t

7、erm 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 befo

8、re 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, h

9、igh 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 scal

10、es) 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

11、 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 s

12、ignal. 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

13、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 , compresses the signal. T

14、his interpretation of scale will be used throughout this text.COMPUTATION OF THE CWTInterpretation of the above equation will be explained in this section. Let x(t) is the signal to be analyzed. The mother wavelet is chosen to serve as a prototype for all windows in the process. All the windows that

15、 are used are the dilated (or compressed) and shifted versions of the mother wavelet. There are a number of functions that are used for this purpose. The Morlet wavelet and the Mexican hat function are two candidates, and they are used for the wavelet analysis of the examples which are presented lat

16、er in this chapter.Once the mother wavelet is chosen the computation starts with s=1 and the continuous wavelet transform is computed for all values of s , smaller and larger than 1. However, depending on the signal, a complete transform is usually not necessary. For all practical purposes, the sign

17、als are bandlimited, and therefore, computation of the transform for a limited interval of scales is usually adequate. In this study, some finite interval of values for s were used, as will be described later in this chapter.For convenience, the procedure will be started from scale s=1 and will cont

18、inue for the increasing values of s , i.e., the analysis will start from high frequencies and proceed towards low frequencies. This first value of s will correspond to the most compressed wavelet. As the value of s is increased, the wavelet will dilate.The wavelet is placed at the beginning of the s

19、ignal at the point which corresponds to time=0. The wavelet function at scale 1 is multiplied by the signal and then integrated over all times. The result of the integration is then multiplied by the constant number 1/sqrts . This multiplication is for energy normalization purposes so that the trans

20、formed signal will have the same energy at every scale. The final result is the value of the transformation, i.e., the value of the continuous wavelet transform at time zero and scale s=1 . In other words, it is the value that corresponds to the point tau =0 , s=1 in the time-scale plane.The wavelet

21、 at scale s=1 is then shifted towards the right by tau amount to the location t=tau , and the above equation is computed to get the transform value at t=tau , s=1 in the time-frequency plane.This procedure is repeated until the wavelet reaches the end of the signal. One row of points on the time-sca

22、le plane for the scale s=1 is now completed.Then, s is increased by a small value. Note that, this is a continuous transform, and therefore, both tau and s must be incremented continuously . However, if this transform needs to be computed by a computer, then both parameters are increased by a suffic

23、iently small step size. This corresponds to sampling the time-scale plane.The above procedure is repeated for every value of s. Every computation for a given value of s fills the corresponding single row of the time-scale plane. When the process is completed for all desired values of s, the CWT of t

24、he signal has been calculated.The figures below illustrate the entire process step by step.Figure 3.3In Figure 3.3, the signal and the wavelet function are shown for four different values of tau . The signal is a truncated version of the signal shown in Figure 3.1. The scale value is 1 , correspondi

25、ng to the lowest scale, or highest frequency. Note how compact it is (the blue window). It should be as narrow as the highest frequency component that exists in the signal. Four distinct locations of the wavelet function are shown in the figure at to=2 , to=40, to=90, and to=140 . At every location,

26、 it is multiplied by the signal. Obviously, the product is nonzero only where the signal falls in the region of support of the wavelet, and it is zero elsewhere. By shifting the wavelet in time, the signal is localized in time, and by changing the value of s , the signal is localized in scale (frequ

27、ency).If the signal has a spectral component that corresponds to the current value of s (which is 1 in this case), the product of the wavelet with the signal at the location where this spectral component exists gives a relatively large value. If the spectral component that corresponds to the current

28、 value of s is not present in the signal, the product value will be relatively small, or zero. The signal in Figure 3.3 has spectral components comparable to the windows width at s=1 around t=100 ms.The continuous wavelet transform of the signal in Figure 3.3 will yield large values for low scales a

29、round time 100 ms, and small values elsewhere. For high scales, on the other hand, the continuous wavelet transform will give large values for almost the entire duration of the signal, since low frequencies exist at all times.Figure 3.4Figure 3.5Figures 3.4 and 3.5 illustrate the same process for th

30、e scales s=5 and s=20, respectively. Note how the window width changes with increasing scale (decreasing frequency). As the window width increases, the transform starts picking up the lower frequency components.As a result, for every scale and for every time (interval), one point of the time-scale p

31、lane is computed. The computations at one scale construct the rows of the time-scale plane, and the computations at different scales construct the columns of the time-scale plane.Now, lets take a look at an example, and see how the wavelet transform really looks like. Consider the non-stationary sig

32、nal in Figure 3.6. This is similar to the example given for the STFT, except at different frequencies. As stated on the figure, the signal is composed of four frequency components at 30 Hz, 20 Hz, 10 Hz and 5 Hz.Figure 3.6Figure 3.7 is the continuous wavelet transform (CWT) of this signal. Note that the axes are translation and scale, not time and frequency. However, translation is strictly related to time, sin