外文翻译---多分辨率分析＆连续小波变换(编辑修改稿)

外文翻译---多分辨率分析＆连续小波变换(编辑修改稿)内容摘要：

uency ponents for short durations and low frequency ponents for lo ng durations. Fortunately, the signals that are encountered in practical applications are often of this type. For example, the following shows a signal of this type. It has a relatively low frequency ponent throughout the entire signal and relatively hi gh frequency ponents for a short duration somewhere around the middle. THE CONTINUOUS WAVELET TRANSFORM The continuous wavelet transform was developed as an alternative approach to the short time Fourier transform to overe 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 puted separately for different segments of the timedomain 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, ., negative frequencies are not puted. 2. The width of the window is changed as the transform is puted for every single spectral ponent, which is probably the most significant characteristic of the wavelet transform. The continuous wavelet transform is defined as follows Equation As 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 ( pactly 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 Scale The 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 nondetailed 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 Fortunately 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 presses a signal. Larger scales correspond to dilated (or stretched out) signals and small scales correspond to pressed signals. All of the signals given in the figure are derived from the same cosine signal, ., they are dilated or pressed versions of the same function. In the above figure, s= 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 (pressed) version of f(t) if s 1 and to an expanded (dilated) version of f(t) if s 1 . However, in the definition of the wavelet transform, the scaling term is used in the denominator, and therefore, the opposite of the above statements holds, ., scales s 1 dilates the signals whereas scales s 1 , presses the signal. This interpretation of scale will be used throughout this text. COMPUTATION OF THE CWT Interpretation 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 are used are the dilated (or pressed) 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 later in this chapter. Once the mother wavelet is chosen the putation starts with s=1 and the continuous wavelet transform is puted for all values of s , smaller and larger than ``139。 39。 . However, depending on the signal, a plete transform is usually not necessary. For all practical purposes, the signals are bandlimited, and therefore, putation 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 continue for the increasing values of s , ., the analysis will start from high frequencies and proceed towards low frequencies. This first value of s will correspond to the most pressed wavelet. As the value of s。