【導(dǎo)讀】MRA，如它的名字一樣，分析了不同分辨率不同頻率的信號(hào)。譜分量不能得到同樣的解決是因?yàn)樵赟TFT的情況下。MRA是為了在高頻率時(shí)，能夠得到良好的時(shí)間分辨率和較差的頻率分辨率，幸運(yùn)的是，在實(shí)際應(yīng)用中所遇到的信號(hào)往往是這種類型。顯示了這種類型的信號(hào)。它有一個(gè)貫穿整個(gè)信號(hào)相對(duì)較低的頻率分量，而在信號(hào)。中間有一個(gè)短暫的、相對(duì)較高的頻率成分。連續(xù)小波變換作為一種替代快速傅里葉變換辦法來(lái)發(fā)展，克服分析的問(wèn)題。{\它的小波}，類似的STFT的窗口功能，并轉(zhuǎn)換為不同分段的時(shí)域信號(hào)。psi為轉(zhuǎn)化功能，它被稱為母小波。這個(gè)詞意味著小波浪。程中起主要作用，或者叫母小波。小的比例，和S=1是最大的比例。然而，在小波變換的定義，縮放詞是在分母，因此，上述聲明相反的成立，后對(duì)所有次積分。此過(guò)程反復(fù)進(jìn)行，直到小波到達(dá)信號(hào)結(jié)束。顯然，該產(chǎn)品是非零只有在信號(hào)的下降，對(duì)小波支持區(qū)域，它是零。應(yīng)到目前的價(jià)值不在于信號(hào)目前，產(chǎn)品的價(jià)值會(huì)比較少，或零。

　　

【正文】 n Figure , lower scales (higher frequencies) have better scale resolution (narrower in scale, which means that it is less ambiguous what the exact value of the scale) which correspond to poorer frequency resolution . Similarly, higher scales have scale frequency resolution (wider support in scale, which means it is more ambitious what the exact value of the scale is) , which correspond to better frequency resolution of lower frequencies. The axes in Figure and are normalized and should be evaluated accordingly. Roughly speaking the 100 points in the translation axis correspond to 1000 ms, and the 150 points on the scale axis correspond to a frequency band of 40 Hz (the numbers on the translation and scale axis do not correspond to seconds and Hz, respectively , they are just the number of samples in the putation). TIME AND FREQUENCY RESOLUTIONS In this section we will take a closer look at the resolution properties of the wavelet transform. Remember that the resolution problem was the main reason why we switched from STFT to WT. The illustration in Figure is monly used to explain how time and frequency resolutions should be interpreted. Every box in Figure corresponds to a value of the wavelet transform in the timefrequency plane. Note that boxes have a certain nonzero area, which implies that the value of a particular point in the timefrequency plane cannot be known. All the points in the timefrequency plane that falls into a box is represented by one value of the WT. Figure Let39。s take a closer look at Figure : First thing to notice is that although the widths and heights of the boxes change, the area is constant. That is each box represents an equal portion of the timefrequency plane, but giving different proportions to time and frequency. Note that at low frequencies, the height of the boxes are shorter (which corresponds to better frequency resolutions, since there is less ambiguity regarding the value of the exact frequency), but their widths are longer (which correspond to poor time resolution, since there is more ambiguity regarding the value of the exact time). At higher frequencies the width of the boxes decreases, ., the time resolution gets better, and the heights of the boxes increase, ., the frequency resolution gets poorer. Before concluding this section, it is worthwhile to mention how the partition looks like in the case of STFT. Recall that in STFT the time and frequency resolutions are determined by the width of the analysis window, which is selected once for the entire analysis, ., both time and frequency resolutions are constant. Therefore the timefrequency plane consists of squares in the STFT case. Regardless of the dimensions of the boxes, the areas of all boxes, both in STFT and WT, are the same and determined by Heisenberg39。s inequality . As a summary, the area of a box is fixed for each window function (STFT) or mother wavelet (CWT), whereas different windows or mother wavelets can result in different areas. However, all areas are lower bounded by 1/4 \pi . That is, we cannot reduce the areas of the boxes as much as we want due to the Heisenberg39。s uncertainty principle. On the other hand, for a given mother wavelet the dimensions of the boxes can be changed, while keeping the area the same. This is exactly what wavelet transform does. THE WAVELET THEORY: A MATHEMATICAL APPROACH This section describes the main idea of wavelet analysis theory, which can also be considered to be the underlying concept of most of the signal analysis techniques. The FT defined by Fourier use basis functions to analyze and reconstruct a function. Every vector in a vector space can be written as a linear bination of the basis vectors in that vector space , ., by multiplying the vectors by some constant numbers, and then by taking the summation of the products. The analysis of the signal involves the estimation of these constant numbers (transform coefficients, or Fourier coefficients, wavelet coefficients, etc). The synthesis, or the reconstruction, corresponds to puting the linear bination equation. All the definitions and theorems related to this subject can be found in Keiser39。s book, A Friendly Guide to Wavelets but an introductory level knowledge of how basis functions work is necessary to understand the underlying principles of the wavelet theory. Therefore, this information will be presented in this section. Basis Vectors Note: Most of the equations include letters of the Greek alphabet. These letters are written out explicitly in the text with their names, such as tau, psi, phi etc. For capital letters, the first letter of the name has been capitalized, such as, Tau, Psi, Phi etc. Also, subscripts are shown by the underscore character _ , and superscripts are shown by the ^ character. Also note that all letters or letter names written in bold type face represent vectors, Some important points are also written in bold face, but the meaning should be clear from the context. A basis of a vector space V is a set of linearly independent vectors, such that any vector v in V can be written as a linear bination of these basis vectors. There may be more than one basis for a vector space. However, all of them have the same number of vectors, and this number is known as the dimension of the vector space. For example in twodimensional space, the basis will have two vectors. Equation Equation shows how any vector v can be written as a linear bination of the basis vectors b_k and the corresponding coefficients nu^k . This concept, given in terms of vectors, can easily be generalized to functions, by replacing the basis vectors b_k with basis functions phi_k(t), and the vector v with a function f(t). Equation then bees Equation The plex exponential (sines and cosines) functions are the basis functions for t