【正文】 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 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 width at s=1 around t=100 ms. The continuous wavelet transform of the signal in Figure will yield large values for low scales around 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 Figure Figures and illustrate the same process for the 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 ponents. As a result, for every scale and for every time (interval), one point of the timescale plane is puted. The putations at one scale construct the rows of the timescale plane, and the putations at different scales construct the columns of the timescale plane. Now, let39。39。39。 THE CONTINUOUS WAVELET TRANSFORM MULTIRESOLUTION ANALYSIS Although the time and frequency resolution problems are results of a physical phenomenon (the Heisenberg uncertainty principle) and exist regardless of the transform used, it is possible to analyze any signal by using an alternative approach called the multiresolution analysis (MRA) . MRA, as implied by its name, analyzes the signal at different frequencies with different resolutions. Every spectral ponent is not resolved equally as was the case in the STFT. MRA is designed to give good time resolution and poor frequency resolution at high frequencies and good frequency resolution and poor time resolution at low frequencies. This approach makes sense especially when the signal at hand has high frequency 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 suppor