THE WAVELET TUTORIAL PART III by ROBI POLIKAR Page 1 of 28 THE WAVELET TUTORIAL PART III MULTIRESOLUTION ANALYSIS THE CONTINUOUS WAVELET TRANSFORM by Robi Polikar 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 component 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 http users rowan edu polikar WAVELETS WTpart3 html 11 10 2004 THE WAVELET TUTORIAL PART III by ROBI POLIKAR Page 2 of 28 good frequency resolution and poor time resolution at low frequencies This approach makes sense especially when the signal at hand has high frequency components for short durations and low frequency components for long 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 component throughout the entire signal and relatively high frequency components 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 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 follows http users rowan edu polikar WAVELETS WTpart3 html 11 10 2004 THE WAVELET TUTORIAL PART III by ROBI POLIKAR Page 3 of 28 Equation 3 1 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 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 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 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 http users rowan edu polikar WAVELETS WTpart3 html 11 10 2004 THE WAVELET TUTORIAL PART III by ROBI POLIKAR Page 4 of 28 Figure 3 2 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 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 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 i e scales s 1 dilates the signals whereas scales s 1 compresses the signal This interpretation of scale will be used throughout this text http users rowan edu polikar WAVELETS WTpart3 html 11 10 2004 THE WAVELET TUTORIAL PART III by ROBI POLIKAR Page 5 of 28 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 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 later 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