THE WAVELET TUTORIAL Page 1 of 10 THE WAVELET TUTORIAL PART IV by ROBI POLIKAR MULTIRESOLUTION ANALYSIS THE DISCRETE WAVELET TRANSFORM Why is the Discrete Wavelet Transform Needed Although the discretized continuous wavelet transform enables the computation of the continuous wavelet transform by computers it is not a true discrete transform As a matter of fact the wavelet series is simply a sampled version of the CWT and the information it provides is highly redundant as far as the reconstruction of the signal is concerned This redundancy on the other hand requires a significant amount of computation time and resources The discrete wavelet transform DWT on the other hand provides sufficient information both for analysis and synthesis of the original signal with a significant reduction in the computation time The DWT is considerably easier to implement when compared to the CWT The basic concepts of the DWT will be introduced in this section along with its properties and the algorithms used to compute it As in the previous chapters examples are provided to aid in the interpretation of the DWT THE DISCRETE WAVELET TRANSFORM DWT The foundations of the DWT go back to 1976 when Croiser Esteban and Galand devised a technique to decompose discrete time signals Crochiere Weber and Flanagan did a similar work on coding of speech signals in the same year They named their analysis scheme as subband coding In 1983 Burt defined a technique very similar to subband coding and named it pyramidal coding which is also known as multiresolution analysis Later in 1989 Vetterli and Le Gall made some improvements to the subband coding scheme removing the existing redundancy in the pyramidal coding scheme Subband coding is explained below A detailed coverage of the discrete wavelet transform and theory of multiresolution analysis can be found in a number of articles and books that are available on this topic and it is beyond the scope of this tutorial http users rowan edu polikar WAVELETS WTpart4 html 11 10 2004 THE WAVELET TUTORIAL Page 2 of 10 The Subband Coding and The Multiresolution Analysis The main idea is the same as it is in the CWT A time scale representation of a digital signal is obtained using digital filtering techniques Recall that the CWT is a correlation between a wavelet at different scales and the signal with the scale or the frequency being used as a measure of similarity The continuous wavelet transform was computed by changing the scale of the analysis window shifting the window in time multiplying by the signal and integrating over all times In the discrete case filters of different cutoff frequencies are used to analyze the signal at different scales The signal is passed through a series of high pass filters to analyze the high frequencies and it is passed through a series of low pass filters to analyze the low frequencies The resolution of the signal which is a measure of the amount of detail information in the signal is changed by the filtering operations and the scale is changed by upsampling and downsampling subsampling operations Subsampling a signal corresponds to reducing the sampling rate or removing some of the samples of the signal For example subsampling by two refers to dropping every other sample of the signal Subsampling by a factor n reduces the number of samples in the signal n times Upsampling a signal corresponds to increasing the sampling rate of a signal by adding new samples to the signal For example upsampling by two refers to adding a new sample usually a zero or an interpolated value between every two samples of the signal Upsampling a signal by a factor of n increases the number of samples in the signal by a factor of n Although it is not the only possible choice DWT coefficients are usually sampled from the CWT on a dyadic grid i e s0 2 and 0 1 yielding s 2j and k 2j as described in Part 3 Since the signal is a discrete time function the terms function and sequence will be used interchangeably in the following discussion This sequence will be denoted by x n where n is an integer The procedure starts with passing this signal sequence through a half band digital lowpass filter with impulse response h n Filtering a signal corresponds to the mathematical operation of convolution of the signal with the impulse response of the filter The convolution operation in discrete time is defined as follows A half band lowpass filter removes all frequencies that are above half of the highest frequency in the signal For example if a signal has a maximum of 1000 Hz component then half band lowpass filtering removes all the frequencies above 500 Hz The unit of frequency is of particular importance at this time In discrete signals frequency is expressed in terms of radians Accordingly the sampling frequency of the signal is equal to 2 radians in terms of radial frequency Therefore the highest frequency component that exists in a signal will be radians if the signal is sampled at Nyquist s rate which is twice the maximum frequency that exists in the signal that is the Nyquist s rate corresponds to rad s in the discrete frequency domain Therefore using Hz is not appropriate for discrete signals However Hz is used whenever it is needed to clarify a discussion since it is very common to think of frequency in terms of Hz It should always be remembered that the http users rowan edu polikar WAVELETS WTpart4 html 11 10 2004 THE WAVELET TUTORIAL Page 3 of 10 unit of frequency for discrete time signals is radians After passing the signal through a half band lowpass filter half of the samples can be eliminated according to the Nyquist s rule since the signal now has a highest frequency of 2 radians instead of radians Simply discarding every other sample will subsample the signal by two and the signal will then have half the number of points The scale of the signal is now doubled Note that the lowpass filtering removes the high frequency information but leaves the scale unchanged Only the subsampling process changes the scale Resolution on the other hand is related to the amount of information in the signal and therefore it is affected by the filtering operations Half band lowpass filtering removes half of the frequencies which can be interpreted as losing half of the information Therefore the resolution is halved after the filtering operation Note however the subsampling operation after filtering does not affect the resolution since removing half of the spectral components from the