[Design Application]
Eliminate Signal Noise With Discrete Wavelet Transformation
Modern DSP and communications applications are beginning to use wavelet transforms in critical algorithms.
The wavelet transform is a mathematical tool that's becoming quite useful for analyzing many types of signals. It has been proven especially useful in data compression, as well as in adaptive equalizer and transmultiplexer applications.
A wavelet is a small, localized wave of a particular shape and finite duration. Several families, or collections of similar types of wavelets, are in use today. A few go by the names of Haar, Daubechies, and Biorthogonal. Wavelets within each of these families share common properties. For instance, the Biorthogonal wavelet family exhibits linear phase, which is an important characteristic for signal and image reconstruction.
Wavelet analysis is simply the process of decomposing a signal into shifted and scaled versions of a particular wavelet. An important property of wavelet analysis is perfect reconstruction, which is the process of reassembling a decomposed signal or image into its original form without loss of information. By examining wavelet theory as it applies to three specific applications, we find that it works so well because these examples rely on perfect reconstruction for their fundamental operation.
There are no set rules for the choice of the mother wavelet used in wavelet analysis. The choice depends on the properties of the mother wavelet, the properties of the signal to be examined, and the requirements of the analysis. For this reason, it's convenient to have tools that let you easily explore and experiment with many different wavelets and input signals. The following examples use MATLAB, the Wavelet Toolbox, and Simulink to make exploration of wavelet concepts convenient.
In this article, the wavelet we use as an example (called the "mother" wavelet) is the Daubechies wavelet, db4. The 4 in the name represents the order of the filter, which corresponds to eight coefficients.
The Discrete Wavelet Transform (DWT) is commonly employed using dyadic multirate filter banks, which are sets of filters that divide a signal frequency band into subbands. These filter banks are comprised of low-pass, high-pass, or bandpass filters. If the filter banks are wavelet filter banks that consist of special low-pass and high-pass wavelet filters, then the outputs of the low-pass filter are the approximation coefficients. Also, the outputs of the high-pass filter are the detail coefficients.
The process of obtaining the approximation and detail coefficients is called decomposition. Termed multilevel decomposition, this process can be repeated, with successive approximations (the output of the low-pass filter in the first bank) being decomposed in turn, so that one signal is broken down into a number of components.
A two-level decomposition is shown in Figure 1. In this illustration, a2 represents the approximation coefficients, while d2 and d1 represent the detail coefficients resulting from the two-level decomposition. After each decomposition, we employ decimation by two to remove every other sample and, therefore, reduce the amount of data present.
The Inverse Discrete Wavelet Transform (IDWT) reconstructs a signal from the approximation and detail coefficients derived from decomposition. The IDWT differs from the DWT in that it requires upsampling and filtering, in that order. Upsampling, also known as interpolating, means the insertion of zeros between samples in a signal. The right side of the figure shows an example of reconstruction.
Another way to interpret the figure is that the analysis filter bank on the left reduces the rate of an input signal and produces multiple output signals with varying rates. The analysis filter bank performs the DWT represented by the decomposition. The synthesis filter bank on the right increases the rates of multiple input signals while combining them into a single output signal. It performs the IDWT represented by the reconstruction.
The Filters Are The Key Now one might ask, what's unique about wavelet filter banks? The magic is in the filters themselves. By choosing filters that are intimately related for both decomposition and reconstruction processes, the effects of aliasing, which can be introduced by the decimation, are removed.
When the signal is reconstructed, it doesn't exhibit any aliasing or distortion (right side of Fig. 1). As a result, the output is said to be a perfect reconstruction.
Wavelet filters have finite length. They aren't truncated versions of infinitely long filter re-sponses. Because of this property, wavelet filter banks can perform local analysis, or the examination of a localized area of a larger signal. Local analysis is an important consideration when dealing with signals that have discontinuities. Wavelet transforms can be applied to these kinds of signals with excellent results. This is due to their ability to locate short-time (local) high-frequency features of a signal and resolve low-frequency behavior at the same time.
As stated earlier, perfect reconstruction is an important property of wavelet filter banks. When the analysis filter bank output is connected to the synthesis filter bank input and the proper delays for alignment are used, as in Figure 1, then the output of the entire system is identical to the input. If a threshold operation is applied to the output of the DWT and wavelet coefficients that are below a specified value are removed, then the system will perform a "de-noising" function.
Two different threshold operations can be viewed in Figure 2. In the first, hard thresholding, coefficients whose absolute values are lower than the threshold are set to zero. Hard thresholding is extended by the second technique, soft thresholding, by shrinking the remaining nonzero coefficients toward zero.
you have written a lot about wavelet but you didnt give the most important thing that are the coefficients .I couldnot find any where the inverse coefficients of DB4 Can u please send it to me I really need them.
Anonymous -February 18, 2006
nice article.
fuzzyguy -November 25, 2005
It is a useful tutorial but I want more information about the filter co-efficient calculation for the wavelet transformation. Because I am going to do my mini project in the filter co-efficient calculataion instead of using the in-built function named dwt in signal processing.
TIRUPPATHI RAJAN G -September 27, 2004
I think this is a very concise article and very helpful for a basic overview of wavelets, especially DWT. This will be helpful with my project.
April -September 14, 2004
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