Wavelets: a mathematical tool for signal processing


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Edith Cowan University. La Trobe University. Macquarie University. Monash University. Queensland University of Technology. RMIT University. The University of Melbourne. The University of Queensland. The University of Sydney. University of Technology Sydney. This is a side effect of the evolutionary nature of these web pages. Rather than having a single bibliography, the references to literature books, articles and web pages are scattered among the web pages.

Many of the basic references can be found on the web page that discusses the applicaton of the the Haar transform to time series information. Fourier analysis, using the Fourier transform, is a powerful tool for analyzing the components of a stationary signal a stationary signal is a signal that repeats. For example, the Fourier transform is a powerful tool for processing signals that are composed of some combination of sine and cosine signals.

The Fourier transform is less useful in analyzing non-stationary data, where there is no repetition within the region sampled. Wavelet transforms of which there are, at least formally, an infinite number allow the components of a non-stationary signal to be analyzed. Wavelets also allow filters to be constructed for stationary and non-stationary signals. Although Haar wavelets date back to the beginning of the twentieth century, wavelets as they are thought of today are new. Wavelet mathematics is less than a quarter of a century old.

Some techniques, like the wavelet packet transform are barely ten years old. This makes wavelet mathematics a new tool which is slowly moving from the realm of mathematics into engineering. For example, the JPEG standard is based on the wavelet lifting scheme.


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The Fourier transform shows up in a remarkable number of areas outside of classic signal processing. Even taking this into account, I think that it is safe to say that the mathematics of wavelets is much larger than that of the Fourier transform. In fact, the mathematics of wavelets encompasses the Fourier transform. The size of wavelet theory is matched by the size of the application area.

Wavelets: A Mathematical Tool for Signal Analysis - Charles K. Chui - Google книги

Initial wavelet applications involved signal processing and filtering. However, wavelets have been applied in many other areas including non-linear regression and compression. An offshoot of wavelet compression allows the amount of determinism in a time series to be estimated. All of the wavelet algorithms that I am aware of must be applied to a data set a time series or a signal that with a power of two number of elements e.

All of the algorithms discussed on these web pages are ordered wavelet transforms. This means that the result consists of a wavelet scaling function value also known as a smooth value or a low pass filter value , followed by bands of wavelet function values sometimes called wavelet coefficients , in increasing frequency. The sizes of these wavelet coefficient bands are ordered in increasing powers of two. If there are N elements in the data set where N is a power of two The coefficient bands following the scaling value will have sizes 2 0 , 2 1 , 2 This is the earliest web page I wrote on wavelets.

EE123 Digital Signal Processing, SP'16 L12 - Discrete Wavelet Transform

It provides some background. The ordered Haar wavelet transform code that is linked to on this web page is not as simple and elegant as the ordered Haar transform implemented via the lifting scheme. This page collects these links. This is useful in understanding the Daubechies wavelet transform. This web page discusses the Daubechies D4 wavelet transform. This web page publishes two versions of the Daubechies transform, one implemented via the Lifting Scheme.

The wavelet Lifting Scheme has been developed by Wim Sweldens and others relatively recently e. When wavelet algorithms are expressed in a lifting scheme structure they are more efficient no temporaries are required and they are simpler and easier to understand. Spectral Analysis and Filtering with the Wavelet Transform. This web page discusses wavelet based spectral analysis and filtering. Wavelet spectral analysis plots provide a graphical representation of the filtering behavior of a particular wavelet transform.

This allows a wavelet filter to be constructed that selects particular wavelet coefficient bands for a given filter. It discusses some serious problems with the polynomial interpolation wavelet. There are many processes which have a random stochastic component, but also exhibit some predictability between one element and the next.

In statistics this is sometimes described as autocorrelation the correlation of a data set with with shifted versions of the data set. Autocorrelation is one measure of whether a past value can be used to predict a future value. A random process that has some degree of autocorrelation is referred to as a long memory process or long range dependence. River flow shows this kind of long term dependence. The height of the river waters follows a random pattern, which is seasonally influenced.

Within a season the trends are influenced by rainfall. A hydrologist, named Hurst, studied Nile river flows and reservoir modeling. The Hurst exponent discussed here takes its name from its discoverer. The Hurst exponent is related to the fractal dimension and appears in the analysis of self-similar 1-D data sets. A number of people have applied the Hurst exponent to financial data sets, which may show some amount of long range dependence.

I originally hoped to use the Hurst exponent to verify the results I got from using wavelet compression as an estimator for predictability in financial time series. The Hurst exponent is intellectually interesting because it occurs in a surprising number of areas in applied mathematics. Also, the Hurst exponent can be calculated using the Wavelet transform. As you might guess from these web pages, I find applications of the wavelet transform difficult to resist. The Hurst exponent has found practical application in computer network traffic simulation and analysis.


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I will warn you up front that it seems to be a dead end for financial modeling and prediction. To develop and verify the wavelet Hurst exponent calculation code, a fair amount of infra-structure had to be built. This support code is discussed on the web pages that are listed below:. The autocorrelation function, standard deviation and probability density function calculation. Generating a random walk data set or fractional brownian motion which is scaled for a particular Hurst exponent requires a source of high quality Gaussian pseudo-random numbers.

This web page discusses some techniques for calculating Gaussian distributed pseudo-random numbers. Simulating Fractional Brownian Motion.

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Histogram Smoothing via the Wavelet Transform. Todd Ogden and in a set of papers on database query optimization, wavelet techniques are applied to histogram creation. I have found that the coverage of wavelets applied to histograms is obscure.


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This web page explores wavelet techniques applied to histograms and finds that wavelets seem to be more computationally expensive than other techniques, without providing any obvious advantages. A more promising application of wavelets is density estimation, which is related on a basic level to histogram creation. In many cases a data sample does not include enough elements to display the smooth probability distribution that would be apparent in a larger sample.

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A data sample may also contain noise, which obscures the distribution. Wavelets are one technique where a non-parametric function can be fit to the data distribution. This fit will remove or partially remove noise and can approximate the underlying distribution of the data.

Wavelets: a mathematical tool for signal processing Wavelets: a mathematical tool for signal processing
Wavelets: a mathematical tool for signal processing Wavelets: a mathematical tool for signal processing
Wavelets: a mathematical tool for signal processing Wavelets: a mathematical tool for signal processing
Wavelets: a mathematical tool for signal processing Wavelets: a mathematical tool for signal processing
Wavelets: a mathematical tool for signal processing Wavelets: a mathematical tool for signal processing
Wavelets: a mathematical tool for signal processing Wavelets: a mathematical tool for signal processing
Wavelets: a mathematical tool for signal processing Wavelets: a mathematical tool for signal processing
Wavelets: a mathematical tool for signal processing Wavelets: a mathematical tool for signal processing

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