Daubechies Wavelet Filter Research Articles

Discrete Wavelet Transform and Long Memory in Stock Market Prices

The discrete wavelet transform is applied to the fractional differencing processes to estimate the long memory parameter by establishing a log-linear relationship between the variance of the wavelet coefficient from the long-memory process and the scaling parameter. The wavelet analysis can remove both the dense covariance matrix the GPH semiparametric method and the MLE create and the disadvantage of Fourier analysis that frequency information can only be extracted for the complete duration of a signal function. The wavelet OLS estimator yields a consistent estimator of the fractional differencing parameter. All the composite stock indices of the Dow Jones, Nasdaq, S&P 500, FTSE, Dax, Mib 30 and Korea's Kospi have found to be generated by the fractional differencing process. These findings are robust because the values were estimated based on the Daubechies wavelet and scaling filters of different widths, the least asymmetric wavelet and scaling filters of different widths, and the coiflet wavelet and scaling filters of width 6.

VLSI architectures of the 1-D and 2-D discrete wavelet transforms for JPEG 2000

This paper proposes efficient VLSI architectures for computation of the one-dimensional (1-D) and two-dimensional (2-D) discrete wavelet transforms (DWTs) for joint photographic experts group 2000. The proposed 1-D DWT architecture computes the wavelet lowpass and highpass output sequences using the lowpass filter architecture alone, whereas the conventional architectures compute the lowpass and highpass output sequences using both lowpass and highpass filter architectures. The proposed architecture is effectively applied to computation of the Daubechies 4-tap wavelet transform using the relationships between the Daubechies wavelet filter coefficients. The proposed architecture does not need memory and control units whereas conventional architectures need a complex control unit, memory unit, and so on. The two proposed VLSI architectures for the 2-D DWT are constructed based on block-based computation. Each M×N (N×M) block DWT is performed along the row and column directions simultaneously, where M and N denote the number of filter taps and the number of columns (rows), respectively, thus the required extra processing units of the proposed architectures are much smaller than those of the conventional architectures.

Constraint-selected and search-optimized families of Daubechies wavelet filters computable by spectral factorization

A unifying algorithm has been developed to systematize the collection of compact Daubechies wavelets computable by spectral factorization of a symmetric positive polynomial. This collection comprises all classes of real and complex orthogonal and biorthogonal wavelet filters with maximal flatness for their minimal length. The main algorithm incorporates spectral factorization of the Daubechies product filter into analysis and synthesis filters. The spectral factors are found for search-optimized families by examining a desired criterion over combinatorial subsets of roots indexed by binary codes, and for constraint-selected families by imposing sufficient constraints on the roots without any optimizing search for an extremal property. Daubechies wavelet filter families have been systematized to include those constraint-selected by the principle of separably disjoint roots, and those search-optimized for time-domain regularity, frequency-domain selectivity, time-frequency uncertainty, and phase nonlinearity. The latter criterion permits construction of the least and most asymmetric and least and most symmetric real and complex orthogonal filters. Biorthogonal symmetric spline and balanced-length filters with linear phase are also computable by these methods. This systematized collection has been developed in the context of a general framework enabling evaluation of the equivalence of constraint-selected and search-optimized families with respect to the filter coefficients and roots and their characteristics. Some of the constraint-selected families have been demonstrated to be equivalent to some of the search-optimized families, thereby obviating the necessity for any search in their computation.

The discrete wavelet transform and the scale analysis of the surface properties of sea ice

The formalism of the one-dimensional discrete wavelet transform (DWT) based on Daubechies wavelet filters is outlined in terms of finite vectors and matrices. Both the scale-dependent wavelet variance and wavelet covariance are considered and confidence intervals for each are determined. The variance estimates are more accurately determined with a maximal-overlap version of the wavelet transform. The properties of several Daubechies wavelet filters and the associated basis vectors are discussed. Both the Mallat orthogonal-pyramid algorithm for determining the DWT and a pyramid algorithm for determining the maximal-overlap version of the transform are presented in terms of finite vectors. As an example, the authors investigate the scales of variability of the surface temperature and albedo of spring pack ice in the Beaufort Sea. The data analyzed are from individual lines of a Landsat TM image (25-m sample interval) and include both reflective (channel 3, 30-m resolution) and thermal (channel 6, 120-m resolution) data. The wavelet variance and covariance estimates are presented and more than half of the variance is accounted for by scales of less than 800 m. A wavelet-based technique for enhancing the lower-resolution thermal data using the reflected data is introduced. The simulated effects of poor instrument resolution on the estimated lead number density and the mean lead width are investigated using a wavelet-based smooth of the observations.