From the construction of wavelets based on derivatives of the Gaussian function to the synthesis of filters with a finite impulse response

Abstract

For continuous wavelet transformation, wavelets based on derivatives of the Gauss function are used, and for multiscale analysis, Daubechies wavelets are used. The development of algorithms for forward and inverse continuous wavelet transform in the frequency domain made it possible in this work to synthesize digital filters with a finite impulse response (FIR) different from existing methods. The quality of the synthesized filters was checked by decomposition and subsequent reconstruction of the signals. To do this, several filters were synthesized that completely cover the frequency range of the signal. Since wavelets are bandpass filters, the authors called the filters wavelets. The more precisely the reconstructed signal repeats the shape of the original signal, the better the wavelet constructed by one method or another. A comparison of the accuracy of signal reconstruction shows that the best conversion result is obtained by using synthesized wavelets. The impulse responses of the FIR filters are synthesized so that their frequency response are similar to the frequency responses of wavelets based on derivatives of the Gaussian function of a large order. The greater the filter order, the closer the frequency response is to a square-wave shape. Algorithms for forward and inverse wavelet transformation of a signal in the frequency domain using wavelets based on derivatives of the Gauss function are proposed. Profiling of the program shows that the time of the wavelet transforms using the fast Fourier transform is 15,000 times less than with the direct numerical integration for sampling the signal of 32,768 samples. These algorithms can be used for wavelets with a square-wave frequency response. At the same time, the numerical calculation time is halved. The accuracy of the reconstruction was compared for wavelets based on second-order derivatives, Daubechies wavelets, and wavelets with a square-wave frequency response. The reconstruction accuracy is highest for the latest wavelets. The use of the wavelet construction method is preferable since this method is relatively simple and it is easy to synthesize multiband filters with any form of frequency response. If, when synthesizing using existing methods, a short transition band can be obtained only for long impulse responses, while the transition band using the method of constructing wavelets is absent even for filters of very small orders. The paper presents the impulse responses of twoband, three-band digital filters and their frequency response. FIR digital filters with a square-wave frequency response have a higher delay-band attenuation coefficient compared to existing filters, do not have transition band, and can be used to process one-dimensional and two-dimensional signals.

Keywords

• wavelet transform
• impulse response
• reconstruction
• multiscale analysis
• mallat algorithm
• frequency response
• synthesis
• digital filter