Section 2 presents the theory of wavelet filter banks, designed to obtain perfect reconstruction. Although this technique has been studied in several digital signal processing applications, so far it has not been widely explored in digital circuit theory, which is the main focus of this investigation, primarily the digital filter synthesis, showing the good choice for wavelets in digital filtering applications. The selective filtering method proposed here designs a wavelet family that decomposes the signal to be filtered into sub-bands fitted in amplitude, obtaining good quantitative results in terms of accuracy of filtering and simultaneously with a low computational cost. This apparent duality could be better described using spectral factorization, which allows separating polynomial roots into two corresponding sequences of low-pass filters in a wavelet filter bank, according to a criterion of perfect reconstruction. This could be achieved using low-pass or high-pass filters, associated with wavelets respectively with few or many vanishing moments. Since signals can show different amplitude levels in both time and wavelet transform domains, it is interesting to partition the energy into several frequency sub-bands for several applications. The wavelet filter banks provide the advantage of separating the signal under consideration into two or more signals, in the frequency domain. In this scenario of drawbacks, the frequency partitioning into sub-bands, obtained by the wavelet transform, could be useful: in the wavelet decomposition of a given signal, frequency sub-bands are obtained with peculiar amplitude values, which could be explored in selective filtering techniques. On the other hand, the FIR filters are more powerful than the IIR ones, but they also require more processing power. Within this context, one of the major challenges of the IIR filters is determining its respective coefficients: in a digital form, the IIR filters can be designed from their analog versions, a procedure that is not easily performed. Simulations of the filter impulse response for the proposed method are presented, displaying the good behavior of the method with respect to the transition bandwidth of the involved filters.įilter design has been extensively explored in circuit synthesis and signal processing, as a part of circuit theory. This algorithm presented superior results when compared to windowed FIR digital filter design, in terms of the intended behavior in its transition band. This approach resulted in an energy partitioning across scales of the wavelet transform that enabled a superior filtering performance, in terms of its behavior on the pass and stop bands. Exploring such motivation, the method involves the design of a perfect reconstruction wavelet filter bank, of a suitable choice of roots in the Z-plane, through a spectral factorization, exploring the orthogonality and localization property of the wavelet functions. Since many approaches to the circuit synthesis using the wavelet transform have been recently proposed, here we present a digital filter design algorithm, based on signal wavelet decomposition, which explores the energy partitioning among frequency sub-bands. On the other hand, finite impulse response (FIR) digital filters are more flexible than the analog ones, yielding higher quality factors. In this context, filtering schemes, such as infinite impulse response (IIR) filters, are described by linear differential equations or linear transformations, in which the impulse response of each filter provides its complete characterization, under filter design specifications. In digital filters theory, filtering techniques generally deal with pole-zero structures.
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