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Wavelet-like decomposition based on filtering in domains of discrete trigonometric transforms – case study

  Over two last decades the wavelet signal decomposition and reconstruction schemes, originally based on the Mallat approach [11], have become a standard technique for signal processing. Typically used finite impulse response (FIR) filter banks, forming a structure referred to as the subband coding [14, 15, 18, 19, 20], are considered in relation to the Fourier transform, thus decompose signals into subbands defined in the frequency domain. As some recent results indicate that the FIR convolution filtering may be efficiently replaced with similar filtering concept based on filters defined in domains of the discrete trigonometric transforms (DTTs) [4-8], the relevant adaptation of the wavelet scheme to this new filtering technique emerged in a natural way. The DTTs [3] offer doubled frequency accuracy, as compared to the Fourier tool with the same signal length, and do not require computations with complex numbers. The DTTs in an orthogonal form, as described in [6], are additionally suited to simplified matrixbased computing. The filtering with the generalized convolution [5, 8] should be rather compared with the circular FIR Fourier-based convolution [12, 14], whose feature in the case of the discussed decomposition scheme is an additional advantage, namely that the edge effect, normally resulting from the regular convolution of the limited signals, is not present, and thus has not got to be handled. A discussion about the adaptation of the wavelet scheme to the DTTs, i.e., replacement of FIR filtering by the generalized convolution, has to be preceded by the relevant study on the whole scheme with the ideal DTT-based filters realized in the transform domain. Such study had to be compared with the existing techniques, gradually developed from the cosine [16, 17] up to any orthogonal transform [9, 10] and also described in [2, 13]. In the described research 16 DTTs have been analyzed, namely eight cosine and eight sine transfo[...]

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