By O. Kulhánek
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Additional info for Introduction to Digital Filtering in Geophysics
8] reveals that the output sequence y is given in terms of two convolution summations. While the first right-hand summation is a convolution of the numerator of H(z) and the input, the second summation is generally a convolution of the denominator of H(z) and the output delayed one time-unit. 8] may be considered as a feedback system. Making use of the z-transform representation we have: 42 r(z) DESIGN PRINCIPLES OP DIGITAL - //( 2) - x(z) +. 1 1 B+ I Z - +L B L Z FILTERS m or: The principles of convolution and recursive filtering are shown in Fig.
G. Koopmans, 1974, p. 185). The amplitude of side lobes may be influenced by the type of truncation, or tapering, of the time window. Smooth termination usually reduces side lobes. Properly selected windows smooth out sharp changes in H(z) but otherwise preserve the characteristics of the filter. Simple time windows like Hamming’s (Blackman and Tukey, 1958): for | Λ | = 0, 1, 2 , . . 50 cos ( /˝) Win - I I> for ) | - 0, 1, 2 , . , ˝ \ I 0 for I I> TV usually satisfy the requirements mentioned above.
2b. 15] may be interpreted in terms of a parallel of two first-order subsystems. Both have a common input and the arrangement sum of the two outputs provides the final output sequence y (see Fig. 2c). g. by Ackroyd (1973). In the block diagrams presented in Fig. 2, several simplifications have been made. Firstly, the numerator of H(z) has been substituted by a constant a 0. This is of course a special case. Generally, the numerator is a polynomial in z - 1 and consequently the block diagrams contain delay blocks also in the forward paths.