By Trevor J. Terrell (auth.)
In this revised and up to date version specific awareness has been paid to the sensible implementations of electronic filters, protecting such subject matters as microprocessors-based filters, single-chip DSP units, laptop processing of 2-dimensional signs and VLSI sign processing.
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Respectively. The design of a recursive digital filter centres around finding the pulse transfer function, G(Z), which satisfies a given performance specificatioh. This design process involves finding the filter coefficients -the a;s and b;s of G(Z), thereby yielding a pulse transfer function which is a rational function in z- 1 . A number of useful design methods are discussed in this chapter; each one is basically a mathematical method of obtaining a solution to the problem of approximating to a desired filter characteristic.
18 that it is impractical to try to evaluate the frequency spectrum for all w, and therefore only a finite set is generally considered. It is convenient to choose the set of N frequencies defined by Wr w 5r = 21Tr = ---~- NT N where r = 0,1 ,2, ... , (N- 1). Therefore . 41) The process of calculating theN values of G( eiwrT) is called the discrete Fourier transform (DFT). 41 is often written in the form Gr = r N-1 ~ i=O . 42) g(i)T W' =0,1 ,2, ... , (N- 1), where Gr=G(eiwrT) and W=e-j2rr/N However, Wr = w 5 r/N and therefore the DFT is a periodic function in the frequency domain, with a period of 21T/T rad/s.
Cooley and J. W. Tukey, 'An Algorithm for the Machine Calculation of Complex Fourier Series',MathematicsofComputing, 19 (1965) 297-301. 2. J. F. Kaiser, 'Design Methods for Digital Filters', Proceedings of the First Allerton Conference on Circuit and System Theory, (1963) 221--36. 3. R. Rabiner and C. M. Rader, Digital Signal Processing (IEEE Press, New York, 1972). 4. H. D. Helms and J. F. Kaiser, Literature in Digital Signal Processing (IEEE Press, New York, 1975). 5. J. A. Cadzow, Discrete Time Systems: An Introduction with Interdisciplinary Applications (Prentice-Hall, Englewood Cliffs, NJ, 197 3).