Linear Phase Low-Pass Filters with Improved Selectivity
Synthesis of constant group delay low-pass polynomial filters in a maximally flat manner (having all derivatives in the origin equal to zero) is straightforward. One uses the Bessel polynomials with the complex variable introduced instead of the usual real x. That is described in this chapter and tables of coefficients and poles of the first ten Bessel polynomials are given. A design example is given too. The next issue visited is approximation of constant group delay by a polynomial filter in equi-ripple manner. An iterative algorithm is proposed allowing for this problem to be solved for arbitrary values of the order of the filter and the maximum group delay error assigned in percentage of the nominal one. Influence of the approximation error to the properties of the filter is studied. Both, maximally flat and equi-ripple group delay approximants exhibit poor selectivity. To improve that an algorithm is described and an example is given as to how transmission zeros at the ω-axis may be introduced. Comparisons are made (in the frequency and time domain) between the two linear phase polynomial solutions being extended with transmission zeros. Finally, advice is given as to how to synthesize low-pass all-pass functions.
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