Passive, active, and digital filters (3ed., CRC, 2009) - tiera.ru

19-40 Passive, Active, and Digital FiltersBy subjecting h to the usual unit norm condition h T h ¼ 1, the optimum filter coefficients thatminimize the cost function are the elements of the eigenvector of the matrix P corresponding to theminimum eigenvalue. The eigenfilter method can solve constrained filter design problem. However,to obtain equiripple filters, a weight adaptive procedure is needed.19.2.5.2 Weighted Chebyshev (L 1 -Norm) CriterionAn IIR filter can also be formulated on weighted Chebyshev criterion. Filter coefficients are chosen suchthat its weighted Chebyshev (minimax) error between desired and actual frequency response is minimized.The iterative cost function in Equation 19.89 is now evaluated on Chebyshev criterion asE ¼ maxv2VW(v)jX k 1 (e jv )j jD(ejv )X k (e jv ) Y k (e jv )j¼ max W k(v)jD(e jv ) þ x T k sj (19:103)v2Vwheres ¼ [D(e jv )c 0c 1 ] Tx k is the coefficient vector previously defined in Equation 19.92.The solution of the minimax problem in Equation 19.103 can be found by solving the followingequivalent linear programming problemsubject tominx ke k (19:104)W k (v)jD(e jv ) þ x T k sje k (19:105)With a stability constraint, the above linear programming problem can be arranged in standard formof linear programming technique and solved using off-the-shelf linear programming software.19.2.6 Stability IssuesAn IIR filter can be unstable if there are some poles outside the unit circle. However, in case that thephase response is not important in the design, an unstable IIR filter obtained from a design algorithm canbe stabilized by conjugate reciprocal substitution of the unstable factor without changing its amplitude.If, however, phase response is a part of design specification, some other techniques must be used. Thereare four major approaches that can be applied for filter stabilization when optimization techniquespreviously described are used. The following is a summary. More details and references can be foundin Ref. [6].1. First approach is proposed by Deczky [3]. In this method, a standard gradient-based optimizationis modified so that the searching trajectory is only inside the border of stability. However, thealgorithm has high computation complexity. A standard optimization tool cannot be used.2. In the second approach, the target frequency response is chosen such that the desired filter isstable as in Ref. [7]. The target filter is restricted and may be too difficult to obtain filterstabilization.