COMPLEX CHEBYSHEV-APPROXIMATION FOR FIR FILTER DESIGN

The alternation theorem is at the core of efficient real Chebyshev app roximation algorithms. In this paper, the alternation theorem is exten ded from the real-only to the complex case. The complex FIR filter des ign problem is reformulated so that it clearly satisfies the Haar cond ition of Chebyshev approximation. An efficient exchange algorithm is d erived for designing complex FIR filters in the Chebyshev sense. By tr ansforming the complex error function, the Remez exchange algorithm ca n be used to compute the optimal complex Chebyshev approximation. The algorithm converges to the optimal solution whenever the complex Cheby shev error alternates; in all other cases, the algorithm converges to the optimal Chebyshev approximation over a subset of the desired bands . The new algorithm is a generalization of the Parks-McClellan algorit hm, so that arbitrary magnitude and phase responses can be approximate d. Both causal and noncausal filters with complex or real-valued impul se responses can be designed. Numerical examples are presented to illu strate the performance of the proposed algorithm.