Complex Chebyshev Approximation for FIR Filter Design

Lina J. Karam, James H. McClellan

Research output: Contribution to journalArticlepeer-review

152 Scopus citations


The alternation theorem is at the core of efficient real Chebyshev approximation algorithms. In this paper, the alternation theorem is extended from the real-only to the complex case. The complex FIR filter design problem is reformulated so that it clearly satisfies the Haar condition of Chebyshev approximation. An efficient exchange algorithm is derived for designing complex FIR filters in the Chebyshev sense. By transforming the complex error function, the Remez exchange algorithm can be used to compute the optimal complex Chebyshev approximation. The algorithm converges to the optimal solution whenever the complex Chebyshev 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 algorithm, so that arbitrary magnitude and phase responses can be approximated. Both causal and noncausal filters with complex or real-valued impulse responses can be designed. Numerical examples are presented to illustrate the performance of the proposed algorithm.

Original languageEnglish (US)
Pages (from-to)207-216
Number of pages10
JournalIEEE Transactions on Circuits and Systems II: Analog and Digital Signal Processing
Issue number3
StatePublished - Mar 1995
Externally publishedYes

ASJC Scopus subject areas

  • Signal Processing
  • Electrical and Electronic Engineering


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