### Abstract

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 language | English (US) |
---|---|

Pages (from-to) | 207-216 |

Number of pages | 10 |

Journal | IEEE Transactions on Circuits and Systems II: Analog and Digital Signal Processing |

Volume | 42 |

Issue number | 3 |

DOIs | |

State | Published - Mar 1995 |

Externally published | Yes |

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### ASJC Scopus subject areas

- Signal Processing
- Electrical and Electronic Engineering

### Cite this

*IEEE Transactions on Circuits and Systems II: Analog and Digital Signal Processing*,

*42*(3), 207-216. https://doi.org/10.1109/82.372870

**Complex chebyshev approximation for FIR filter design.** / Karam, Lina; McClellan, James H.

Research output: Contribution to journal › Article

*IEEE Transactions on Circuits and Systems II: Analog and Digital Signal Processing*, vol. 42, no. 3, pp. 207-216. https://doi.org/10.1109/82.372870

}

TY - JOUR

T1 - Complex chebyshev approximation for FIR filter design

AU - Karam, Lina

AU - McClellan, James H.

PY - 1995/3

Y1 - 1995/3

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=0029270767&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0029270767&partnerID=8YFLogxK

U2 - 10.1109/82.372870

DO - 10.1109/82.372870

M3 - Article

VL - 42

SP - 207

EP - 216

JO - IEEE Transactions on Circuits and Systems II: Express Briefs

JF - IEEE Transactions on Circuits and Systems II: Express Briefs

SN - 1057-7130

IS - 3

ER -