## Complex Chebyshev Approximation for FIR Filter Design (1995)

Venue: | IEEE Transactions on Circuits and Systems II |

Citations: | 5 - 2 self |

### BibTeX

@ARTICLE{Karam95complexchebyshev,

author = {Lina J. Karam and James H. Mcclellan},

title = {Complex Chebyshev Approximation for FIR Filter Design},

journal = {IEEE Transactions on Circuits and Systems II},

year = {1995},

volume = {42},

pages = {207--216}

}

### OpenURL

### 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 ParksMcClellan algorithm, so that arbitrary magnitude and phase responses can be approximated. Both causal and non-causal filters with complex or real-valued impulse responses can be ...

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Citation Context ...ashoff and Roleff [14] restated the complex Chebyshev approximation problem as a linear-programming problem in the presence of an infinite number of constraints. They also formulated the dual problem =-=[15]-=- with only a finite number of constraints. An approximate discretized method was presented to solve the primal problem. Independently, Streit and Nuttall [7] introduced a design algorithm which solves... |

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Citation Context ...s organized as follows. Section 2 states the complex Chebyshev approximation problem. New theoretical results are presented and proved in Section 3. In particular, the fundamental alternation theorem =-=[25, 26]-=- is extended from the real-only to the complex case (Theorems 1, 2 and 3). In Section 4, the complex FIR filter design problem is reformulated as an approximation problem with real-valued basis functi... |

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Citation Context ...h is the best approximation of f with respect to Vn on B. Proof. (12) states that the error alternates on n + 1 extremal points fx i g n+1 i=1 . Using a theorem due to Rivlin and Shapiro (Appendix A) =-=[27, 28, 22]-=-, it is sufficient to prove that we can find n+1 positive numbers w 1 ; : : : ; wn+1 , P n+1 i=1 w i = 1, such that 1 n+1 X i=1 w i [f(x i ) \Gamma h(x i )]h j (x i ) = 0; j = 1; : : : ; n (13) where ... |

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Citation Context ... designed. Numerical examples are presented to illustrate the performance of the proposed algorithm. I. Introduction Exchange algorithms are efficient tools for the design of linear-phase FIR filters =-=[1, 2, 3]-=-. The linear-phase restriction converts the filter design problem into a real approximation problem. However, linear-phase filters with short transition bands introduce large delays [4]. Moreover, the... |

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Citation Context ...s organized as follows. Section 2 states the complex Chebyshev approximation problem. New theoretical results are presented and proved in Section 3. In particular, the fundamental alternation theorem =-=[25, 26]-=- is extended from the real-only to the complex case (Theorems 1, 2 and 3). In Section 4, the complex FIR filter design problem is reformulated as an approximation problem with real-valued basis functi... |

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Citation Context ...h is the best approximation of f with respect to Vn on B. Proof. (12) states that the error alternates on n + 1 extremal points fx i g n+1 i=1 . Using a theorem due to Rivlin and Shapiro (Appendix A) =-=[27, 28, 22]-=-, it is sufficient to prove that we can find n+1 positive numbers w 1 ; : : : ; wn+1 , P n+1 i=1 w i = 1, such that 1 n+1 X i=1 w i [f(x i ) \Gamma h(x i )]h j (x i ) = 0; j = 1; : : : ; n (13) where ... |

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Citation Context ... filters [1, 2, 3]. The linear-phase restriction converts the filter design problem into a real approximation problem. However, linear-phase filters with short transition bands introduce large delays =-=[4]-=-. Moreover, the linear-phase restriction is not needed in the stopband. Imposing the linear-phase requirement only in the passband results in a complex approximation problem. Complex approximation is ... |

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Citation Context ...et al. [14] and Nuttall et al. [7] for the complex FIR filter design problem. This method is computationally intensive and solves only an approximate discretized version of the original problem. Tang =-=[16]-=- introduced a single exchange algorithm for the complex Chebyshev problem. His algorithm basically solves the dual problem formulated in [14], but only IEEE Trans. Circuits and Systems--Part II, 1995 ... |

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Citation Context ...requirement only in the passband results in a complex approximation problem. Complex approximation is also needed for the design of filters with nonlinear phase characteristics such as FIR equalizers =-=[5, 6]-=-, beamformers [7], and seismic migration filters [8, pp. 359-- 361]. The frequency response H(!) of a length--N FIR digital filter is in general a complex-valued function of the normalized frequency !... |

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Citation Context ...l, strictly positive and continuous function on B. Due to the large number of applications [8, 9], several algorithms have been proposed for designing a Chebyshev optimal complex FIR filter. Cuthbert =-=[10]-=- introduced a design algorithm for non-linear phase FIR filters based on separately approximating the real and imaginary parts of the filter frequency response. Holt et al. [11] followed Cuthbert 's d... |

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Citation Context ...ficantly higher speed of convergence, compared to Tang's algorithm [16, 17]. Iterative reweighted least squared (IRLS) error algorithms have been also proposed for the complex Chebyshev approximation =-=[21, 22, 23]-=-. These methods are based on Lawson's algorithm which exhibits a slow convergence compared to the other discussed methods. Recently, Tseng [24] introduced a relatively efficient implementation of Laws... |

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Citation Context ...the passband results in a complex approximation problem. Complex approximation is also needed for the design of filters with nonlinear phase characteristics such as FIR equalizers [5, 6], beamformers =-=[7]-=-, and seismic migration filters [8, pp. 359-- 361]. The frequency response H(!) of a length--N FIR digital filter is in general a complex-valued function of the normalized frequency !: H(!) = N2 X n=N... |

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Citation Context ...x FIR filter. Cuthbert [10] introduced a design algorithm for non-linear phase FIR filters based on separately approximating the real and imaginary parts of the filter frequency response. Holt et al. =-=[11]-=- followed Cuthbert 's design approach but used an iterative procedure based upon the classical Remez exchange algorithm [12, 2] for the two real approximations. Barrodale et al. [13] adapted a linear-... |

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Citation Context ...true best approximation. Steiglitz [5] approximated the Chebyshev FIR filter design by a linear-programming problem with a specified set of constraints on the magnitude and phase. Glashoff and Roleff =-=[14]-=- restated the complex Chebyshev approximation problem as a linear-programming problem in the presence of an infinite number of constraints. They also formulated the dual problem [15] with only a finit... |

3 | Rational Chebyshev Approximations in the Complex Plane - Ellacott, Williams - 1976 |

2 |
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Citation Context ... designed. Numerical examples are presented to illustrate the performance of the proposed algorithm. I. Introduction Exchange algorithms are efficient tools for the design of linear-phase FIR filters =-=[1, 2, 3]-=-. The linear-phase restriction converts the filter design problem into a real approximation problem. However, linear-phase filters with short transition bands introduce large delays [4]. Moreover, the... |

2 |
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Citation Context ...requirement only in the passband results in a complex approximation problem. Complex approximation is also needed for the design of filters with nonlinear phase characteristics such as FIR equalizers =-=[5, 6]-=-, beamformers [7], and seismic migration filters [8, pp. 359-- 361]. The frequency response H(!) of a length--N FIR digital filter is in general a complex-valued function of the normalized frequency !... |

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Citation Context ...onse. Holt et al. [11] followed Cuthbert 's design approach but used an iterative procedure based upon the classical Remez exchange algorithm [12, 2] for the two real approximations. Barrodale et al. =-=[13]-=- adapted a linear-programming method to approximate the real and imaginary parts simultaneously. Approximating the real and imaginary parts separately does not, in general, give the optimal solution f... |

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Citation Context ...magnitude/phase equalization. Preuss [6] introduced a heuristic simultaneous exchange algorithm for the design of FIR filters with complex-valued coefficients. This algorithm was improved by Schulist =-=[19]-=-. The algorithm by Preuss and Schulist does not always converge [17]. Even when convergence is achieved, the optimality of the solution cannot be determined. Burnside and Parks [20] introduced an impl... |

1 |
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Citation Context ... the real and imaginary parts of the filter frequency response. Holt et al. [11] followed Cuthbert 's design approach but used an iterative procedure based upon the classical Remez exchange algorithm =-=[12, 2]-=- for the two real approximations. Barrodale et al. [13] adapted a linear-programming method to approximate the real and imaginary parts simultaneously. Approximating the real and imaginary parts separ... |

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Citation Context ...oblem. His algorithm basically solves the dual problem formulated in [14], but only IEEE Trans. Circuits and Systems--Part II, 1995 1 c fl IEEEswith real parameters. Alkhairy et al. [17] and Schulist =-=[18]-=- adapted Tang's algorithm for the design of FIR filters with real coefficients. Tang's algorithm for the dual problem led to significant improvements in terms of accuracy, speed, and storage requireme... |

1 |
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Citation Context ...ficantly higher speed of convergence, compared to Tang's algorithm [16, 17]. Iterative reweighted least squared (IRLS) error algorithms have been also proposed for the complex Chebyshev approximation =-=[21, 22, 23]-=-. These methods are based on Lawson's algorithm which exhibits a slow convergence compared to the other discussed methods. Recently, Tseng [24] introduced a relatively efficient implementation of Laws... |

1 |
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Citation Context ...ficantly higher speed of convergence, compared to Tang's algorithm [16, 17]. Iterative reweighted least squared (IRLS) error algorithms have been also proposed for the complex Chebyshev approximation =-=[21, 22, 23]-=-. These methods are based on Lawson's algorithm which exhibits a slow convergence compared to the other discussed methods. Recently, Tseng [24] introduced a relatively efficient implementation of Laws... |

1 |
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Citation Context ...proposed for the complex Chebyshev approximation [21, 22, 23]. These methods are based on Lawson's algorithm which exhibits a slow convergence compared to the other discussed methods. Recently, Tseng =-=[24]-=- introduced a relatively efficient implementation of Lawson's algorithm and applied it to the complex Chebyshev FIR filter design problem. However, none of the existing algorithms exploits the charact... |