The design of FIR digital filters with a complex-valued desired frequency response using the Chebyshev error is investigated. The complex approximation problem is converted into a real approximation problem which is nearly equivalent to the complex problem. A standard linear programming algorithm for the Chebyshev solution of overdetermined equations is used to solve the real approximation problem. Additional constraints are introduced which allow weighting of the phase and/or group delay of the approximation. Digital filters are designed which have nearly constant group delay in the passbands. The desired constant group delay which gives the minimum Chebyshev error is found to be smaller than that of a linear phase filter of the same length. These filters, in addition to having a smaller, approximately constant group delay, have better magnitude characteristics than exactly linear phase filters with the same length. The filters have nearly equiripple magnitude and group delay.

Design of FIR filters in the complex domain

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. >

Complex Chebyshev approximation for FIR filter design

We present a systematic procedure for the design of filters intended for multirate systems. This procedure Is motivated by viewing the equiripple design of filters in linear time-invariant systems as a process of obtaining optimum minimax filters for a class of bounded energy input signals. The philosophy of designing optimum minimax filters for classes of input signals is extended to multirate systems, which are not time-invariant. We develop a generalized Fourier analysis appropriate for linear periodic systems and use it to derive new error criteria for multirate filter design. Using such criteria yields optimum minimax multirate filters for the input signal class. The utility of our method is demonstrated by using it to analyze several multirate systems. We give numerical results on the design of a multirate implementation of a narrowband filter and compare our work to previous work on multirate filter design. Our numerical analysis is based upon a new formulation of the design as a semi-infinite linear programming problem. >

Linear periodic systems and multirate filter design

We present an improved algorithm for an established filter design problem, the design of an FIR filter that best approximates, in the complex Chebyshev sense, a desired complex-valued frequency response. The algorithm is a variant of the simplex algorithm of linear programming, which has an interpretation as an implicit multiple exchange. It is iterative, robust, and exhibits good convergence speed. Global optimum convergence is guaranteed. Both complex and real-valued impulse responses can be designed with it; the design of complex coefficient filters is new. An example is given for each case. The design of noncausal filters is new. In addition to these new applications, we conjecture that this new algorithm may have important advantages over existing techniques, with respect to the maximum filter length possible, speed and stability of convergence, accuracy, and memory requirements. The ability to design long filters is among the more significant improvements over previous work. Filters of length 1000 have been designed with the new method. >

Optimal design of FIR filters with the complex Chebyshev error criteria

We propose a new algorithm for finding best minimax polynomial approximations in the complex plane. The algorithm is the first satisfactory generalization of the well-known Remez algorithm for real approximations. Among all available algorithms, ours is the only quadratically convergent one. Numerical examples are presented to illustrate rapid convergence.

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A fast algorithm for linear complex Chebyshev approximations

A new computational technique is described for the Chebyshev, or minimax, approximation of a given complex valued function by means of linear combinations of given complex valued basis functions. The domain of definition of all functions can be any finite set whatever. Neither the basis functions nor the function approximated need satisfy any special hypotheses beyond the requirement that they be defined on a common domain. Theoretical upper and lower bounds on the accuracy of the computed Chebyshev error are derived. These bounds permit both a priori and a posteriori error assessments. Efforts to extend the method to functions whose domain of definition is a continuum are discussed. An application is presented involving ’’re‐shading’’ a 50‐ element antenna array to minimize the effects of a 10% element failure rate, while maintaining full steering capability and mainlobe beamwidth.

A. H. Nuttall

Papers

A general Chebyshev complex function approximation procedure and an application to beamforming