Abstract not found

The FFT is used widely in signal processing for efficient computation of the Fourier transform (FT) of finitelength signals over a set of uniformly-spaced frequency locations. However, in many applications, one requires nonuniform sampling in the frequency domain, i.e.,a nonuniform FT . Several papers have described fast approximations for the nonuniform FT based on interpolating an oversampled FFT. This paper presents an interpolation method for the nonuniform FT that is optimal in the min-max sense of minimizing the worst-case approximation error over all signals of unit norm. The proposed method easily generalizes to multidimensional signals. Numerical results show that the min-max approach provides substantially lower approximation errors than conventional interpolation methods. The min-max criterion is also useful for optimizing the parameters of interpolation kernels such as the Kaiser-Bessel function.

The usual way of designing a lter is to specify a lter length and a nominal response, and then to nd a lter of that length which best approximates that response. In this paper we propose a di erent approach: specify the lter only in terms of upper and lower limits on the response, nd the shortest lter length which allows these constraints to be met, and then nd a lter of that order which is farthest from the upper and lower constraint boundaries in a mini-max sense. Previous papers have described methods for using an exchange algorithm for nding a feasible linear-phase FIR lter of a given length if one exists, given upper and lower bounds on its magnitude response. The resulting lters touch the constraint boundaries at many points, however, and are not good nal designs because they do not make best use of the degrees of freedom in the coe cients. We use the simplex algorithm for linear programming to nd a best linear-phase FIR lter of minimum length, as well as to nd the minimum feasible length itself. The simplex algorithm, while much slower than exchange algorithms, also allows us to incorporate more general kinds of constraints, such as concavity constraints (which can be used to achieve very at magnitude characteristics). We give examples that illustrate how the proposed and the usual approaches di er, and how the new approach can be used to design lters with at passbands, lters which meet point constraints, minimum phase lters, and bandpass lters with controlled transition band behavior. 1.

Sparse arrays have been proposed for two-dimensional arrays for three-dimensional ultrasound imaging in order to reduce the number of channels in the system. Such arrays have been designed by picking array elements in a random fashion, either according to a uniform or a Gaussian distribution. A random array can have large variations in the level of the maximum sidelobe. A method for optimization of the sidelobe level of 1-D sparse arrays has been demonstrated. This shows that weighting can give responses that resemble filled DolphChebyshev arrays. The initial thinning pattern is of less importance for the final result, but the less ideal the unweighted pattern is, the more dynamic range is required from the weight function.

Abstract. A method is described for the computation of the Green’s function in the complex plane corresponding to a set of K symmetrically placed polygons along the real axis. An important special case is a set of K real intervals. The method is based on a Schwarz–Christoffel conformal map of the part of the upper half-plane exterior to the problem domain onto a semi-infinite strip whose end contains K − 1 slits. From the Green’s function one can obtain a great deal of information about polynomial approximations, with applications in digital filters and matrix iterations. By making the end of the strip jagged, the method can be generalized to weighted Green’s functions and weighted approximations. Key words. Green’s function, conformal mapping, Schwarz–Christoffel formula, polynomial approximation,

We introduce a new model that can be used in the perceptual optimization of standard color image coding algorithms (JPEG/MPEG). The human visual system model is based on a set of oriented filters and incorporates background luminance dependencies, luminance and chrominance frequency sensitivities, and luminance and chrominance masking effects. The main problem in using oriented filter-based models for the optimization of coding algorithms is the difference between the orientation of the filters in the model domain and the DCT block transform in de coding domain. We propose a general method to combine these domains by calculating a local sensitivity for each DCT (color) block. This leads to a perceptual weighting factor for each DCT coefficient in each block. We show how these weighting factors allow us to use advanced techniques for optimal bit allocation in JPEG (e.g. custom quantization matrix design and adaptive thresholding). With the model we propose it is possible to calculate a perceptually weighted mean squared error (WMSE) directly in the DCT color domain, although the model itself is based on a directional frequency band decomposition. 1.

This paper describes an exchange algorithm for the frequency domain design of linear-phase FIR equiripple filters where the Chebyshev error in each band is specified. The algorithm is a hybrid of the algorithm of Hofstetter, Oppenheim and Siegel and the Parks-McClellan algorithm. The paper also describes a modification of the Parks-McClellan algorithm where either the passband or the stopband ripple size is specified and the other is minimized.

We examine the problem of approximating a complex frequency response by a real-valued FIR filter according to the L 2 norm subject to additional inequality constraints for the complex error function. Starting with the Kuhn-Tucker optimality conditions which specialize to a system of nonlinear equations we deduce an iterative algorithm. These equations are solved by Newton's method in every iteration step. The algorithm allows arbitrary tradeoffs between an L 2 and an L1 design. The L 2 and the L1 solution result as special cases. Wir untersuchen das Problem der Approximation eines komplexen Frequenzganges mittels eines reellwertigen nichtrekursiven Filters nach der L 2 -Norm mit zusatzlichen Ungleichungsbedingungen fur die komplexe Fehlerfunktion. Ausgehend von den Kuhn-Tucker Optimalitatsbedingungen, die auf ein nichtlineares Gleichungssystem fuhren, leiten wir einen iterativen Algorithmus her. Diese Gleichungen werden in jedem Iterationsschritt mittels des Newton-Verfahrens gelost. D...

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 ...

FIR filter design problems in the frequency domain are nonlinear (semi-infinite) optimization problems. In practice, however, these almost always have not been approached directly, but been solved in a simplified form and/or only under restricting assumptions. In this paper, quite general mathematical formulations of the four main design approximation problems in the frequency domain are presented, which enable the derivation of theoretical results (collected here from [31], [32]) and the application of general-purpose optimization procedures to their direct solution. For the actual solution, a nonlinear semi-infinite programming method from the thesis [9] of the first author is discussed and applied to several specific design problems. In some cases, the computed solution of the nonlinear problem is compared with that of a convex approximation of the problem.