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The concept of mask operation in fractional Fourier domains is a generalization of the conventional Fourier-based filtering in the frequency domain. It is known that simultaneously employing multiple mask operations in multiple different fractional Fourier domains can lead to significant performance advantages when compared with just employing a single mask operation in a single fractional Fourier domain. However, there is no systematic scheme for optimal joint design of the discrete fractional Fourier transform (DFrFT) matrices and the corresponding sets of mask coefficients. In this paper, we consider this design problem and construct a formulation that does not depend on the knowledge of noise statistics. We then develop an iterative algorithm, which is a hybrid descent (HD) approach, to solve the formulated optimization problem. For this HD approach, a gradient descent method is supplemented by a modified simulated annealing algorithm. It is employed to find the global optimal rotation angles of the DFrFT matrices. During the iterative process, the corresponding sets of mask coefficients can be constructed analytically. Simulation results demonstrate that the proposed scheme is highly effective.

This correspondence investigates the least squares and minimax design problems for allpass variable fractional delay (VFD) filters. A two stage optimization approach is proposed to solve the resulting minimax optimization problem. This approach includes a combination of a one-dimensional global search method and an adaptive scheme to refine the discretization points. In addition, the paper investigates the design of allpass VFD filters which minimizes the weighted integral squared error subject to constraints on peak error deviation from the desired response. By using approximations, the design problem can be formulated as a quadratic optimization problem. Design examples show that a tradeoff between the weighted integral squared error and the peak error deviation can be achieved. In addition, the integral squared error can be reduced significantly from the minimax solution while maintaining approximately the same peak error deviation. Similarly, the peak error deviation can be significantly reduced from the least squares solution while maintaining approximately the same integral squared error.

The optimum design of a uniform finite impulse response filter bank can be formulated as a nonlinear semi-infinite optimization problem. However, this optimization problem is nonconvex with infinitely many inequality constraints. In this paper, we propose a new hybrid approach for solving this highly challenging nonlinear, nonconvex semi-infinite optimization problem. In this approach, a gradient-based method is used in conjunction with a filled function method to determine a global minimum of the problem. This new hybrid approach finds an optimal result independent of the initial guess of the solution. The method is applied to some existing examples. The results obtained are superior to those obtained by other existing methods.

A new and practical approach using the cepstrum technique is proposed in the design of minimum-phase digital filters as the sum of two allpass functions. The desired magnitude response is specified in the frequency domain. Its corresponding minimum-phase response is then obtained from the desired magnitude response. The desired phases for the two allpass filters are obtained from the magnitude and phase responses. For both filters to be stable, the corresponding denominator polynomials are minimum phase. The filter coefficients are obtained from the desired phases using the cepstrum technique. Design examples show that the method works well for both classical filter specification and general magnitude specification in the frequency domain.