Design of Minimum-Phase Filters Using Optimization

Abstract

A method for the design of minimum-phase FIR digital filters using spectral factorization is described. In the proposed method, the required digital filter is designed by formulating a set of nonlinear equations that represent the design problem at hand and then solving it by using the Levenberg-Marquardt optimization method. By using an exact formulation of the Jacobian and Hessian matrix in the Levenberg-Marquardt method, significant performance improvement and fast convergence is achieved. Comparisons show that the new method leads to a significant reduction in the maximum value and L2 norm of the residual error relative to those in state-ofthe-art competing methods.

On the Computation of a Minimum Phase Factor

“…The O-W equations have implications for the implementation of a minimum phase transformation on a digital computer. This was demonstrated by Kidambi and Antoniou in 2017 [11], and [11] presented results for the design of minimum phase finite impulse response (FIR) Chebyshev filters, reporting residual errors orders of magnitude smaller than those obtained through competing methods [12], [13], [14], [8]. In terms of computational requirements, the Hilbert transform and the cepstrum are both based on the discrete Fourier transform (DFT) 1 and require very long FFT's to obtain good FIR filter performance.…”

“…A linear phase filter g presented as example 1 in [11] is deployed in this section. The 25 tap settings for vector g is shown in Table II.…”

“…In Section II factorization in the discrete time domain is proposed, based on locally Toeplitz matrices that are defined. Section III presents numerical results for the factorization of a 25 tap filter presented in [11], and it is demonstrated that based on 64 bit MATLAB software the Chebyshev minimum phase FIR filter residual error is several orders of magnitude smaller than the error reported in [11]. Section IV presents numerical results for the transformation of a given arbitrary phase FIR to a minimum phase FIR, and numerical results based on the proposed algorithm are compared to results obtained through MMSE design.…”

“…In terms of computational requirements, the Hilbert transform and the cepstrum are both based on the discrete Fourier transform (DFT) 1 and require very long FFT's to obtain good FIR filter performance. On the other hand, [11] showed that solving the O-W equations is very efficient. The methodology and results proposed in [11] have not been improved upon since its publication in 2017, and there are no methods available in the open literature able to yield a Chebyshev minimum phase filter with a smaller residual error than that reported in [11].…”

“…On the other hand, [11] showed that solving the O-W equations is very efficient. The methodology and results proposed in [11] have not been improved upon since its publication in 2017, and there are no methods available in the open literature able to yield a Chebyshev minimum phase filter with a smaller residual error than that reported in [11].…”