An efficient least-squares design of IIR all-pass filter

Abstract

Least-squares design of digital filters is generally achieved by solving a system of linear equations. The matrices involved in the set of linear equations can be formulated as a Toeplitz-plus-Hankel form such that a matrix inversion is avoided with effectiveness. In this paper, some trigonometric properties are further exploited to obtain the closed-form expressions required for the system associated matrices in the least-squares design of IIR all-pass filters. Simulation results confirm that the proposed method achieves efficiency and good performance.

An Improved All-Pass Filter Design Using Closed-Form Toeplitz-plus-Hankel Matrix

“…complexity. Su et al [17,18] further exploited trigonometric identities and uniformly sampled the frequency band of interest to compute the sum of a series of Toeplitz-plus-Hankel matrix. The improved computing-efficient least-squares algorithm consequently yields the same performance as that of Kidambi's method [9] while markedly reduces the computational requirements.…”

“…In this paper, the discrete error phase response of the IIR all-pass filters in [9,17,18] is reformulated as an integral square error form [19]. Therefore, the allpass filter coefficients are obtained by solving a system of linear equations involves a simple and compact Toeplitz-plus-Hankel matrix.…”

“…Therefore, the allpass filter coefficients are obtained by solving a system of linear equations involves a simple and compact Toeplitz-plus-Hankel matrix. The elements of the Toeplitz-puls-Hankel matrix can be explicitly derived when the passband edge frequencies are specified based on trigonometric identities without sampling the band of interest like [17,18]. The proposed closed-form technique is consequently indicated from simulation results to improve designed performance and preserve computational efficiency as compared to those of methods [17,18].…”

“…The objective function for solving the system of linear equations can be easily formulated as a leastsquares error form over the discrete frequency [9,17,18] E = a T s 1 l…”

“…For a N N Toeplitz-plus-Hankel matrix, only N and 2N 1 distinct elements needed to be computed for the Toeplitz and Hankel matrix, respectively. Several In previous works [17,18], the authors exploited trigonometric identities and sampled the frequency band of interest uniformly to derive the close-form expressions for the Toeplitz-plus-Hankel matrix. Consequently, the closed-form expressions result in the same performance as that of Kidambi's method [9] while substantially reduces the computational complexity.…”