All-pass filter design can be generally achieved by solving a system of linear equations. The associated matrices involved in the set of linear equations can be further formulated as a Toeplitz-plus-Hankel form such that a matrix inversion is avoided. Consequently, the optimal filter coefficients can be solved by using computationally efficient Levinson algorithms or Cholesky decomposition technique. In this paper, based on trigonometric identities and sampling the frequency band of interest uniformly, the authors proposed closed-form expressions to compute the elements of the Toeplitz-plus-Hankel matrix required in the least-squares design of IIR all-pass filters. Simulation results confirm that the proposed method achieves good performance as well as effectiveness.

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.

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This paper extends a neural network based architecture for the weighted least-squares design of IIR all-pass filters. The error difference between the desired phase response and the phase of the designed all-pass filter is formulated as a Lyapunov error criterion. The filter coefficients are obtained when neural network achieves convergence by using the corresponding dynamic function. Furthermore, a weighted updating function is proposed to achieve good approximation to the minimax solution. Simulation results indicate that the proposed technique is able to achieve good performance in a parallelism mannerKeywords-all-pass, IIR, Lyapunov, weighted least-squares. I. INTRODUCTIONDigital all-pass filters have received much attention in many signal applications such as digital communications, notch filtering, phase equalization, multirate filtering system, construction of a wavelet, etc. [1]-[4]. The design methods can be generally classified into maximally flat design [5], leastsquares approximation [6]-[8], and minimax approximation [9]-[11]. Weighted least-squares (WLS) methods [7]-[9], [13], [15]have attracted much research attention due to their flexible utilization for any type of filter designs. Moreover, the WLS methods can obtain an optimal solution analytically with an appropriate weighting function is used, and the equiripple (i.e., optimal in the minimax sense) requirement is therefore achieved.The approaches mentioned above generally required by solving a set of linear equations, using a linear programming method or a generalized exchange algorithm. Thus, the computation is burdensome as the filter length is long and the applications are quite limited to real-time design.Recently in the literature, several authors [12]-[16] have used neural network based algorithms to design different kinds of digital filters. In our previous researches [15],[16], a simplification of the feedback neural network [12],[13]for yielding reduced computation while achieving equivalent results has been developed. In this paper, we extend the unified and simplified structure to implement the WLS design of IIR all-pass filters with good performance.

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