Weighted least-squares design of IIR all-pass filters using a Lyapunov error criterion

Abstract

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.

Neural network‐based design of 2‐channel quadrature mirror filter banks

“…[30][31][32][33][34][35][36][37][38][39] In literature, 35,37 the Lyapunov energy function is exploited to design IIR all-pass filters in a parallel manner with good performance. [30][31][32][33][34][35][36][37][38][39] In literature, 35,37 the Lyapunov energy function is exploited to design IIR all-pass filters in a parallel manner with good performance.…”

“…To conquer the obstacle, several literature 30,35,37,38 assumes that θ i (ω) ≈ θ d, i (ω) when the presented algorithm tends to reach the desired phases to simplify the design procedure. Therefore, (8) can be rewritten as…”

“…30,35,37,38 In this paper, the Hopfield neural networks are exploited to implement the optimal solution based on dynamic equations. 30,35,37,38 In this paper, the Hopfield neural networks are exploited to implement the optimal solution based on dynamic equations.…”

“…The aforementioned numerical methods may fail to achieve an optimal design because of the high nonlinearity of the objective function. In the literature, 35,37,44,46 the authors applied the feedback neural networks [49][50][51][52] to greatly reduce computational burdens and exactly achieve the same design results as those of methods. In the literature, 35,37,44,46 the authors applied the feedback neural networks [49][50][51][52] to greatly reduce computational burdens and exactly achieve the same design results as those of methods.…”

“…Least-squares (LS) 9,26,[30][31][32][33][34][35][36][37][38] and minimax approximations 36,39 are frequently investigated for the design of IIR all-pass filters. The aforementioned numerical methods may fail to achieve an optimal design because of the high nonlinearity of the objective function.…”