Low‐complexity design framework of all‐pass filters with application in quadrature mirror filter banks design

Abstract

Digital all-pass filters design problem can be simplified by solving a set of linear equations associated with a Toeplitzplus-Hankel matrix in the least-squares sense. Consequently, the Cholesky decomposition or split Levinson technique can be appropriately used to obtain the optimal solution. In this study, the determination of the set of linear equations to calculate the all-pass-based quadrature mirror filter banks is achieved by exploiting some trigonometric identities and the frequency sampling method. The proposed simplification allows for expressing a sum of sinusoids by a single expression. The simulation results indicate that the presented new and simpler closed-form expressions of the Toeplitz-plus-Hankel associated matrices can achieve accurate performance with a considerable reduction in computational complexity.

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

“…11,12,15,25,26 Two neural networks with different number of neurons and Hopfield-related parameters are concurrently designed to obtain the IIR all-pass filters a 1 (n) and a 2 (n). To evaluate the design performance of the proposed neural-based method, the all-pass-based QMF bank designs set the same specifications as those of methods.…”

“…The presented approach yields excellent stopband attenuation than methods. 11,15,26,30 In this table, the efficient weighted LS algorithm 30 using Toeplitz-plus-Hankel matrix to the design of IIR all-pass filters is also included for FIGURE 4 The design procedure of IIR all-pass-based QMF banks using neural networks comparison. 11,15 The designed magnitude error response of QMF banks surpasses those of methods 11,15 which can be seen from Figure 5D.…”

“…The design of QMF banks using IIR all-pass filters possesses the merits of approximately linear-phase response, less computational burden, efficient polyphase implementation, and without yielding magnitude distortion. The analysis filters of the 2-channel QMF banks shown below can be alternatively implemented based on the combination of IIR all-pass filters A 1 (z 2 ) and A 2 (z 2 ) 9,12,15,19,25,26 : The analysis filters of the 2-channel QMF banks shown below can be alternatively implemented based on the combination of IIR all-pass filters A 1 (z 2 ) and A 2 (z 2 ) 9,12,15,19,25,26 :…”

“…To satisfy that H 0 (e jω ) and H 1 (e jω ) are low-pass and high-pass analysis filters, respectively, several researches 15,25,26 suggest that the 2 phase constraints shown below can meet the design requirement and stability. To satisfy that H 0 (e jω ) and H 1 (e jω ) are low-pass and high-pass analysis filters, respectively, several researches 15,25,26 suggest that the 2 phase constraints shown below can meet the design requirement and stability.…”

“…The all-pass filter coefficients a i (m) are corresponding to the output state v i,m of the Hopfield neural network, ie, a i (m) = v i,m for 1 ≤ m ≤ N i . Further using frequency sampling method, 26 the computations of Hopfield-related strength can be explicitly derived. Table 1 compares the required computational computations for associated matrices with different algorithms.…”