In this paper, a new design method for digital bandpass and bandstop infinite impulse response filters with nearly linear-phase response is proposed. In this method, the phase response of an all-pass filter is optimized in the frequency domain to yield less passband error ${\text{Er}}_{p}$ and stopband error ${\text{Er}}_{s}$ with optimal stopband attenuation $A_{s}$ . To achieve high accuracy in passband and stopband regions, fractional derivative (FD) constraints are evaluated in the respective regions, and the filter coefficients are computed using the Lagrange multiplier method. The behavior of fidelity parameters measured in terms of ${\text{Er}}_{p}{,\text{Er}}_{s}$ , and phase error ${\text{Er}}_{{\text{ph}}}$ is multimodel w.r.t. FD values. Therefore, modern heuristic technique, known as cuckoo search optimization is used for determining the optimal value of FDs and reference frequency simultaneously to minimize the fitness function, which is constructed as a sum of the squared error in passband and stopband. The designed filter yields up to 60% reduction in ${\text{Er}}_{p}$ and ${\text{Er}}_{{\text{ph}}}$ in the case of bandpass filter. Meanwhile, the response of filter is not degraded due to the finite word length effect.

Design of Bandpass and Bandstop Infinite Impulse Response Filters Using Fractional Derivative

This paper presents a new efficient method for implementing the Hilbert transform using an all-pass filter, based on fractional derivatives (FDs) and swarm optimization. In the proposed method, the squared error difference between the desired and designed responses of a filter is minimized. FDs are introduced to achieve higher accuracy at the reference frequency ( $\boldsymbol {\omega }_{\mathbf {0}}$ ), which helps to reduce the overall phase error. In this paper, two approaches are used for finding the appropriate values of the FDs and reference frequencies. In the first approach, these values are estimated from a series of experiments, which require more computation time but produce less accurate results. These experiments, however, justify the behavior of the error function, with respect to the FD and $\boldsymbol {\omega }_{\mathbf {0}}$ , as a multimodal and nonconvex problem. In the second approach, a variant of the swarm-intelligence-based multimodal search space technique, known as the constraint-factor particle swarm optimization, is exploited for finding the suitable values for the FD and $\boldsymbol {\omega }_{\mathbf {0}}$ . The performance of the proposed FD-based method is measured in terms of fidelity aspects, such as the maximum phase error, total squared phase error, maximum group delay error, and total squared group delay error. The FD-based approach is found to reduce the total phase error by 57% by exploiting only two FDs.

Hilbert Transform Design Based on Fractional Derivatives and Swarm Optimization

In signal processing applications, it is often required to have multi-notch filters (MNFs) which simultaneously possess an excellent quality factor, ‘Q’, and a brief transient response. Though, the duration of transient response can be improved by decreasing ‘Q’ which, however, also degrades the quality of the filter. Due to their contradictory nature, dealing with ‘Q’ and transient response becomes a difficult task. This paper is a sincere attempt to address this problem for MNFs by exploring time-varying pole-radius infinite impulse response MNF designs employing a hyperbolic tangent sigmoid (HTS) function-based pole-radius variation. Two-notch and three-notch filters, based on HTS function, are designed, analyzed for stability and tested for transient suppression, selectivity and noise removal. Experimental validations supported by simulations on sinusoids and ECG signals are performed in the LabVIEW
 $$^{\mathrm{TM}}$$
 environment and have been presented as complete performance assessment. In contrast to the recently reported and traditional MNF designs, significant performance enhancements were recorded for the proposed designs. Therefore, based on the presented investigations, it is concluded that the proposed MNFs are efficient designs for removing harmonics from sinusoids and ECG signals.

Time-Varying Pole-Radius IIR Multi-Notch Filters with Improved Performance

Design of feedback-structured IIR notch filter with transient suppression using gain variation

A novel all-pass-based infinite impulse response (IIR) digital multiple-notch filter design method is presented in this paper. The proposed method ensures that, for an arbitrary pass-bands ripple value, specifications regarding positions of notch, left- and right-hand cutoff frequencies are exactly met, which leads to symmetric magnitude responses about notch frequencies. Although iterative in its nature, proposed method requires a small number of iterations to find coefficients of the multiple-notch filter, making it very useful for designers. Since the magnitude response of the all-pass-based IIR multiple-notch filter is more sensitive in the stop-bands than in the pass-bands, sensitivity analysis along with simulation in limited word-length environment is performed.

Design of IIR Multiple-Notch Filters with Symmetric Magnitude Responses About Notch Frequencies

This study exploits a computationally efficient least-squares algorithm for designing multiple notch filters that can be applied to cancel the power line interference in electrocardiogram signal. The simulation results verify that proposed filter design framework achieves preferable performance for the preprocessing of electrocardiogram signal.

Interference elimination of electrocardiogram signals using IIR multiple notch filters

Digital all-pass filters design problem can be simplified by solving a set of linear equations associated with a Toeplitz-plus-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.

https://ietresearch.onlinelibrary.wiley.com/doi/epdf/10.1049/iet-spr.2016.0213

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

Least-squares design of two-channel quadrature mirror filter banks can be efficiently solved using infinite impulse response all-pass filters without yielding magnitude distortion. This paper exploits a neural network-based Lyapunov energy function to relate the phase objective function of the all-pass-based quadrature mirror filter banks. Applying the neural network architecture and suitable Hopfield-related parameters, the optimal all-pass filter coefficients can be obtained. By further using the parallel combination of the all-pass filters, the two-channel quadrature mirror filter banks can be efficiently designed. Simulation results demonstrate that the proposed approach achieves accurate performance in both reconstruction error and group delay.

Zhan-Pei Lin

Papers

Interference elimination of electrocardiogram signals using IIR multiple notch filters

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

Design of all-pass-based quadrature mirror filter banks using a Lyapunov error criterion

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