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Topic

1-D and 2-D Digital Filters Design Using Model Reduction and Optimization Methods for Broadband Beamforming and Interference Rejection

Department of Electrical and Computer Engineering

Date & location

• Friday, March 7, 2025

• 10:00 A.M.

• Virtual Defence

Reviewers

Supervisory Committee

• Dr. Panajotis Agathoklis, Department of Electrical and Computer Engineering, University of Victoria (Co-Supervisor)

• Dr. Dale Shpak, Department of Electrical and Computer Engineering, UVic (Co-Supervisor)

• Dr. Jens Bornemann, Department of Electrical and Computer Engineering, University of Victoria (Member)

• Dr. Yang Shi, Department of Mechanical Engineering, UVic (Outside Member) 

External Examiner

• Dr. Majid Ahmadi, Department of Electrical and Computer Engineering, University of Windsor 

Chair of Oral Examination

• Dr. Daniela Constantinescu, Department of Mechanical Engineering, UVic

   

Abstract

This thesis presents several design algorithms for nearly linear-phase one-dimensional (1-D) and two-dimensional (2-D) infinite impulse response (IIR) digital filters. Optimization techniques as well as model order reduction (MOR) filter design methods are considered in this study.

For 1-D, finite impulse response (FIR) filters can achieve perfectly linear phase which makes them important in applications such as the field of audio signal processing where a flat delay characteristic may be desired. However, in most applications a perfectly linear phase response is not required and filters that have nearly linear phase response are quite acceptable. In such cases, IIR filters are more attractive than FIR filters. The design of IIR filters is more challenging than that of FIR filters because it results in a highly nonlinear objective function that requires sophisticated optimization methods. The 1-D optimization method proposed here solves the problem of approximating specified magnitude and linear-phase responses simultaneously. Since IIR filters can be designed to have nearly linear phase response in the passband, their passband group delay is usually considerably smaller than the delay of linear phase FIR filters with equivalent magnitude responses. Meeting a required minimum stopband attenuation or a minimum deviation from the desired magnitude and phase responses in the passbands are common design constraints that can be handled by the proposed optimization method for 1-D IIR filter. Also, an important constraint in the design of IIR filters is the prescription of a maximum pole radius, which allows to guarantee the stability margin and low coefficient selectivity for the obtained filter for finite-precision implementations. These design specifications are consistent with the constraints which often arise in practical filter design problems. In this research work, an optimization method for solving this constrained 1-D IIR design problem is presented.

The above optimization method used for designing 1-D IIR filters is extended to 2-D separable-denominator IIR digital filters with nearly linear phase in the passband. During the development of the proposed design techniques for 2-D digital filters, a special emphasis has been placed on their computational efficiency and a method for the design of 2-D IIR digital filters based on a balanced realization (BR) model order reduction technique is proposed. In this method, the initial design is a linear phase 2-D FIR filter realized in a 2-D state space model, which leads to a stable 2-D separable-denominator IIR filter with nearly linear phase in the passband. The model reduction method is based on structured controllability Ps and structured observability Qs gramians. These gramians are block-diagonal positive-definite matrices satisfying 2-D Lyapunov inequalities. An efficient general algorithm is developed to compute these matrices by minimizing the trace of Ps and the trace of Qs under Linear Matrix Inequalities (LMI) constraints. The use of these gramians ensures that the resulting 2-D IIR filter is a 2-D stable filter. Furthermore, the obtained nearly linear-phase 2-D IIR filter is more economical and computationally more efficient than the original 2-D FIR filter. Numerical examples using MATLAB show that the proposed method provides a good compromise between the filter selectivity and computational complexity when compared to existing techniques, making the results of this dissertation directly applicable to many practical applications. For example, in the field of array signal processing, 2-D digital filters having a fan-shaped filter in the passband emerge as powerful tools, particularly when employed as beamformers in scenarios where the Direction of Arrivals (DOAs) of the desired broadband Plane Waves (PWs) are known. In such cases, the designed 2-D FIR and IIR filters having a fan-shaped filter passband in the 2-D frequency domains are used as beamformers. Benefiting from the knowledge of DOAs of the desired broadband PWs, these filters are used to extract the signal of interest (SOI), suppress the interference, and reduce the noise corrupting the SOI. The successful implementation of 2-D FIR and IIR fan filters as beamformers not only enhances the rejection of the interference but also demonstrates its capability to reduce the effect of AWGN. This dual functionality holds significant implications for practical applications in digital signal processing, in which robustness against interference and noise is important. Simulation results demonstrate a good performance of the proposed beamformers and confirmed that the filters obtained using the proposed methods are capable of extracting and enhancing the desired 2-D broadband signals according to their directions of arrival under severe interference and noise.