# Nearly Linear-Phase 2-D Recursive Digital Filters Design Using Balanced Realization Model Reduction
###### Abstract  
This paper presents a new method for the design of separable-denominator 2-D IIR filters with nearly linear phase in the passband. The design method is based on a balanced realization model reduction technique. The nearly linear-phase 2-D IIR filter is designed using 2-D model reduction from a linear-phase 2-D FIR filter, which serves as the initial filter. The structured controllability and observability Gramians \(P^{\mathrm{o}}\) and \(Q^{\mathrm{s}}\) serve as the foundation for this technique. onal positive-definite matrices that satisfy 2-D Lyapunov equations. An efficient method is used to compute these Gramians by minimizing the traces of \(P^{\mathrm{o}}\) and \(Q^{\mathrm{s}}\) under linear matrix inequality (LMI) constraints. The use of these Gramians ensures that the resulting 2-D IIR filter preserves stability and can be implemented using a separable-denominator 2-D filter with fewer coefficients than the original 2-D FIR filter. Numerical examples show that the proposed method compares favorably with existing techniques. 2-D IIR digital filters; structured Gramians; Lyapunov inequalities; linear matrix inequalities (LMI); balanced truncation +
Footnote †: journal: Accepted: 22 November 2023  
For 2-D IIR filters with a separable denominator \(D(z_{1},z_{2})=D_{1}(z_{1})D_{2}(z_{2})\), the stability is much easier to guarantee than the stability of non-separable-denominator filters [9; 26]. Two-dimensional (2-D) IIR filters with separable denominators have received more attention because, in addition to fewer stability constraints, the complexity of the implementation is reduced due to fewer coefficients with a separable denominator than with a nonseparable denominator of the same order [9; 27; 28]. The process of designing separable filters often results in filters that have high order. For various reasons, it is desirable to replace a high-order filter with a lower-order one to reduce the implementation cost and improve computational efficiency. Many academics have been studying balanced realization model reduction strategies for 2-D digital filters over the past few years as a solution to this challenge, and the findings have been shown to be helpful in the design of digital filters [8; 29]. Since the controllability and observability Gramians of the system determine a balanced realization in the main, different types of Gramians can be appropriately described for a given 2-D system. As a result, for a given 2-D discrete system, there are many balanced realization types that result in various balanced approximations [30]. This paper is concerned with a balanced realization model reduction of 2-D digital filters based on the structured controllability Gramian \(P^{s}\) and the structured observability Gramian \(Q^{s}\), where \(P^{s}\) and \(Q^{s}\in\mathbb{R}^{n\times n}\). In other words, to use a balanced truncation, the controllability Gramian \(P\) and the observability Gramian \(Q\) are needed. Linear matrix inequality (LMI) methods in the MATLAB environment can be employed to compute these Gramians. The theory of LMI has captured the interest of various research communities, particularly those working in the field of control systems [31; 32; 33]. Given that linear programming problems are easily handled by computers, the idea of LMI and its applications is founded on this fact [34]. Some of the earlier design work using LMI includes the work of [PERSON] and [PERSON] [35], and [PERSON] [36]. In [36], frequency-weighted balanced and closed-loop balanced truncation are used to solve the same problem. In addition, block-diagonal solutions of LMI are used in model reduction of uncertain systems, where the state partitions correspond to different frequency variables [37]. One of the most important steps in creating a coordinate system with balanced realization is to calculate the leading diagonal block matrices of the Gramians of the supplied 2-D system [38]. There have been many 2-D IIR filter design techniques that satisfy magnitude specifications using various methods [21; 22].