# Nearly Linear-Phase 2-D Recursive Digital Filters Design Using Balanced Realization Model Reduction

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## Abstract

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## 1. Introduction

## 2. Gramians of 2-D Discrete Systems

## 3. Proposed Computation Method for Structured Gramians

## 4. Balanced Realization/Truncation Technique for 2-D IIR Digital Filters

#### Design Procedures

- Design a linear-phase 2-D FIR digital filter that approximates the required frequency response.
- Realize the designed 2-D FIR filter $H({z}_{1},{z}_{2})$ in state space using Roesser’s model [6,41] as follows:$$\left[\begin{array}{c}{X}^{h}(i+1,j)\\ {X}^{v}(i,j+1)\end{array}\right]=\left[\begin{array}{c}{A}_{11}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}{A}_{12}\\ {A}_{21}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}{A}_{22}\end{array}\right]\left[\begin{array}{c}{x}^{h}(i,j)\\ {x}^{v}(i,j)\end{array}\right]+\left[\begin{array}{c}{b}_{1}\\ {b}_{2}\end{array}\right]u(i,j)$$$$y(i,j)=\left[\begin{array}{c}{c}_{1}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}{c}_{2}\end{array}\right]\left[\begin{array}{c}{x}^{h}(i,j)\\ {x}^{v}(i,j)\end{array}\right]+du(i,j),$$
- Compute the structured controllability Gramian ${P}^{s}$ and the structured observability Gramian ${Q}^{s}$ using the LMI-based algorithm proposed in Section 3. Note that since either ${A}_{12}$ or ${A}_{21}$ is zero, the equations are simplified and the computational cost of these Gramians is reduced. The obtained structured controllability and structured observability Gramians are block-diagonal matrices, i.e., ${P}^{s}=diag({p}_{1},{p}_{2})$ and ${Q}^{s}=diag({q}_{1},{q}_{2})$.
- Find the invertible matrices ${T}_{1}$ and ${T}_{2}$ such that$$\begin{array}{cc}\hfill {T}_{i}^{-1}{p}_{i}{\left({T}_{i}^{-1}\right)}^{T}& ={\Sigma}_{i}={T}_{i}^{T}{q}_{i}{T}_{i}\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& =diag({\sigma}_{1i},{\sigma}_{2i},\dots {\sigma}_{{n}_{i}i}),\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}i=1,2.\hfill \end{array}$$
- Compute the matrices ${M}_{1i}={p}_{i}{q}_{i},\phantom{\rule{4pt}{0ex}}i=1,2$.
- Decompose the matrices ${M}_{1i}$ as $[{u}_{i},{s}_{i},{v}_{i}]=svd\left({M}_{1i}\right),\phantom{\rule{4pt}{0ex}}i=1,2.$where the full singular value decomposition of an m-by-n matrix M involves:
- m-by-m matrix u.
- m-by-n matrix s.
- n-by-n matrix v.

- Compute the matrices ${T}_{i}={u}_{i}\sqrt{{s}_{i}},\phantom{\rule{4pt}{0ex}}i=1,2$.

- Obtain the similarity transformation matrix$T:=diag({T}_{1},{T}_{2})$.
- Form the balanced realization model ${\Sigma}_{b}=({T}^{-1}AT,{T}^{-1}b,cT)$.
- Obtain the reduced-order $({r}_{1},{r}_{2})$ filter by partitioning the balanced realization obtained in the above step ${\Sigma}_{r}=({A}_{r},{b}_{r},{c}_{r})$.

## 5. Illustrative Examples and Numerical Evaluation

#### 5.1. 2-D Lowpass Filters

#### 5.2. 2-D Bandpass Filter

#### 5.3. Two-Dimensional Fan Filter

#### 5.4. Fan Filtering of Plane Wave Image

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

1-D | One-Dimensional |

2-D | Two-Dimensional |

FIR | Finite Impulse Response |

IIR | Infinite Impulse Response |

LMI | Linear Matrix Inequality |

MOR | Model Order Reduction |

PB | Passband |

SB | Stopband |

SDP | Semi-Definite Problem |

tr | Trace |

diag | Diagonal |

Min | Minimum |

Max | Maximum |

PW | Plane Wave |

U | Unit Circle |

## Appendix A

#### Appendix A.1

**Example A1.**

**Example A2.**

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**Figure 2.**Magnitude responses and group delays of 2-D lowpass FIR and IIR filters discussed in Example 1 of Section 5.1: (

**a**) 2-D FIR lowpass filter of order (24,24); (

**b**) 2-D IIR of reduced order (13,13) in [6]; (

**c**) 2-D IIR of reduced order (13,13) using LMI; (

**d**) magnitude contour of reduced-order filter; (

**e**) group delay ${\tau}_{1}$ of reduced order (13,13); (

**f**) group delay ${\tau}_{2}$ of reduced order (13,13).

**Figure 3.**Magnitude responses and group delays of the reduced 2-D IIR bandpass filter described in Section 5.2.

**Figure 4.**2-D FIR and IIR fan filters described in Section 5.3: (

**a**) initial 2-D FIR fan filter of order (49,49); (

**b**) reduced 2-D IIR fan filter of order (34,34); (

**c**) group delay ${\tau}_{1}$ of reduced 2-D IIR fan filter; (

**d**) group delay ${\tau}_{2}$ of reduced 2-D IIR fan filter; (

**e**) impulse response of 2-D IIR fan filter; (

**f**) magnitude contour of the reduced fan filter.

**Figure 5.**Magnitude response of 2-D FIR and IIR fan filters described in Section 5.4.

**Figure 7.**Original and filtered plane wave images using reduced-order 2-D IIR fan filter and their spectrum.

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**MDPI and ACS Style**

Omar, A.; Shpak, D.; Agathoklis, P.
Nearly Linear-Phase 2-D Recursive Digital Filters Design Using Balanced Realization Model Reduction. *Signals* **2023**, *4*, 800-815.
https://doi.org/10.3390/signals4040044

**AMA Style**

Omar A, Shpak D, Agathoklis P.
Nearly Linear-Phase 2-D Recursive Digital Filters Design Using Balanced Realization Model Reduction. *Signals*. 2023; 4(4):800-815.
https://doi.org/10.3390/signals4040044

**Chicago/Turabian Style**

Omar, Abdussalam, Dale Shpak, and Panajotis Agathoklis.
2023. "Nearly Linear-Phase 2-D Recursive Digital Filters Design Using Balanced Realization Model Reduction" *Signals* 4, no. 4: 800-815.
https://doi.org/10.3390/signals4040044