# Nearly Linear-Phase 2-D Recursive Digital Filters Design Using Balanced Realization Model Reduction
## Appendix A  
In this Appendix, two examples of obtaining the structured controllability and structured observability Gramians are presented. They include 2-D system matrices in the non-separable and separable characteristic polynomial. **Example A1**. : _Consider a 2-D discrete system described by [PERSON]'s model, \((m,n)=(2,2)\), as follows:_  
\[A= \begin{bmatrix}-0.5000&0.7500&0.3895&0.0389\\ 0&0&0&0\\ 0.1423&0&-0.4000&0.0200\\ -0.0342&0&-0.6000&0.0300\end{bmatrix} \]  
_where \(x^{h}\in\mathbb{R}^{2}\) and \(x^{v}\in\mathbb{R}^{2}\)._  
\[b= \begin{bmatrix}1&0&1&0\end{bmatrix} ^{T}\]\[c=\quad \begin{bmatrix}1&0&0&0\end{bmatrix} \]  
Applying the proposed LMI algorithm to solve the Lyapunov inequalities, the structured controllability and the structured observability Gramians are found to be positive-definite block-diagonal matrices as given below:  
\[ \begin{array}{cccc}\mathrm{P}^{s}=&\begin{bmatrix}2.6744536&0.0000005&0&0\\ 0.0000005&0.0000018&0&0\\ 0&0&1.7237207&0.1361198\\ 0&0&0.1361198&0.7926782\end{bmatrix}\end{array} \]  
\[ \begin{array}{cccc}\mathrm{Q}^{s}=&\begin{bmatrix}1.6523883&-0.8820146&0&0\\ -0.8820146&1.2662409&0&0\\ 0&0&1.2060296&0.0671541\\ 0&0&0.0671541&0.0173983\end{bmatrix}\end{array} \]  
**Example A2**. : _In this example, we consider the following 2-D model \(\Sigma=(A_{11},A_{12},A_{21},A_{22},\)\(b_{1},b_{2},c_{1},c_{2})\), also presented as an illustrative example in Section 3.4 in [22]. The corresponding system matrices are given below:_  
\[A_{11}=\quad \begin{bmatrix}0&1&0&0\\ 0&0&1&0\\ 0&0&0&1\\ -0.282145&0.551205&-0.875599&1.361780\end{bmatrix} ,\]  
\[A_{12}=0_{4\times 4}\;,\]  
\[A_{21}=\quad \begin{bmatrix}-0.020056&0.114892&-0.167819&0.227100\\ 0.011559&-0.054709&0.120890&-0.150348\\ 0.026037&0.035530&-0.034800&0.047409\\ 0.032347&-0.055488&0.045669&0.077340\end{bmatrix} ,\]  
\[A_{22}=\quad \begin{bmatrix}0&0&0&-0.090322\\ 1&0&0&0.199219\\ 0&1&0&-0.390625\\ 0&0&1&0.750000\end{bmatrix} ,\]  
\[b_{1}=\quad \begin{bmatrix}0&0&0&1\end{bmatrix} ^{T},\]  
\[b_{2}=\quad \begin{bmatrix}0.368103&-0.315109&0.227313&-0.073000\end{bmatrix}  ^{T},\]  
\[c_{1}=\quad \begin{bmatrix}0.261041&-0.127614&-0.033891&0.424881\end{bmatrix} ,\]  
\[c_{2}=\quad \begin{bmatrix}0&0&0&1\end{bmatrix} ,\]  
\[d=0.6520.\]  
_Here, \(x^{h}\in\mathbb{R}^{4}\) and \(x^{v}\in\mathbb{R}^{4}\). Note that since \(A_{12}\) is a zero matrix, this is a separable-denominator system where the denominator of the corresponding transfer function can be factored as the product of a polynomial in \(z_{1}\) by a polynomial in \(z_{2}\). The structured controllability matrix \(\mathrm{P}^{s}\) and the structured observability matrix \(\mathrm{Q}^{s}\) are found to be symmetric positive-definite and block-diagonal matrices where \(\mathrm{P}^{s}=\text{diag}(p_{1},p_{2})\) and \(\mathrm{Q}^{s}=\text{diag}(q_{1},q_{2})\) and these sub-matrices \(p_{i}\) and \(q_{j}\)for \(i=1,2\) are given as below:_\[p_{1}= \begin{bmatrix}8.1089&6.7562&4.6871&3.0064\\ 6.7562&7.5461&6.2880&4.3242\\ 4.6871&6.2880&7.0435&5.8058\\ 3.0064&4.3242&5.8058&6.4723\\ \end{bmatrix} ,\]  
\[p_{2}= \begin{bmatrix}2.5173&-1.8049&1.0641&0.5113\\ -1.8049&4.2626&-2.7316&1.7339\\ 1.0641&-2.7316&4.2820&-1.9780\\ 0.5113&1.7339&-1.9780&5.2563\\ \end{bmatrix} ,\]  
\[q_{1}= \begin{bmatrix}0.6672&-0.7158&0.7194&-0.7046\\ -0.7158&1.5485&-1.5435&1.7515\\ 0.7194&-1.5435&2.3619&-2.7027\\ -0.7046&1.7515&-2.7027&4.8899\\ \end{bmatrix} ,\]  
\[q_{2}= \begin{bmatrix}3.0759&1.8324&0.6877&0.3077\\ 1.8324&2.7993&1.6562&0.5947\\ 0.6877&1.6562&2.5770&1.5290\\ 0.3077&0.5947&1.5290&2.3714\\ \end{bmatrix} ,\]  
_For the same system presented in the above example, we consider now a non-separable denominator case where the matrix \(A_{12}\) is given by:_  
\[A_{12}= \begin{bmatrix}-0.0050&0.0287&-0.0420&0.0568\\ 0.0029&-0.0137&0.0302&-0.0376\\ 0.0065&0.0089&-0.0087&0.0119\\ 0.0081&-0.0139&0.0114&0.0193\\ \end{bmatrix}