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In S. Amari's paper - Synthesis of cross-coupled resonator filters using an analytical gradient-based optimization technique(2000), for reducing a filter's unnecessary cross-couplings, the matrix \$[A]\$ given by

\$[A] = [M]-j[R]+ \omega[U]\$ , where all matrices are of order \$N\$. Whereas, in Richard J. Cameron's paper - General coupling matrix synthesis methods for Chebyshev filtering functions(1999), all matrices are defined as (\$N+2\$)th order, which raises my primary concern.

The major issue is this - I defined the cost function from S.Amari's paper as given, but the optimized matrix \$M\$ created both as \$N\$th and (\$N+2\$)th order matrices show wrong results.

e.g., initial \$M\$ = \$ \left[\begin{array}{cc} -0.1332&0.5551&0.7227&0.3135 \\0.5551&0.8960&-0.2958&-0.0021 \\0.7227&-0.2958&0.1296&-0.9113 \\0.3135&-0.0021&-0.9113&-0.1332 \end{array} \right]\$

optimized \$M\$ = \$ \left[\begin{array}{cc} 0.4510&0.5047&0.0050&0.6747 \\0.5047&0.5501&0.5500&0.5499 \\0.0050&0.5500&0.5499&0.5499 \\0.6747&0.5499&0.5499&0.5047 \end{array} \right]\$ , in \$Nth\$ order.

To make it \$(N+2)\$th order, a source-coupling is introduced with one resonator and load-coupling with another resonator as

\$M_{N+2}\$ = \$ \left[\begin{array}{cc} 0&1&0&0&0&0 \\1&0.4510&0.5047&0.0050&0.6747&0 \\0&0.5047&0.5501&0.5500&0.5499&0 \\0&0.0050&0.5500&0.5499&0.5499&0 \\0&0.6747&0.5499&0.5499&0.5047&1 \\0&0&0&0&1&0 \end{array} \right]\$ , with source and load terminations on \$1 \Omega\$.

However, in both cases, the S21 and S11 plots are undesirable. I am using the MATLAB fmincon() solver with interior-point algorithm using bound constraints. Usage of global optimizers such as GlobalSearch leads to even worse unrealizable couplings. Reducing the matrix with Similarity Transformations(as in Cameron's paper) gives a practical solution. Is there anything wrong with the solver or the cost function definition(code) ?

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