# Gradient based optimization for microwave filter design

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) ?