A Modified Vector Fitting Technique to Extract Coupling Matrix from S-parameters

Question

1. What is the Cauchy method in microwave filter design?

2. How do S-parameters affect VF fitting?

3. What is the purpose of pole forcing in S-Parameters VF process?

4. How does NLP optimization determine the coupling matrix topology?

Accepted Answer

The Cauchy method is a technique used for generating polynomial interpolants from measurements of passive devices. It is employed for the polynomial model synthesis of the S-parameters of microwave filters. The solution of the Cauchy method can be used as coefficients of characteristic polynomials needed for interpolating or extrapolating the complex S-parameter data or the synthesis of the coupling matrix. However, the Cauchy method faces the illconditioning problem when dealing with high-order microwave filters or diplexers. It is one of the rational function approximation methods used in microwave filter design.

Accepted Answer

S-parameters affect VF fitting by making it less suitable for small magnitude values compared to Y-parameters. This results in the rational function not fitting all poles within the passband and mistakenly identifying spurious near the passband as poles outside the passband. To improve fitting accuracy, a focus range is applied to the lowpass response, excluding the outer band that might contain irregular spurious or harmonics. The recommended focus range is within +-1.2 rad/s to +-2 rad/s. However, this approach may cause a magnitude drop in the out-band due to irregular poles for S-parameters. This drawback can be reduced using S-parameters pole forcing in the next section.

Accepted Answer

Pole forcing in S-Parameters VF process aims to ensure that the poles of all S-parameters are identical. This is achieved by running pole relocation iterations on the S21 parameter to obtain a set of resultant poles, denoted as 'a n'. These resultant poles are then used for all S-parameters. The process involves calculating the residues of each S-parameter using overdetermined linear matrix equations and solving them using the least square method. By reusing the poles, the VF process is shortened, and the resultant residues are obtained directly. This enhancement ensures that the poles satisfy the unitary condition, even when the focus range is applied. Overall, pole forcing improves the accuracy and efficiency of the S-Parameters VF process.

Accepted Answer

Non-linear polynomial (NLP) optimization is used to determine the coupling matrix with its desired topology. This method allows filters to achieve any topology without performing matrix rotations. The NLP optimization method can be unconstrained or finitely bounded. To skip matrix rotation, the objective function is based on comparing reconstructed S-parameter polynomials and the desired topology. Frequency points of maximum and minimum S-parameters values within the [-1, 1] range are selected to improve accuracy and speed of optimization. These frequency points are used to evaluate the resulting VF and desired-topology of S-parameter polynomials. After comparing and solving the polynomials, the template matrix elements can be found. The coupling matrix template is not limited to inline topology, and (7) is just one example of the coupling matrix extraction approach. S-parameters can be generated based on the optimized coupling matrix [7]. The result is validated by comparing the S-parameter of the coupling matrix to the frequency response of the original filter.

Accepted Answer

The Chebyshev bandpass filter is designed using EM simulation from ANSYS 3D High Frequency Simulation Software (HFSS). The filter is realized as a waveguide filter based on inline topology. The conventional VF is not able to fit the in-band poles with high return loss. The four initial poles are selected and the overdetermined linear matrix is formed based on the initial poles, frequency points, and their corresponding S-parameters. The least square approximation is used to solve the linear matrix. The S-parameter polynomials are formed using the resultant poles and residues. The inline coupling matrix is obtained using preconditioned conjugate gradient (PCG) as the NLP optimization method. The final fitting response is shown in Fig. 8.

Accepted Answer

The fifth-order Chebyshev waveguide filter design impacts in-band poles by achieving a 5.11% of FBW and a center frequency of 2.445 GHz. However, conventional vector fitting struggles to fit the in-band poles with high return loss, as shown in Fig. 9. To address this, five initial poles are required for vector fitting (4). Instead of using a zero initial pole, a small value is used to prevent iteration failure. After applying focus fitting pre-processing with 2 rad/s and poles forcing, the S-parameter polynomials are shown in (12a), (12b), and (12c). The comparison of simulation results with and without pole forcing reveals a small discrepancy in the out-of-band frequency around 4 GHz and 5.5 GHz. The full vector fitting and inline coupling matrix obtained using NLP optimization are shown in (13). The mean absolute error (MAE) is used to calculate the difference between the fitted filter response, with the in-band response of S11 and S22 parameters greatly improved using this variation of VF. However, there is a slight trade-off on the S21 parameter, which is more noticeable at the start and end of the measured fifth-order Chebyshev filters' response. Overall errors of every S-parameter are still below 7%, as summarized in Table 1.

Accepted Answer

NLP optimization is used to extract the coupling matrix from S-parameters by utilizing a pre-defined template matrix topology. This technique involves analyzing simulated or measured S-parameter data to determine the coupling matrix. By implementing NLP optimization, the synthesis and reconfiguration of the filter coupling matrix can be achieved without matrix rotation limits. Additionally, incorporating preprocessing methods such as focus range selection and pole forcing into the procedure enhances the accuracy of rational system function extraction from S2P files of simulated and fabricated filter responses. The results demonstrate that this method improves accuracy with minimal discrepancies at out-of-band frequencies.

A Modified Vector Fitting Technique to Extract Coupling Matrix from S-parameters

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Most frequently asked questions

1. What is the Cauchy method in microwave filter design?

2. How do S-parameters affect VF fitting?

3. What is the purpose of pole forcing in S-Parameters VF process?

4. How does NLP optimization determine the coupling matrix topology?

5. How is the Chebyshev bandpass filter designed?

6. How does fifth-order Chebyshev waveguide filter design impact in-band poles?

7. How does NLP optimization extract coupling matrix from S-parameters?

References

Microstrip filters for RF/microwave applications

Rational approximation of frequency domain responses by vector fitting

Microwave Filters for Communication Systems: Fundamentals, Design and Applications

Improving the pole relocating properties of vector fitting

Artificial neural networks for RF and microwave design - from theory to practice

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