How to Generate the Coupling Matrix

For this guide, we will use a BPF case as an example for our customer requirements.


Step 1: Input the Customer specification to generate the specification “Mask”



Tip: Users don’t need to input a wideband rejection specification; it will squeeze the overall filter performance and result in inaccurate results.

Step 2: Input the frequency specification and the Return Loss (RL)

  1. Filter order and Unloaded Q are experience-based parameters. Users may optimize these parameters until the specifications can be achieved. Initially, we can set the unloaded Q value as infinite.
  2. RL is usually set 3dB lower than the customer specification for design margin purposes.
  3. The “Shift” function is also called fine-tune function. It is used for fine-tuning the filter performance when users incorporate the temperature drift analysis. Initially we can set these values to 0.

Tip: Click the blue radio button to switch different Frequency input formats

Step 3: Design the transmission zeros to meet the specification

Tip: Click “NOR” or “GHZ” to switch Frequency input formats

  • Initially the transmission zero (TZ) value can start with the rejection frequency either at the beginning or end of the rejection specification.
  • All TZ values need to be tuned and optimized, step by step, based on the customer’s specification. Sometimes more TZs need to be added to provide sharp isolation performance to satisfy the rejection requirement.

    Step 4: Transmission zeros knowledge—The Filter Transfer Function

  • F(s) is a polynomial with real coefficients, and its roots lie along the imaginary axis as conjugate pairs; P(s) is a pure even polynomial with real coefficients. Its roots lie on the imaginary axis in conjugate pairs.
  • Filter requirements call for low loss in the passband and high loss in other frequency bands. Such a requirement can best be achieved by assigning all the zeros of F(s) to the j𝜔 axis in the passband region and all zeros of P(s) to the j𝜔 axis in the high loss frequency bands.
  • For some applications, the zeros of P(s) have a non-j𝜔-axis location. This results in an improved phase and group delay response in the passband at the expense of attenuation in the stopband. Such a trade-off is sometimes beneficial for the overall system requirements.


  • In the normalized format, the “Complex Pair Zero” transmission zeros is always in pairs. This is true with symmetric and asymmetric filter responses. Therefore the number of real zeros and the number of complex zeros must be even numbers.
  • For “Pure Finite Zeros” transmission zeros, the number of transmission zeros is even for symmetric responses or odd for asymmetric responses.

Step 5: Define the physical topology (if necessary)


  • SynMatrix defaults to the folded-type topology
  • If the user-defined topology is not synthesizable, SynMatrix will either return the optimized result or an error message
  • SynMatrix now supports up to 90% of all topologies which can be used for a wide variety of engineering applications


Step 6: Click "Calculate All" to start the synthesis process
  • After adjusting the filter order and TZs position, the final filter order is 9 and the TZ number is 4
  • The unloaded Q value can be based on experience or the simulation
  • To mitigate the risks from structural design and post-tuning workflow issues, the margin rules need to be applied as follows:
    • Set RL to be less than 3dB from the customer’s specification
    • The rejection level needs to be at least 3dB less than the customer specification
  • A lower RL level requires higher coupling energy, which may result in impractical coupling coefficients (ex. Planar and coplanar structures). Likewise, the lower rejection level may result in a small coupling coefficient value that can not be realized in real life.
  • Additionally, lower RL and transmission zeros levels may cause difficulties during test and tuning

Step 7: Fine-tune the matrix and keep the proper design margins
drawing drawing
  • The red square box means the current design fails to meet the thermal drift variation during ambient change
  • The fine-tune function will be applied to either expand/narrow the BW or shift up/down the frequency to balance the right/left frequency margins
  • During fine-tuning, the TZ position may be optimized to obtain the best performance and achieve the desired frequency margins

Tip: The first value indicates the frequency shift and the second is BW tuning; all units for both parameters are in “MHz"


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