To keep up with recent developments in mobile communications, researchers urgently need to develop microwave and millimeter-wave (mm-wave) components that are economical, efficient, and compact in size, all while maintaining good performance. Thus, as an important part of any communication system, the filter, is attracting a great deal of research interest. Emerging applications in the field of wireless communications have engendered more advanced requirements for RF/microwave filters, such as smaller size, lighter weight, lower cost, multiband, and better performance. Taking the substrate integrated waveguide (SIW) filter as an example, cutting the equivalent magnetic wall is an effective miniaturization method <xref ref-type="bibr" rid="ref1" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">[1]</xref> – <xref ref-type="bibr" rid="ref2" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"/> <xref ref-type="bibr" rid="ref3" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"/> <xref ref-type="bibr" rid="ref4" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"/> <xref ref-type="bibr" rid="ref5" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"/> <xref ref-type="bibr" rid="ref6" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"/> <xref ref-type="bibr" rid="ref7" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"/> <xref ref-type="bibr" rid="ref8" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">[8]</xref> , but it can cause a small amount of radiation leakage because of the cut in the boundary <xref ref-type="bibr" rid="ref9" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">[9]</xref> . Multilayer technology can avoid radiation leakage <xref ref-type="bibr" rid="ref10" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">[10]</xref> – <xref ref-type="bibr" rid="ref11" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"/> <xref ref-type="bibr" rid="ref12" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"/> <xref ref-type="bibr" rid="ref13" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">[13]</xref> , but the manufacturing process is relatively complex. In addition, the available miniaturization methods involve one cavity multimode <xref ref-type="bibr" rid="ref14" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">[14]</xref> – <xref ref-type="bibr" rid="ref15" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"/> <xref ref-type="bibr" rid="ref16" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"/> <xref ref-type="bibr" rid="ref17" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">[17]</xref> and loading technologies <xref ref-type="bibr" rid="ref18" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">[18]</xref> – <xref ref-type="bibr" rid="ref19" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"/> <xref ref-type="bibr" rid="ref20" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">[20]</xref> . Among the different options for these technologies, as a metamaterial structure, complementary split-ring resonators (CSRRs) have attracted the attention of many scholars for their unique characteristics.