# Ultra-compact quad-band half-mode SIW bandpass filter for Sub-6 GHz 5G applications

Reza Asgharivaskasi, Valiollah Mashayekhi, Nima Azadi-Tinat, Mohsen Koohestani

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Reza Asgharivaskasi, Valiollah Mashayekhi, Nima Azadi-Tinat, Mohsen Koohestani. Ultra-compact quad-band half-mode SIW bandpass filter for Sub-6 GHz 5G applications. Scientific Reports, 2025, 15 (1), pp.30077. 10.1038/s41598-025-14222-6 . hal-05234839

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# Ultra-compact quad-band half-mode SIW bandpass filter for Sub-6 GHz 5G applications

Reza Asgharivaskasi<fnref n="1" />, Valiollah Mashayekhi<fnref n="1" />✉, Nima Azadi-Tinat<fnref n="1" />✉ & Mohsen Koohestani²,³

This paper presents the design of an ultra-compact, low-loss, multi-band bandpass filter (BPF) tailored for sub-6 GHz 5G applications. The filter is based on a half-mode substrate-integrated waveguide (HMSIW) structure integrated with metamaterial-inspired unit cells, consisting of three circular and two symmetrical serrated complementary split-ring resonators (CSRRs). This configuration enables efficient and stable wave propagation even below the HMSIW cutoff frequency. The proposed structure significantly reduces the overall footprint while supporting multiple passbands. The filter occupies a compact area of 14.1 × 14.1 mm², corresponding to less than 0.004 λg² (with λg being the guided wavelength at 0.97 GHz), making it one of the smallest quad-band designs reported to date. It is highly suitable for seamless integration into wireless communication systems, including WiFi, WiMAX, WLAN, 5G, and other sub-6 GHz applications. The filter exhibits four distinct passbands centered at 0.97 GHz (0.87–1.11 GHz), 2.58 GHz (2.45–2.65 GHz), 4.5 GHz (4.05–4.6 GHz), and 5.6 GHz (5.45–5.65 GHz), with corresponding insertion (return) losses of – 0.38 dB (30 dB), – 1.1 dB (35 dB), – 0.84 dB (25 dB), and – 1.1 dB (22 dB), respectively. Additionally, the design offers reconfigurability, allowing easy adaptation to different frequency bands with minimal structural modifications. To validate the proposed concept, a prototype was fabricated and experimentally characterized. The measured results show strong agreement with simulations, confirming the efficiency and robustness of the design.

**Keywords** Ultra-compact, Band pass filter, Half-mode SIW, Sub-6 GHz applications, Wireless communication.

The rapid evolution of wireless technology has driven huge demand for compact, low-power consumption, and low-loss components with multi-band functionality, particularly for sub-6 GHz applications. The sub-6 GHz frequency range has been extensively used in modern wireless communication systems, including 5G networks. To meet these demands, compact and high-performance multi-band filters have gained significant research interest<fnref n="1" />⁻⁶.

Recent advancements in multi-band passive microwave filters have focused on achieving compact size and low insertion loss. However, conventional design approaches, such as standard or multi-layer microstrip configurations, often lead to increased dimensions at lower frequency bands and integration challenges. To overcome these limitations, substrate integrated waveguide (SIW) structures have emerged as a promising alternative, offering advantages such as low cost, low loss, ease of fabrication, and seamless integration with other planar components⁷,⁸. By mimicking traditional rectangular waveguides through metalized via-holes connecting the top and bottom metal plates, SIW structures effectively address the drawbacks of conventional filter designs.

Several notable studies have explored different SIW-based multi-band filter configurations. For instance, the study in⁹ presented a dual-band SIW BPF based on CSRRs at resonant frequencies of 2.4 GHz and 5.75 GHz for WiMAX and WLAN applications, achieving a compact size of around 0.03 λg²@2.4GHz; however, this structure suffers from high insertion losses – 2.4 dB, and – 2 dB at mentioned frequencies, respectively. Moreover, the study in¹⁰ proposed two dual-band SIW BPFs based on open-loop ring resonators (OLRRs), achieving sizes of 0.012 λg²@1.5GHz and 0.041 λg²@1.75GHz for L-band, C-band, 5G, and satellite applications.

While SIW components are more compact than standard microwave structures, their dimensions remain relatively large for compact electronic devices. To address this limitation, the half-mode substrate integrated waveguide (HMSIW) technique has been introduced, effectively reducing the size of SIW structures by

<footnote n="1">¹Department of Electrical Engineering, Shahrood University of Technology, Shahrood, Iran. ²École Supérieure d'Electronique de l'Ouest, Angers, France. ³Institut d'Electronique et des Technologies du numeRique (IETR), Université de Rennes, Rennes, France. ✉email: vmashayekhi@shahroodut.ac.ir; azadi@shahroodut.ac.ir</footnote>
approximately 50% while preserving their fundamental waveguide properties. This modification provides a practical miniaturization solution without compromising performance, making HMSIW an attractive platform for designing compact and high-performance microwave gadgets²,⁷.

To further reduce filter size, the study in² proposed an HMSIW triple-band BPF incorporating metamaterial (CSRR) unit cells on both the top and bottom layers, resulting in a compact structure with low insertion loss and an electrical length of 0.01 $λ_g^2@2.35GHz$. Furthermore, the study in¹¹ introduced a triple-band SIW BPF employing two quarter-mode SIW (QMSIW) cavities, though it suffers from a high electrical size of 0.05 $λ_g^2@2.1GHz$. Additionally, the study in¹² presented a quad-band BPF based on a substrate integrated coaxial cavity (SICC), which not only exhibits high insertion losses (–1.9 dB at 1.9 GHz, –1.7 dB at 2.62 GHz, –2.1 dB at 3.63 GHz, and –2.2 dB at 4.63 GHz), but also features a multi-layer structure that limits its integration capability with other circuits, similar to the study in¹³. These limitations highlight and emphasize the demand for a multi-band, miniaturized, planar, and low-loss filter architecture and design concept.

This study presents the design of an ultra-compact, low-loss, multi-band BPF optimized for sub-6 GHz 5G applications. To address the limitations of prior designs, such as excessive electrical size, limited operational bands, high insertion and low return losses, and poor circuit integration, we propose a novel BPF with a HMSIW structure integrated with metamaterial-inspired unit cells. The proposed design addresses the aforementioned limitations by utilizing a single-layer HMSIW structure with embedded simple design metamaterial-inspired (specifically, serrated and concentric circular structures). By exploiting evanescent-mode propagation, this configuration enables a highly integrated solution operating at four practical passbands with suitable 3 dB bandwidth and center frequencies of 0.97 GHz (0.87–1.11 GHz), 2.58 GHz (2.45–2.65 GHz), 4.5 GHz (4.05–4.6 GHz), and 5.6 GHz (5.45–5.65 GHz) compared to those presented in²,³,⁹⁻¹¹, and¹³. The filter achieves low insertion loss (–0.38 dB, –1.1 dB, –0.84 dB, and –1.1 dB), and high return loss (30 dB, 35 dB, 25 dB, and 22 dB, respectively) across all bands and ultra-compact area compared to those proposed in⁹,¹⁰,¹²,¹³. Furthermore, the design occupies an ultra-compact area of less than 0.004 $λ_g^2$ (with $λ_g$ being the guided wavelength at 0.97 GHz) and reduces the size constraints of recent studies in²,⁹⁻¹², and¹³ by approximately 60%, 86%, 66%, 92% 96%, and 91%. In addition, the filter resonant frequencies can be flexibly tuned by minor modifications to specific structural parameters, providing enhanced design adaptability for desired frequency applications. The term “reconfigurability” refers to the design-level reconfigurability of the proposed filter, achieved through minimal structural modifications during the design or fabrication stage. Specifically, it means that each passband’s resonant frequency can be independently and precisely adjusted by making minimal modifications to certain geometrical parameters of the filter. These include the dimensions of the circular CSRRs, the length and width of the serrated CSRR elements, and angular configurations. This approach enables the filter to be easily adapted to different frequency bands and application requirements with minimal structural redesign. These attributes make the proposed filter a promising candidate for WiFi, WiMAX, WLAN, 5G, next-generation, and any other sub-6G GHz wireless communication systems that demand very compact, high-performance, and multi-band filtering solutions.

The rest of the paper is organized as follows: Section II presents the working principle of the HMSIW bandpass filter, the design evolution of the proposed filter, and the role of each resonator in generating the desired passbands. In addition, this section provides a comprehensive analysis, including surface current distributions, and parametric studies of key geometric parameters. Section III offers the simulation and measurement results of a fabricated prototype and a comparative evaluation with recent works to validate the performance of the proposed design. Section IV concludes the paper by summarizing the main contributions and potential applications.

## Quad-band filter design and configuration

### Geometry

Figure 1 illustrates the configuration, design parameters, and a fabricated prototype of the proposed HMSIW filter. The structure comprises three circular CSRRs and two symmetric serrated CSRRs, implemented on a 20-mil Rogers RT/Duroid 5880 substrate ($ε_r = 2.2, tanδ = 0.0009$). The patch layer is connected to the ground plane through fourteen metallic vias, each with a diameter of 0.5 mm and a center-to-center spacing of 1 mm (Table 1).

<figure><img src="image_3.png" /><figcaption>Fig. 1. The structure of the proposed filter: (a) prototype photograph, (b) schematic geometry used in the simulation. All metallic parts shown in yellow in the numerical models were considered to be copper.</figcaption></figure>
The filter is excited via an SMA connector soldered to a 50-ohm feedline with a 1.5 mm strip, which was analyzed through full-wave simulations using Ansys HFSS. The proposed filter occupies an ultra-compact size of only 14.1 × 14.1 mm² (Fig. 1), making it highly suitable for integration into array and MIMO antenna configurations. Its miniaturized design ensures high performance while maintaining low cost, reduced complexity, and seamless compatibility with modern wireless communication systems.

## Operation mechanism

The Half-Mode Substrate Integrated Waveguide (HMSIW) is a compact variant of the conventional Substrate Integrated Waveguide (SIW), formed by bisecting the structure along its magnetic wall. This modification reduces the physical size of the SIW by approximately 50% while preserving its electromagnetic performance. In the design of SIW technologies and their derivatives (e.g., HMSIW), the vias diameter and vias spacing are critical parameters that directly influence wave confinement and overall electromagnetic performance. As the spacing between adjacent vias decreases, the SIW structure more closely approximates the behavior of a conventional rectangular waveguide. Radiation losses due to energy outflow (out of the vias gap), which are significantly affected by both via diameter and spacing, play a crucial role in determining the efficiency of the SIW. Minimizing these losses is essential for achieving optimal performance. This can be accomplished by carefully selecting the via dimensions based on established design guidelines. Specifically, to minimize energy leakage, the design must satisfy $d < \frac{\lambda_g}{5}$ and $p \le 2d$, where $\lambda_g$ is the guided wavelength in the SIW¹⁴. Assuming a lossless medium, the propagation characteristics of the proposed filter can be described by the following key parameters¹⁵:

$$ \gamma = \alpha + j\beta \quad \Rightarrow \quad \gamma = j\beta = j\sqrt{K_0^2 - K_c^2} \quad (1) $$

$$ K_0 = \frac{2\pi f}{C_0} \quad (2) $$

$$ E_z = E_0 e^{-\gamma z} \quad (3) $$

where $\gamma$, $\alpha$, and $\beta$ denote the propagation, attenuation, and phase constants along the propagation direction, respectively; $C_0$ represents the speed of light in free space. Additionally, $K_0$ and $K_c$ are the free-space wave number and cutoff wave number of the passive component, respectively. Furthermore, $E_0$ and $E_z$ correspond to the initial electric field amplitude and its distribution along the propagation direction $z$. It is important to note that the behavior of the HMSIW depends on the relationship between $K_0$ and $K_c$:

* for $K_0 > K_c$: $\beta$ becomes real, rendering $\gamma = j\beta$ (purely imaginary). The propagation modes are supported, enabling wave transmission with phase variation along $z$, $E_z = E_0 e^{-j\beta z}$.

* for $K_0 < K_c$: $\beta$ becomes imaginary, rendering $\gamma = \alpha$ (purely real). The wave undergoes attenuation, with the field decaying exponentially and no power propagation (so called evanescent mode, $E_z = E_0 e^{-\alpha z}$).

As indicated by the wave equations, conventional Transverse Electric (TE) and Transverse Magnetic (TM) modes cannot propagate when operating below the cutoff frequency of the HMSIW. In this regime, the modes transition into evanescent states, characterized by exponentially decaying field amplitudes ($K_0 < K_c$). Moreover, the propagation constant in a lossy medium ($\alpha \neq 0$) is given by:

$$ \gamma = \alpha + j\beta = \sqrt{j\omega\mu(\sigma + j\omega\epsilon)} \quad (4) $$

$$ \gamma^2 = j\omega\mu(\sigma + j\omega\epsilon) = -\omega^2\mu\epsilon + j\omega\mu\sigma \quad (5) $$

$$ \alpha^2 - \beta^2 + j2\alpha\beta = -\omega^2\mu\epsilon + j\omega\mu\sigma \quad (6) $$

<table><caption>Table 1. Geometrical parameters of the proposed filter.</caption><thead><tr><th>Param</th><th>Values</th><th>Param</th><th>Values</th><th>Param</th><th>Values</th></tr></thead><tbody><tr><td>L1</td><td>14.1 mm</td><td>n</td><td>0.4 mm</td><td>i</td><td>0.3 mm</td></tr><tr><td>L2</td><td>14.1 mm</td><td>k1</td><td>0.4 mm</td><td>j</td><td>0.5 mm</td></tr><tr><td>p</td><td>1 mm</td><td>k2</td><td>0.2 mm</td><td>c1</td><td>3.7 mm</td></tr><tr><td>d</td><td>0.5 mm</td><td>k3</td><td>0.8 mm</td><td>c2</td><td>3.5 mm</td></tr><tr><td>r</td><td>4.6 mm</td><td>k4</td><td>0.2 mm</td><td>c3</td><td>3.2 mm</td></tr><tr><td>x</td><td>8.6 mm</td><td>k5</td><td>0.4 mm</td><td>c4</td><td>3 mm</td></tr><tr><td>w</td><td>0.3 mm</td><td>s</td><td>0.2 mm</td><td>c5</td><td>1.8 mm</td></tr><tr><td>g</td><td>0.1 mm</td><td>s1</td><td>0.5 mm</td><td>f1</td><td>1.5 mm</td></tr><tr><td>θ</td><td>82°</td><td>u</td><td>0.5 mm</td><td>f2</td><td>2.6 mm</td></tr><tr><td>m</td><td>12 mm</td><td>v</td><td>0.8 mm</td><td></td><td></td></tr></tbody></table>
* Real part: $\alpha^2 - \beta^2 = -\omega^2\mu\epsilon$

* Imaginary part: $2\alpha\beta = \omega\mu\sigma$

$$ \alpha = \omega \sqrt{\frac{\mu\epsilon}{2}} \left( \sqrt{1 + \left(\frac{\sigma}{\omega\epsilon}\right)^2} - 1 \right)^{1/2} \quad (7) $$

$$ \beta = \omega \sqrt{\frac{\mu\epsilon}{2}} \left( \sqrt{1 + \left(\frac{\sigma}{\omega\epsilon}\right)^2} + 1 \right)^{1/2} \quad (8) $$

where $\epsilon$, $\mu$, and $\sigma$ denote the relative permittivity, permeability, and conductivity of the dielectric substrate, respectively.

To establish effective passbands, two key conditions must be satisfied: (i) minimized attenuation constant ($\alpha$), ensuring minimal signal decay, and (ii) nonzero and smooth phase constant ($\beta$) across the operating frequency range, enabling consistent wave propagation.

As illustrated in the dispersion diagram (Fig. 2), the integration of an HMSIW (with a cutoff frequency of 3.8 GHz$^{14,16}$) and metamaterial-inspired unit cells confirms the existence of four distinct passbands. Within these bands, the attenuation constant $\alpha$ remains negligibly small (though non-zero), minimizing signal loss, whereas the phase constant $\beta$ is continuous and nonzero, ensuring efficient and stable wave propagation.

This behavior validates the ability of the proposed structure to support low-loss transmission across multiple frequency bands, making it suitable for multi-band waveguiding applications.

## Design evolution process

The evolution of the filter design was meticulously shaped through a series of structural modifications, each carefully devised to achieve optimal performance within the constraints of limited space. The overarching goal was to enhance functionality without compromising size, weight, or complexity. These intricate processes of miniaturization, multi-band operation, and low loss operation were realized through the strategic integration of well-established techniques and geometries. A paramount challenge lay in preserving the filter compactness while operating practically in multiple frequency bands, requiring both innovative design strategies and deep technical expertise.

As depicted in Fig. 3, the foundational design, Filter-1, features a circular CSRR with an angular opening $\phi$, which is a crucial parameter governing the resonator’s resonance characteristics. This initial configuration serves as the benchmark for performance evaluation. In Filter-1, with $\phi$ set to 170 degrees, the resonant frequency emerges at approximately 1.87 GHz. By gradually reducing $\phi$, the effective electrical length of the CSRR extends, resulting in a downward shift of the resonant frequency. In Filter-2, where $\phi$ is halved to 85 degrees, the resonance is observed at 1.5 GHz. A further reduction to 1 degree in Filter-3 extends the electrical length even further, lowering the resonance to 1.25 GHz (Fig. 4). This transformation induces a significant frequency shift of 620 MHz (33%) from Filter-1 to Filter-3, underscoring the profound influence of $\phi$ on spectral characteristics. Finally, Filter-3 attains a primary resonance and transmission zero at 1.25 GHz and 2.8 GHz, characterized by a return loss exceeding 30 dB, an insertion loss as low as – 0.2 dB, and a 3 dB fractional bandwidth (FBW) of 27%, as simulated through full-wave electromagnetic simulations using Ansys HFSS (Fig. 4).

While the lower resonance was successfully established at around 1 GHz, the design necessitated an additional resonator to accommodate higher-frequency passbands within the sub-6 GHz frequencies. To this end, serrated CSRR structures were introduced in Filter-4 and Filter-5, serving as a resonator for extending the operational upper band. The implementation of a single serrated CSRR in Filter-4 precipitated the emergence of a second resonance at 6.6 GHz (Fig. 4). Recognizing the need to refine this upper resonance while preserving the integrity of the first, asymmetric serrated CSRRs was incorporated in Filter-5, effectively bringing the second resonance down to 5.8 GHz. Thus, the resultant Filter-5 operates at 1.25 GHz and 5.8 GHz, exhibiting return

<figure><img src="image_4.png" /><figcaption>Fig. 2. Dispersion and attenuation diagrams for the proposed quad-band BPF.</figcaption></figure>
<figure><img src="image_5.png" /><figcaption>Fig. 3. Schematic geometry of various filter involved in the design evolution process.</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>Fig. 4. Effect of structural modifications on the proposed filter frequency response: (a) $|S_{11}|$, (b) $|S_{21}|$.</figcaption></figure>

losses surpassing 30 dB and insertion losses below – 0.25 dB for both bands, and 3 dB FBWs of 26.4% and 23.1% for the respective resonances (Fig. 4).

The necessity to integrate a mid-band resonance, particularly to encompass WiFi and other similar frequencies, prompted the development of Filter-6 and Filter-7. These iterations leveraged an innovative nested CSRR topology, wherein two concentric CSRRs were arranged to engineer dual resonances at approximately 2.65 GHz and 4.6 GHz. Filter-7 operates at 1.15 GHz, 2.65 GHz, 4.6 GHz, and 5.65 GHz, demonstrating return losses exceeding 20 dB across all the operational frequency bands (Fig. 4a), with insertion losses constrained to below – 0.3 dB, – 1 dB, – 0.65 dB, and – 0.55 dB, respectively (Fig. 4b). The final stage of refinement, shown in Filter-8, involved the strategic addition of two slots, effectively extending the electrical length and shifting the first resonance downward to 1 GHz. The culmination of this evolutionary process yielded a quad-band BPF, covering 1 GHz, 2.6 GHz, 4.6 GHz, and 5.65 GHz, with return losses exceeding 20 dB across all the operational frequency bands (Fig. 4a). Additionally, low insertion losses of – 0.2 dB, – 0.65 dB, – 0.67 dB, and – 0.57 dB were obtained, respectively, while achieving 3 dB FBWs of 31%, 8.5%, 9.3%, and 8.3% (Fig. 4b).

As depicted in the summary of the design progression in Table 2, the evolution of the filter design clearly demonstrates a trade-off among multi-band capability, achievable bandwidth, and electrical size. The transition from Filter-1 to Filter-3 results in a reduction of the 3 dB FBW from 58.8% to 27%, corresponding to an approximate 54% decrease. Meanwhile, the design becomes approximately twice as compact in terms of electrical size ($0.08 \times 0.08 \lambda_g^2$), highlighting a deliberate effort toward miniaturization. Nevertheless, these early-stage filters still exhibit relatively large electrical footprints and are limited to single-band responses, underscoring the need to increase the number of passbands for more versatile performance. The transition from Filter-3 to
<table><caption>**Table 2.** Design progression from single-band (Filter-1) to quad-band (Filter-8).</caption><thead><tr><th rowspan="2">Filter</th><th colspan="2">First band</th><th colspan="2">Second band</th><th colspan="2">Third band</th><th colspan="2">Fourth band</th></tr><tr><th>f<sub>r1</sub><br>(GHz)</th><th>FBW<br>(%)</th><th>f<sub>r2</sub><br>(GHz)</th><th>FBW<br>(%)</th><th>f<sub>r3</sub><br>(GHz)</th><th>FBW<br>(%)</th><th>f<sub>r4</sub><br>(GHz)</th><th>FBW<br>(%)</th></tr></thead><tbody><tr><td>1</td><td>1.87</td><td>58.8</td><td>×</td><td>×</td><td>×</td><td>×</td><td>×</td><td>×</td></tr><tr><td>2</td><td>1.5</td><td>54</td><td>×</td><td>×</td><td>×</td><td>×</td><td>×</td><td>×</td></tr><tr><td>3</td><td>1.25</td><td>27</td><td>×</td><td>×</td><td>×</td><td>×</td><td>×</td><td>×</td></tr><tr><td>4</td><td>1.25</td><td>25.6</td><td>6.6</td><td>20.7</td><td>×</td><td>×</td><td>×</td><td>×</td></tr><tr><td>5</td><td>1.25</td><td>26.4</td><td>5.8</td><td>23.1</td><td>×</td><td>×</td><td>×</td><td>×</td></tr><tr><td>6</td><td>1.25</td><td>22.4</td><td>3</td><td>0.01</td><td>5.7</td><td>19.1</td><td>×</td><td>×</td></tr><tr><td>7</td><td>1.15</td><td>18.2</td><td>2.65</td><td>5.3</td><td>4.6</td><td>10</td><td>5.65</td><td>9</td></tr><tr><td>8</td><td>1</td><td>31</td><td>2.6</td><td>8.5</td><td>4.6</td><td>9.3</td><td>5.65</td><td>8.3</td></tr></tbody></table>

Filter-4 and Filter-5 marks a significant step toward practical multi-band operation. While the 3 dB FBW of the first band (at 1.25 GHz) slightly decreases a second passband emerges, with center frequencies at 6.6 GHz for Filter-4 and 5.8 GHz for Filter-5. Notably, the second passband in Filter-5 exhibits a 2.4% wider 3 dB FBW, approximately a 11% improvement over Filter-4, indicating enhanced bandwidth performance in the higher band. This advancement highlights the efficient utilization of the filter structure and represents a meaningful progression toward compact and versatile multi-band filter architectures. However, Filter-6, which introduces a tri-band response at 1.25 GHz, 3 GHz, and 5.7 GHz, illustrates the increasing complexity associated with maintaining consistent performance across multiple bands. While the first and third bands maintain reasonable 3 dB FBWs of 22.4% and 19.1%, respectively, the second band at 3 GHz exhibits an unstable response. This instability underscores the design challenges in achieving uniform bandwidth and insertion loss performance as the number of passbands increases.

Ultimately, Filter-7 achieves the target quad-band operation, with passbands centered at 1.15 GHz, 2.65 GHz, 4.6 GHz, and 5.65 GHz, marking a major advancement in the design evolution. However, this milestone comes with narrower bandwidths, particularly in the lower-frequency bands. Specifically, the 3 dB FBW of the first band in Filter-7 is reduced by approximately 69%, 66%, 32.6%, 28.9%, 31.1%, and 18.7%, compared to Filter-1 through Filter-6, respectively. For the second band, Filter-7 shows reductions of approximately 74.4% and 77% in 3 dB FBW, compared to Filter-4 and Filter-5, respectively, though it demonstrates an improvement over Filter-6, in which the second band was unstable. In terms of miniaturization, Filter-7 achieves a substantial size reduction, by approximately 62%, 40.5%, and 14.5% compared to Filter-1, Filter-2, and Filters-3 (through Filter-6), respectively, reflecting a significant advancement toward compact, multi-band filter design.

Finally, Filter-8 offers a more balanced trade-off between high FBW and miniaturization, maintaining quad-band functionality at 1 GHz, 2.6 GHz, 4.6 GHz, and 5.65 GHz, with moderately wider and more consistent bandwidths (Table 2). The 3 dB FBW of the first band (at 1 GHz) is reduced by approximately 47.2%, 42.5%, compared to Filter-1 and Filter-2, respectively. Meanwhile, it increases the number of passbands to four and achieves a substantial reduction in electrical size, approximately 72% and 56% smaller than Filter-1 and Filter-2, respectively. Furthermore, it clearly improves the 3 dB FBW of the first band by about 14.8%, 21%, and 17.5%, compared to Filter-3, Filter-4, and Filter-5, respectively. Although the 3 dB FBW of the second band in Filter-8 is reduced by approximately 58.9% and 63%, compared to Filter-4 and Filter-5, respectively, it increases the number of passbands and reduces the electrical size of the aforementioned filters by approximately 36%. It also demonstrates a performance improvement over Filter-6, in which the second band is unstable. In addition, Filter-8 outperforms Filter-7 in terms of both bandwidth and performance stability, with improvements of approximately 70% and 60% in the first and second passbands, respectively. The 3 dB FBWs of the third and fourth bands remain relatively consistent with those of Filter-7, ensuring reliable multi-band behavior. In terms of miniaturization, Filter-8 achieves a substantial reduction in electrical size, of approximately 36% and 25% compared to Filter-6 and Filter-7, respectively. This design progression underscores the balancing act among multi-band capability, bandwidth, and electrical size, ultimately culminating in a highly compact and well-balanced quad-band filter.

The transformation from Filter-1 to Filter-8 provides a systematic and progressive enhancement of structure and functionality, culminating in a highly efficient, very compact, and versatile filtering solution. By ingeniously harnessing resonator interactions, this design framework offers unparalleled degrees of freedom in shaping frequency response while preserving minimal dimensions. The resultant filter emerges as a paragon of multi-band engineering, characterized by low loss, compactness, and adaptability, poised to meet the rigorous demands of modern sub-6 GHz wireless communication systems.

## Equivalent circuit model analysis

To analyze the operating principle of the proposed filter from a circuit-level perspective, an equivalent circuit model was developed, as illustrated in Fig. 5a. The HMSIW via structure is represented by an inductance, $L_{via}$, while the circular CSRRs and the two symmetric serrated CSRRs are modeled as resonator tanks. Specifically, the transmission zero due to the outer (with rectangular slots), middle, and inner circular CSRRs are represented by the capacitance-inductance pairs (C1, L1), (C3, L3), and (C5, L5), respectively, while the transmission zero
<figure><img src="image_7.png" /><figcaption>**Fig. 5.** Equivalent circuit model for the proposed structure: (a) schematic, (b) simulated $|S_{11}|$ and $|S_{21}|$ results, and (c) unwrapped phase of $S_{21}$, obtained from HFSS and ADS.</figcaption></figure>

associated with the serrated CSRRs are characterized by $C_7$ and $L_7$. The coupling effects between the resonant elements are incorporated using additional lumped circuit elements. The interaction between the HMSIW structure and the CSRRs is accounted for by $C_c$ and $L_c$, whereas the couplings between adjacent CSRR elements are denoted as $C_2$, $C_4$, and $C_6$, corresponding to the outer-to-middle circular CSRR, middle-to-inner circular CSRR, and inner circular-to-serrated CSRR couplings, respectively. To accurately account for the phase response, a $\pi$-LC network needs to be added to model the input/output excitation line¹⁷. The initial values of the lumped components for the resonating tanks correspond to their respective pole frequencies obtained from full-wave simulations, while the coupling parameters were tuned using Keysight advanced design system (ADS) simulations. The specific values assigned to these circuit elements in the proposed model are as follows: $L_{via} = 6$ nH, $C_1 = 4.5$ pF, $L_1 = 1.9$ nH, $C_2 = 0.005$ pF, $C_3 = 1$ pF, $L_3 = 2.8$ nH, $C_4 = 0.005$ pF, $C_5 = 0.11$ pF, $L_5 = 9.3$ nH, $C_6 = 0.005$ pF, $C_7 = 0.16$ pF, $L_7 = 4.3$ nH, $C_c = 0.2$ pF, $L_c = 0.2$ nH, $C = 0.93$ pF, and $L = 1.9$ nH.

As demonstrated in Fig. 5b and c, the simulated and calculated responses of the quad-band HMSIW filter exhibit a strong correlation between transmission poles and zeros, validating the accuracy of the equivalent circuit model in representing the transmission characteristics of the proposed structure.

## Electric current distribution

To further demonstrate the filter operation and emphasize the role of the resonators, the surface current distributions were analyzed for two filter configurations, Filter-5 and Filter-8, as depicted in Fig. 6. The color scale illustrates the current distribution intensity, transitioning from dark blue (representing low current density) to green, yellow, and red (representing high current density). For comparison, the vector currents in Filter-5 were examined at the resonant frequencies of 1.25 GHz and 5.8 GHz. At 1.25 GHz, the currents concentrate on the outer CSRR, which is prominently excited (Fig. 6a). Similarly, at 5.8 GHz, the currents advance, leading to the excitation of the serrated CSRRs, as illustrated in Fig. 6b.

Additionally, the vector currents in Filter-8 were evaluated at the resonant frequencies of 1 GHz, 2.6 GHz, 4.6 GHz, and 5.65 GHz. At 1 GHz, the excitation of the outer circular CSRR was observed (Fig. 6c), while at 2.6 GHz, the middle circular CSRR was excited (Fig. 6d). At 4.6 GHz, the signal distributes mainly through the inner circular CSRR (Fig. 6e), and finally, at 5.65 GHz, the serrated CSRRs were excited (Fig. 6f). These current distributions validate the operational mechanism of the proposed filters and confirm the role of each CSRR in generating the respective passbands.

## Parametric study

The prominent feature of the proposed filter is its flexibility to be reconfigured for other frequencies through minor design modifications. In the preceding sections, the effects of the angular parameter ($\varphi$) and rectangular slot elements on the resonant frequency and transmission zero were examined. This section provides a comprehensive investigation into the impact of key structural parameters on the operational characteristics of the proposed filter.

A detailed analysis of the results (as depicted in Fig. 7) reveals that specific geometric modifications significantly influence individual resonant frequencies while having minimal impact on other modes. These findings highlight the flexibility of the design in precisely controlling multi-band filtering characteristics;
<figure><img src="image_8.png" /><figcaption>Fig. 6. Surface current distributions: Filter-5 at (a) 1.25 GHz and (b) 5.8 GHz and Filter-8 at (c) 1 GHz, (d) 2.6 GHz, (e) 4.6 GHz, (f) 5.65 GHz, respectively. Scale: logarithmic with 15 subdivisions ranging from 1E0 to 1E3 A/m.</figcaption></figure>

<figure><img src="image_9.png" /><figcaption>Fig. 7. Analysis of function for different key parameters of the proposed BPF: (a) $|S_{11}|$, (b) $|S_{21}|$.</figcaption></figure>

* The inner diameter and width of the middle circular CSRR play a crucial role in determining the second resonant frequency, while their impact on other resonance modes remains minimal. A detailed parametric evaluation indicates that the *k*3 and *k*4 parameters serve as the primary tuning variables governing this frequency shift. For instance, when *k*3 = 0.2 mm and *k*4 = 0.8 mm, the resonance occurs at approximately 2.25 GHz. Increasing *k*3 to 0.5 mm and reducing *k*4 to 0.5 mm, shifts up the resonance to 2.45 GHz (corresponding to a 8.9% increase), while further adjustments to *k*3 = 0.8 mm and *k*4 = 0.2 mm result in a resonance at 2.6 GHz, representing a 15% resonant frequency upshift compared to the initial configuration (*k*3 = 0.2 mm and *k*4 = 0.8 mm). These results demonstrate a strong correlation between the structural parameters and the resonant frequency. By carefully selecting *k*3 and *k*4, the second passband can be precisely tuned to the desired frequency, enhancing design flexibility and adaptability for various multi-band filtering applications.

* Analogous to the influence of $\phi$ discussed in Sect. 2.3, the parameter $\theta$ plays a pivotal role in determining the third resonant frequency, while its impact on other resonance modes remains negligible. As depicted in Fig. 7, variations in $\theta$ effectively modify the electrical length of the CSRR, allowing precise control over the third resonance. Simulation results reveal that for $\theta$ = 52°, the third resonant frequency is observed at approximately 4.3 GHz. Increasing $\theta$ to 82° shifts the resonance to 4.6 GHz, representing a 6.5% upshift, while further increasing $\theta$ to 122° raises the frequency to 5.1 GHz, a 18.6% increase compared to $\theta$ = 52°. These findings
<figure><img src="image_10.png" /><figcaption>Fig. 8. Measurement setup.</figcaption></figure>

<figure><img src="image_11.png" /><figcaption>Fig. 9. Simulated and measured frequency response of the proposed BPF: (a) $|S_{11}|$, (b) $|S_{21}|$.</figcaption></figure>

establish a direct correlation between $\theta$ and the tunability of the third resonant frequency, demonstrating its significance as a key design parameter for optimizing multi-band filter performance.

* The length and width of the serrated CSRR elements play a critical role in determining the fourth resonant frequency, while having negligible influence on the stability of other frequency bands. A detailed parametric evaluation indicates that adjusting these structural elements enables precise control over the fourth passband, allowing for optimized frequency allocation without compromising the filter's overall performance. As shown in Fig. 7, the results for the standard Fig. 1 serrated CSRR elements and the modified configuration with c1 to c4 = 2.5 mm demonstrate a shift in the fourth resonant frequency from 5.65 GHz to 5.85 GHz, corresponding to a 3.54% upshift compared to the standard case presented in Table 1. This confirms that fine-tuning the serrated CSRR dimensions provides an effective approach for frequency adjustment, enhancing the design flexibility for various multi-band applications requiring compactness, low insertion loss, and high selectivity.

## Results and discussion

To assess the effectiveness of the proposed design in achieving size reduction while maintaining high performance, a prototype filter was fabricated. The measured scattering parameters are presented to validate the simulation results. The measurement setup is shown in Fig. 8.

The measured and simulated frequency response of the filter, shown in Fig. 9, exhibits strong agreement, validating the accuracy of the design methodology. The minor discrepancies between the simulated and measured results can be attributed to fabrication tolerances, as the filter includes several delicate features such as thin strips and shorting pins. More significantly, the additional soldering required to metallize the vias may introduce parasitic effects, leading to slight deviations in the S-parameters. Nevertheless, these discrepancies remain within acceptable limits and do not compromise the overall performance trends or the validity of the proposed design. The measured return losses for the passbands centered at 0.97 GHz (0.87–1.11 GHz), 2.58 GHz (2.45–2.65 GHz), 4.5 GHz (4.05–4.6 GHz), and 5.6 GHz (5.45–5.65 GHz) exceed 30 dB, 35 dB, 25 dB, and 22 dB, respectively. Additionally, the measured insertion losses are – 0.38 dB, – 1.1 dB, – 0.84 dB, and – 1.1 dB, highlighting the low-loss characteristic of the proposed filter. The measured 3 dB FBWs for the respective passbands are 24.8%, 7.7%, 12%, and 3.5%, demonstrating well-defined frequency selectivity. Fig. 10 presents the measured
and simulated phase responses of the proposed structure. At first glance, the phase variation with frequency indicates the presence of a propagating mode, and the negative phase observed within the transmission bands confirms forward wave propagation below the cutoff frequency of the HMSIW filter. However, a discrepancy between the measured and simulated phase responses is evident. This phase offset arises from the omission of the SMA-to-N type adapter, which was not included in the VNA calibration. To compensate for this, the input/output transmission line segments in the full-wave simulation were extended by a length equivalent to that of the adapter, leading to a good agreement between the measured and simulated phase responses of the proposed filter.

The fabricated HMSIW quad-band BPF has a compact size of only $14.1 \times 14.1 \text{ mm}^2$ (0.063 $\lambda_g$ × 0.063 $\lambda_g$), where $\lambda_g$ is the wavelength at 0.97 GHz frequency. A comparative analysis with existing multi-band BPFs, presented in Table 3, highlights significantly higher efficiency compared to prior works, reflecting superior spectral utilization and compactness. This improvement is evident in the following aspects:

*   **Miniaturization:** among existing multi-band designs, the concept presented in this work occupies significantly less area, approximately 60%, 86%, 86%, 66%, 90%, 92%, 96%, 91%, 45%, 99%, 98%, 74.7%, 99%, and 99% less area than the designs reported in$^{2,9-I,9-II,10-I,10-II,11-13,18-23}$, respectively. This notable reduction in size is crucial for modern electronic systems, where compactness is essential.

*   **Insertion loss (IL):** the proposed design demonstrates low insertion loss performance, ranging from – 0.38 to –1.1 dB across its operating bands. This IL range is significantly lower than most previously reported works, such as$^{9,12,13,19,20,23}$, which exhibit losses exceeding – 1.7 dB. Although a few studies like$^{2,18}$ report IL values comparable to the proposed design, overall, our results indicate improved efficiency, lower signal attenuation, and superior impedance matching across all operating bands. These improvements contribute to the enhanced filter performance and its suitability for applications demanding minimal insertion loss.

*   **FBW-to-size ratio:** as shown in Table 3, the proposed filter exhibits remarkable improvements in FBW, offering a higher FBW-to-size ratio than previously reported designs. Specifically, it offers 229%, 433%, and 330% higher efficiency for the first passband, and 74%, 213%, and 123% higher efficiency for the second passband when compared to$^{2,10-I}$, and$^{18}$, respectively. Additionally, for the third band, it achieves 335.4% and 411% higher efficiency relative to$^{2,18}$, respectively.

*   **Passband multiplicity and structural simplicity:** unlike$^{2,9-11,13,18,19}$, the proposed filter supports four distinct passbands while maintaining a single-layer design. In contrast to multi-layer approaches such as$^{12,13,23}$, this structure simplifies fabrication and enhances compatibility with planar circuit integration.

These results confirm that the proposed multi-band filter offers several advantages, including ultra-compact size, low insertion loss, ease of fabrication, and seamless integration with planar microwave circuits. The comparative analysis further underscores its superiority, positioning it as a promising candidate for the development of compact, high-performance bandpass filters.

## Conclusion

This paper deals with the design of an ultra-compact, low-loss, multi-band bandpass filter suitable for sub-6 GHz 5G applications. The proposed filter leverages a half-mode substrate-integrated waveguide (HMSIW) structure integrated with metamaterial-inspired unit cells composed of three circular and two symmetrical serrated complementary split-ring resonators (CSRRs). When excited by an axial electric field, this configuration supports efficient and stable wave propagation even below the HMSIW cutoff frequency. Based on this principle, the filter achieves four passbands centered at approximately 0.97 GHz, 2.58 GHz, 4.5 GHz, and 5.6 GHz. Remarkably, the design occupies an area of less than 0.004 $\lambda_g^2$, making it one of the most compact quad-band BPFs reported to date. Simulated and measured results confirm excellent performance, with insertion (return) losses of -0.38 dB (30 dB), -1.1 dB (35 dB), -0.84 dB (25 dB), and -1.1 dB (22 dB) at the respective operating frequencies. The proposed HMSIW quad-band filter offers several advantages: ultra-compact footprint, high return loss, low insertion loss, low-cost fabrication, straightforward integration with planar circuits, and flexibility for frequency reconfiguration with minimal design changes. By addressing common limitations in recent works while

<figure><img src="image_12.png" /><figcaption>**Fig. 10.** Unwrapped phase response.</figcaption></figure>
<table><caption>**Table 3.** Comparison of proposed filter with previous studies.</caption><thead><tr><th>Refs.</th><th>f<sub>o</sub> (GHz)</th><th>IL (dB)</th><th>RL (dB)</th><th>FBW (%)</th><th>Size (λ<sub>g</sub><sup>2</sup>)</th><th>FBW-to-size ratio</th><th>Layer, substrate</th></tr></thead><tbody><tr><td>[9]-I</td><td>2.4, 5.75</td><td>-2.4, -2.0</td><td>23, 14</td><td>5.9, 5.4</td><td>0.23×0.13</td><td>197.3, 180.6</td><td>Single, RO4003C</td></tr><tr><td>[9]- II</td><td>3.2, 5.8</td><td>-0.9, -2.3</td><td>31, 21</td><td>14.2, 8</td><td>0.24×0.12</td><td>493.1, 277.7</td><td>Single, RO4003C</td></tr><tr><td>[10]- I</td><td>1.5, 4.96</td><td>-0.85, -0.9</td><td>20, 23</td><td>14, 7.3</td><td>0.077×0.155</td><td>1173, 611.6</td><td>Single, RT/Duroid 5870</td></tr><tr><td>[10]- II</td><td>1.75, 4.65</td><td>-1.1, -1.15</td><td>14, 21</td><td>14.9, 10.4</td><td>0.22×0.18</td><td>376.7, 262.6</td><td>Single, RT/Duroid 5870</td></tr><tr><td>[18]</td><td>0.35, 0.95, 1.68</td><td>-1.1, -0.77, -0.67</td><td>23, 15, 18</td><td>10.5, 6.2, 4.3</td><td>0.07×0.103</td><td>1456.3, 859.9, 596.4</td><td>Single, RO3010</td></tr><tr><td>[11]</td><td>2.1, 3.98, 6.6</td><td>-0.93, -0.98, -1.2</td><td>16, 18, 22</td><td>17.1, 8.8, 8.4</td><td>0.05</td><td>342, 176, 168</td><td>Single, RT/Duroid 5880</td></tr><tr><td>[19]</td><td>14.4, 16.9, 22.1</td><td>-2.53, -1.23, -1.31</td><td>17.5, 21.3, 16.9</td><td>1.2, 2.4, 2.9</td><td>1.44×1.44</td><td>0.6, 1.2, 1.4</td><td>Single, RT/Duroid 5880</td></tr><tr><td>[13]</td><td>1.83, 2.1, 2.47</td><td>-2.31, -2.25, -3.01</td><td>30, 20, 15</td><td>3.1, 2.9, 2.3</td><td>0.22×0.22</td><td>64, 59, 47.5</td><td>Multi, RT/Duroid 5880</td></tr><tr><td>[2]</td><td>2.35, 3.7, 5.65</td><td>-0.35, -0.6, -0.85</td><td>30, 30, 25</td><td>19, 11, 7</td><td>0.1×0.1</td><td>1900, 1100, 700</td><td>Single, RO4003C</td></tr><tr><td>[20]</td><td>2.54, 3.46, 4.5, 5.2</td><td>-1.58, -1.78, -2.23, -2.45</td><td>>14.2</td><td>6.4, 5.7, 3.9, 4.9</td><td>0.59×0.42</td><td>26, 23, 15.9, 19.8</td><td>Single, RT/Duroid 5880</td></tr><tr><td>[21]</td><td>1.2, 2.52, 3.51, 4.44</td><td>0.034, 0.079, 0.043, 0.085</td><td>30, 25, 20, 20</td><td>4.8, 3.9, 3.9, 2.2</td><td>0.157×0.1</td><td>305.7, 252.3, 252.3, 140.1</td><td>Single, MgO</td></tr><tr><td>[22]</td><td>11.53, 12.51, 14.7, 15.22</td><td>-1.33, -1.22, -1.43, -1.53</td><td>> 25</td><td>3.1, 3.3, 2.4, 2.1</td><td>2.15×1.27</td><td>1.1, 1.2, 0.9, 0.9</td><td>Single, RT/Duroid 5880</td></tr><tr><td>[12]</td><td>1.9, 2.62, 3.63, 4.63</td><td>-1.9, -1.7, -2.1, -2.2</td><td>>15, >20, >15, >15</td><td>3.2, 2.8, 1.4, 1.7</td><td>0.35×0.35</td><td>26, 22.8, 11.4, 13.9</td><td>Multi, RT/Duroid 5880</td></tr><tr><td>[23]</td><td>10.21, 10.77, 11.36, 11.83</td><td>-1.94, -2.04, -2.6, -2.25</td><td>>35, >35, >20, >15</td><td>2.3, 1.8, 1.1, 1.5</td><td>0.87×0.87</td><td>3.1, 2.4, 1.3, 2</td><td>Multi, Tanconic TLY-5</td></tr><tr><td>Proposed</td><td>0.97, 2.58, 4.5, 5.6</td><td>-0.38, -1.1, -0.84, -1.1</td><td>30, 35, 25, 22</td><td>24.8, 7.7, 12, 3.5</td><td>0.063×0.063</td><td>6250, 1915, 3048, 880</td><td>Single, RT/Duroid</td></tr></tbody></table>

maintaining high performance across four bands, this filter presents a compelling solution for modern wireless systems, including WiFi, WiMAX, WLAN, 5G, and other sub-6 GHz applications requiring compact and efficient multi-band filtering. The proposed quad-band filter, featuring concentric and serrated metamaterial-inspired structures on a single-layer HMSIW platform, offers strong potential for advanced RF systems. A key extension could involve reconfigurability through tuning elements such as varactor diodes, enabling real-time frequency adaptation for dynamic environments. Its compact and well-engineered design also makes it suitable for reconfigurable multi-band power dividers or MIMO systems, supporting multifunctional integration in sub-6 GHz wireless, cognitive radio, and IoT applications.

## Data availibility

All data generated or analyzed during this study are included in this published article.

Received: 17 April 2025; Accepted: 30 July 2025

Published online: 17 August 2025

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## Author contributions

R.A. developed the methodology, implemented the software, and prepared the original draft. V.M. contributed to the conceptualization and methodology, and participated in reviewing and editing the manuscript. N.A. was involved in the development of methodology and contributed to the review and editing process. M.K. contributed to the conceptualization and methodology, prepared the original draft, and participated in reviewing and editing the manuscript. All authors contributed to the scientific discussion of the results and approved the final manuscript.

## Declarations

### Competing interests

The authors declare no competing interests.

### Additional information

Correspondence and requests for materials should be addressed to V.M. or N.A.-T.

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