## 2 State-of-the-Art  
This section examines current dielectric resonator cavity filter (DRCF) technology, analyzing operating principles, advantages, and limitations while comparing them with alternative solutions such as microstrip filters. Dielectric resonator cavity filters are widely used in satellite communication systems due to their high performance characteristics. These filters are particularly effective at increasing near-band selectivity as the roll-off factor \(\alpha\) increases, which directly translates to an improvement in the quality factor. This enhanced performance stems from the filters' ability to maintain steep rejection slopes while preserving excellent passband characteristics. These filters feature a high dielectric constant (\(\epsilon_{r}\)) typically ranging from 10 to 30 [19; 20], combined with low insertion loss (IL) enabled by low loss tangent values (\(\tan\delta\)). They achieve high intrinsic quality factors, e.g., \(Q_{i}>\) 10,000 [19; 20; 21], while maintaining a low coefficient of thermal expansion (CTE) below 10 ppm/\({}^{\circ}\)C [20; 21] and excellent mechanical stability under extreme vibration conditions. Common dielectric resonator materials are employed, such as alumina (Al\({}_{2}\)O\({}_{3}\)), lead zirconate titanate (PZT), barium titanate (BaTiO\({}_{3}\)), calcium zirconate (CaZrO\({}_{3}\)), and silicon dioxide (SiO\({}_{2}\)) [19; 20; 21]. These high-permittivity materials enable significant size reduction while providing intrinsic quality factors \(Q_{i}\) ranging from 5000 to 20,000 [19; 20; 21]. The resonators are typically housed in silver-plated aluminum cavities with alodine or bonderite surface treatments [22], achieving low insertion loss and high external quality factors \(Q_{e}\) suitable for satellite communication systems. The performance of dielectric resonators depends critically on their resonant mode, which may be transverse electric (TE), transverse magnetic (TM), or hybrid electromagnetic (HE). Hybrid modes are less common in practical applications due to their combined longitudinal electric and magnetic field components, which complicate both theoretical analysis and performance predictability. In contrast, TE and TM modes exhibit greater stability as they maintain purely transverse field components relative to the propagation direction. This results in more predictable field distributions and reduced sensitivity to manufacturing tolerances, material variations, and environmental conditions. The combination of this modal stability with the inherent high permittivity and low-loss characteristics of ceramic materials makes TE and TM modes preferable for high-performance applications. Figure 4 shows the electric field distribution for the primary modes in a cylindrical DR: TE\({}_{01\delta}\) (the dominant low-frequency mode) and HE\({}_{11}\). Figure 5 shows a third-order dielectric resonator cavity filter implementation. The excitation modes are mainly determined by the resonator's geometry and aspect ratio (thickness-to-diameter relationship) [23]. The design uses mechanical tuning elements for individual dielectric resonator frequency adjustment. Inter-resonator coupling combines iris-coupled cavities with precisely controlled spacing. The resonators are mounted on dedicated pedestals to maintain exact vertical positioning. However, as previously discussed, flexible filter solutions demand not only higher-order designs but also the implementation of both transmission zeros and equalization zeros, especially when operating with highly constrained fractional bandwidths. In DRCF implementations, these zeros can be realized using either probe- or iris-based cross-coupling techniques that extract a portion of the energy from the main transmission path [16; 24], as demonstrated in Figure 6a,b. This approach generates a coupling matrix where element M\({}_{41}\) retains identical magnitude but opposite polarity.