## 2 Materials and Methods  
Mg\({}_{2}\)SiO\({}_{4}\) (MSO) powders were first synthesized by the conventional solid-state method with the raw materials of MgO (99.9%, Shanghai Macklin Biochemical Co., Ltd., Shanghai, China) and SiO\({}_{2}\) (99.99%, Shanghai Macklin Biochemical Co., Ltd., Shanghai, China) according to the Mg\({}_{2}\)SiO\({}_{4}\) stoichiometric ratio calcined at different temperatures (1250 \({}^{\circ}\)C, 1300 \({}^{\circ}\)C, 1350 \({}^{\circ}\)C, and 1400 \({}^{\circ}\)C) for 3 h. (61 wt% Mg\({}_{2}\)SiO\({}_{4}\)-39 wt% Li\({}_{2}\)TiO\({}_{3}\))-8 wt% LiF-x wt% CaTiO\({}_{3}\) ceramics (x = 0, 7.5, 8.5, 9.5, 10.5, and 11.5; MLL, MLLC7.5, MLLC8.5, MLLC9.5, MLLC10.5, and MLLC11.5 for short, respectively) were prepared by the conventional solid-state reaction process using the starting materials including the composite Mg\({}_{2}\)SiO\({}_{4}\), Li\({}_{2}\)TiO\({}_{3}\) (99%, Shanghai Aladdin Biochemical Technology Co., Ltd., Shanghai, China), LiF (99.99%, Shanghai Aladdin Biochemical Technology Co., Ltd., Shanghai, China) and CaTiO\({}_{3}\) (99.5%, Shanghai Aladdin Biochemical Technology Co., Ltd., Shanghai, China). The mixed powders were ball-milled in absolute ethyl alcohol for 24 h. After the slurry was dried, the homogeneous powders were obtained and pressed into pellets, approximately 12 mm in diameter and 6 mm in thickness using 200 MPa pressure and 5% PVA binder. When burning out the binder at 600 \({}^{\circ}\)C for 2 h, all the samples were sintered at 775 \({}^{\circ}\)C, 800 \({}^{\circ}\)C, 825 \({}^{\circ}\)C, and 850 \({}^{\circ}\)C for 4 h in air using a muffle. To verify LTCC application, 40 wt% Ag powders were mixed with the MLLC11.5 powders without sintering, using the magnetic stirring method in absolute ethyl alcohol for 8 h. After mixing well and drying, the homogeneous powders were sintered at 800 \({}^{\circ}\)C for 4 h. The MLLC11.5 green body was coated with silver slurry and then sintered at 800 \({}^{\circ}\)C for 4 h. The GPS antenna was designed using HFSS software and prepared using the screen-printing method.  
The crystal structures and microstructures of all the samples were examined by X-ray diffraction (XRD, SmartLab 3 KW, Rigaku, Tokyo, Japan) and scanning electron microscopy (SEM, Regulus8100, Hitachi, Tokyo, Japan), respectively. The elemental analysis used energy-dispersive X-ray spectroscopy (EDS) included in the SEM. The volume densitieswere obtained according to the Archimedes method, and the theoretical densities (\(D_{t}\)) were calculated according to Equation (1)  
\[D_{t}=1/\sum_{i=1}^{n}\frac{\rho_{wi}}{D_{ti}},(n\geq 2) \tag{1}\]  
where \(\rho_{wi}\) and \(D_{ti}\) are the weight percentage and theoretical density of the \(i\)-th phase, respectively, estimated from the Rietveld refinement results. The relative density is the proportion of the volume density to the theoretical density. \(\varepsilon_{r}\) and \(Q\)\(\times\)\(f\) are evaluated at about 25 \({}^{\circ}\)C by a vector network analyzer (E8363B, Agilent, California, USA). The frequency range is about 8.06 GHz-8.55 GHz. \(\tau_{f}\) is calculated according to Equation (2) by measuring the resonant frequencies at 20 \({}^{\circ}\)C (\(f_{20}\)) and 80 \({}^{\circ}\)C (\(f_{80}\)). All the error bars were obtained from three groups of data.  
\[\tau_{f}=\frac{f_{80}-f_{20}}{f_{20}(80-20)}\times 10^{6}(\mathrm{ppm}/^{ \circ}\mathrm{C}) \tag{2}\]