## 4 Experimental Results and Verification  
### Settling Time Measurement  
The configuration of the phase setting measurement is simplified to a functional block as depicted in Figure 8. A dual-channel arbitrary function generator (AFG; Tektronix AFG3022B) is employed as the phase shifter here. Figure 8 illustrates that the AFG utilizes the 10 MHz OCXO as a reference to generate two identical 10 MHz signals, which are then used to drive the two frequency synthesizers (FSs) for generating two cesium atomic resonant frequencies at 9.192 631 770 GHz. One of the FSs proposed in this paper is denoted as FSA, while the other is a commercial microwave signal synthesizer (FSB; SMB100A, Rohde and Schwarz).  
Figure 8: Simplified configuration for measuring the phase-settling ability.  
The 10 MHz signal generated by the AWG can be phase modulated to simulate the phase step function, allowing the FSs to experience a phase step by utilizing the phase shift produced by the AWG. Phase modulation is applied to FSA to generate the phase step, while maintaining a fixed phase in FSB. Since the two 9.192 GHz signals are simultaneously phase-locked to the same 10 MHz reference source, their phases are precisely synchronized. The two signals are compared in a mixer (Mini-circuits, ZMX-10G+), and the resulting phase error signal is recorded using a digital storage oscilloscope. In accordance with the mixer principle described in Section 3.1, the higher-frequency output from the mixer is eliminated due to the bandwidth limitation of the oscilloscope, while the lower-frequency component is retained. When the phase step signal is applied to FSA, the mixer output signal is recorded, and the settling time of the step response is measured to assess the phase-settling ability.  
When the phase shift signal is applied to the 10 MHz reference source of FSA, the phase response affecting the frequency of 9.192 631 770 GHz is multiplied by 919.2. Consequently, the minimum phase shift of the AWG is 0.01\({}^{\circ}\), resulting in an approximate 9.192\({}^{\circ}\) phase shift at 9.192 631 770 GHz. The remarkable property of the DDS in an FS, as per the DDS principle, is the agility of the phase shift. Therefore, in comparison with the phase shift time, the phase-settling time is insignificantly short and can be disregarded.  
When FSA and FSB are simultaneously phase-locked to the same 10 MHz reference clock, as depicted in Figure 8, the output signal from the mixer becomes a direct current (DC) signal. We adjust the phase of FSA until the DC signal reaches its maximum negative value. The phase step is applied within the range of \(\pi\), resulting in an approximate phase shift of 0.2\({}^{\circ}\) applied to the 10 MHz reference. The measured values are compared with the simulation results under the same conditions as shown in Figure 9. The dotted lines represent the simulation results, while the solid lines depict the results obtained using the configuration described in Figure 8.  
Figure 9 illustrates the phase step response for various values of the loop filter gain \(k_{p1}\). As the \(k_{p1}\) parameter increases from 2 to 4 in step of 2/3, the measured settling times are 1.5 ms, 1.064 ms, 825.5 us, and 644.5 us, respectively. The measured results align with the simulated results, providing evidence that the theoretical calculation for the transfer function of the synthesis and the experimental setup for measuring the phase step response are accurate and reasonable. The parameter adjustment and simulation results in Figure 9 are consistent with those in Figure 5a. Parameter \(k_{p1}\) represents the gain of the PLLI loop filter. In our design, an adjustable sliding rheostat is employed to modify the gain of the PLLI loop filter.  
Figure 9: _Cont._