## 3 Results and Discussion
A negative FTM database between the time difference of a pair of pulses and the frequency of microwave signal was set up as shown in Figure 3b. Through these operations, the FTM relationship was achieved and then the unknown microwave frequency measurement can be looked up through the FTM database by measuring the time different between two pulses. For an unknown signal, a pair of optical pulses was recorded in the oscilloscope and the time difference was calculated to inversely map the unknown microwave signal frequency using the FTM-based relationship shown in Figure 3b. The microwave frequency measurement range ranged from 3.5 to 12.5 GHz depending on the LFM signal performance. A microwave signal from the vector network analyzer acted as unknown signal with the output power of 5 dBm and the frequency of 5 GHz was injected into the mixer to demonstrate the single-frequency performance of the frequency measurement system. The recorded pulse in a single period on the oscilloscope is shown in Figure 4a. As shown in Figure 4a, the duration of the right pulse is shorter than the left pulse. It results from the positive slope of the frequency scanning less than the negative slope in one period as shown in Figure 3a. The real-time frequency distribution of the pulse is shown in Figure 4b using STFT. From Figure 4b, the time difference \(\Delta T\) between a pair of pulses is achieved clearly. By measuring the time difference, the frequency of the unknown microwave signal can be calculated using the FTM relationship shown in Figure 3b. Figure 5a-e shows the recorded unknown signal in the oscilloscope, where the frequency is set as 4-12 GHz with 2 GHz frequency space. The time difference between a pair of pulses was calculated by the STFT as shown in Figure 5f-j. The microwave frequency was measured by the time information inversely mapping to the frequency information in Figure 3b. The input microwave power for our proposed frequency measurement system should be larger than -15 dBm, which is mainly limited by noise floor of the frequency measurement system as well as the efficiency of the frequency mixing between the input microwave signal and the generated frequency scanning signal. The minimum input power of the unknown microwave signal can be further improved by incorporating an electrical amplifier. Besides, the maximum power is up to 20 dBm which is dependent on the 1-dB compression point of the mixer. Hence, the linear dynamic range can be further enhanced by reducing the noise and optimizing the component of the proposed frequency measurement system. Figure 4: Measured result for a single frequency microwave signal at 5 GHz. (**a**) Measured waveform on the oscilloscope. (**b**) Corresponding instantaneous frequency and time relationship using STFT. When the single-frequency microwave signal ranging from 4 to 12 GHz injects into the microwave frequency measurement system, the measurement frequencies and measurement errors at different frequencies are calculated as the result shown in Figure 6a,b. The frequency measurement range is from 4 to 12 GHz, which depends on the frequency scanning range generated by heterodyning of two lasers. Our proposed frequency measurement range can be extended to 15 GHz and even above by increasing the driving voltage. The maximum measured bandwidth is theoretically limited by the bandwidth of photodiode as well as the mixer. The measurement errors for the single-frequency microwave signal in Figure 6b are less than 90 MHz, which mainly comes from the resolution of the oscilloscope and the ambiguity of time difference after STFT. A higher accuracy can be obtained with the higher slope of frequency-to-time function in Figure 3a [11] as well as precise temperature control to fix one wavelength. Besides, in order to degrade the measured error, the FTM relationship in Figure 3b can be acquired by doing statistics over many measurements. The wavelength stability of the laser array is closely related to the operating temperature.