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An approach to microwave frequency measurement with a high resolution and a broad bandwidth is proposed based on three parallel low-speed photonic analog-to-digital converters (ADCs) architecture. Through simultaneously bandpass sampling the input microwave signal with the three photonic ADCs, the input frequency can be calculated from the Fourier frequencies of the photonic ADCs using the proposed frequency recovery algorithm. Theoretical analysis and simulation results indicate that the proposed method is applicable for both single-tone and multi-tone microwave signals. By employing three ~1 GS/s@8 bits photonic ADCs, 0-100 GHz frequency measurement with an error of ± 0.5 MHz and a spur-free dynamic range of 94 dB-Hz2/3 over the full band has been numerically demonstrated. Additionally, a proof-of-concept experiment is carried out to demonstrate the effectiveness of the proposed method, where a frequency measurement range of 0-20 GHz with a measurement error of ± 8 kHz is realized by utilizing three photonic ADCs with sampling rates of 27.690 MS/s, 27.710 MS/s, and 27.730 MS/s. Larger frequency measurement range can be achieved by using an optical modulator with a larger bandwidth.
© 2017 Optical Society of America
1. Introduction
Identification of microwave frequency is of critical importance in modern military and civil radio-frequency (RF) systems such as wireless communications, radar systems, and electronic warfare [1–3]. For example, in an electronic reconnaissance receiver, it is required to quickly recognize the carrier frequency of an unknown microwave signal intercepted by the antennas before its further processing. Currently, this frequency measuring task is generally achieved by electronic techniques, which can realize a high resolution but work in a relative narrow bandwidth. Although the bandwidth can be further broadened through electronic channelization technology, it makes the whole system structurally complex, bulky and high power consumption, which is not conducive to the miniaturization development tendency of the RF systems.
In order to conquer the above-mentioned limitations confronted by electronic approaches, various photonic-assisted microwave frequency measuring methods have been proposed in the past decades due to the prominent advantages of photonic technology such as broad bandwidth, low loss and immune to electromagnetic interference (EMI) [4–13]. Based on the operation principle, the photonic-assisted microwave frequency measuring techniques can be mainly categorized into three types, i.e., frequency-power mapping [4–7], frequency-time mapping [8-9] and photonic channelization [10–13]. One of the most common techniques is the frequency-power mapping scheme, which is realized by monitoring and comparing the group-velocity-dispersion-induced power penalties between two radio-over-fiber (RoF) channels (either distinct physical channels or wavelength division multiplexing ones). Although this scheme can achieve a relatively high resolution (as low as multi-tens of MHz), it is not suitable for measuring multi-tone microwave signals. The frequency-time mapping scheme is realized through employing group-velocity-dispersion-induced relative time delay of either the carrier-suppressed double-sideband modulation signal or the single-sideband modulation one in a RoF link, which is available for measuring multi-tone microwave signals. Nevertheless, the resolution of this scheme is limited by the speed of the optical time gate, and is generally not high enough (larger than hundreds of MHz). The photonic channelization scheme is analogue to the electronic channelization one, which is achieved in the optical domain through an optical comb filter (such as a Fabry-Perot etalon) or a spatial dispersion element (such as an arrayed-waveguide grating, a diffraction grating). This scheme is also suitable for measuring multi-tone microwave signals and avoids the propagation delay in a long fiber, but its resolution is generally low due to the limited dispersive power of the optical dispersion component (usually larger than 1GHz). Therefore, it remains to be a hot topic to explore methods which can achieve a high-resolution microwave frequency measurement in a broad bandwidth and available for multi-tone signals.
In this paper, a new photonic approach to the microwave frequency measurement is proposed based on photonic analog-to-digital converters (ADCs). By utilizing three low-speed optical sampling ADCs with co-prime sampling rates to bandpass sample the microwave signal, the frequency of the signal can be calculated from the three Fourier frequencies using frequency recovery algorithm. In the proposed scheme, the limitation of the measured frequency range is the bandwidth of the Mach-Zehnder modulator (MZM), which is used to load the microwave signal into the photonic ADC. A large frequency measuring range is feasible since a MZM with a bandwidth larger than 100 GHz is available [14]. The simulation results indicate that the proposed scheme achieves both single-tone and multi-tone microwave signal frequency measurement in a large frequency range (0-100 GHz) with a high accuracy ( ± 0.5 MHz). In addition, a frequency measurement range of 0-20 GHz with a measurement error of ± 8 kHz is experimentally demonstrated by utilizing three photonic ADCs with sampling rates of 27.690 MS/s, 27.710 MS/s, and 27.730 MS/s.
2. Operation principle
The proposed microwave frequency measurement scheme based on three photonic ADCs is shown in Fig. 1, where each photonic ADC is composed of a mode-locked laser (MLL), a MZM, a photodetector (PD), an electronic amplifier (AM) and an electronic ADC, respectively. In this scheme, optical pulse trains from three low repetition rate MLLs are modulated by an unknown microwave signal via three MZMs biased at the quadrature points, respectively. Then the modulated optical pulse trains are converted to broadened electronic pulses by the narrow-band PDs. At the output of each PD, the signal is amplified by an AM to match the full scale of the electronic ADC, which guarantees taking full use of the quantization level of the electronic ADC during the digitization process. Finally, the digital data from the three optical sampling ADCs are sent to the digital signal processor (DSP) to calculate the frequency of the input microwave signal.
Fig. 1 Schematic diagram of the proposed photonic-assisted microwave frequency measurement. MLL: mode-locked laser, MZM: Mach-Zehnder modulator, PD: photodetector, AM: amplifier, ADC: analog to digital converter, DSP: digital signal processor, RF: radio frequency.
The three low-speed optical sampling ADCs, which can bandpass sample a microwave signal with a high frequency, are the guarantee for the frequency measurement in a broad bandwidth. The operation principle can be described as follows. For the first optical sampling ADC (i.e., the one composed of MLL1, MZM1, AM1, PD1 and ADC1), the frequency spectrum of MLL1 can be written as
where is the unit impulse function. and are the central frequency and repetition frequency of MLL1, respectively. is the Fourier expansion coefficient of the pulse train from MLL1. is an integer determined by the spectral width of MLL1. Since the output optical pulses of MLL1 are ultra-short ones whose spectral width is very huge (generally multi- or multi-tens of nm), is large enough to ensure that covers a frequency range of hundreds or thousands of GHz. This wide spectrum characteristic guarantees the bandpass sampling (i.e., down-converted sampling) of a high-frequency microwave signal in conjunction with the following MZM1 and PD1. In order to simplify the analysis, a single-tone RF signal is supposed to be the input signal of MZM1, whose frequency spectrum can be represented bywhere is the frequency of the input RF signal to be measured. Thanks to the wide RF bandwidth of an electro-optic MZM (e.g., the polymer MZM can achieve more than 100 GHz bandwidth) and the low jitter of ultra-short pulses from MLL (generally lower than 20 fs) [14–16], the input RF signal in a broad frequency range can be optically sampled with a high accuracy. Considering MZM1 is biased at the quadrature point and working in the condition of small signal modulation, the frequency spectrum of the modulated optical pulses can be calculated aswhere is the modulation index. It should be pointed out that the high-order harmonics have been neglected in Eq. (3) since they have no contribution to the frequency measurement. Furthermore, the high-order harmonics should be doubtlessly suppressed to avoid ambiguous frequency recovery, which will be further discussed in the simulation section. Then, the modulated optical pulse sequence is sent to PD1, where optical heterodyne mixing occurs. The generated frequency components in the RF frequency range can be expressed aswhere and are the amplitude coefficient of the corresponding frequency component. It can be clearly seen from Eq. (4) that the input signal with a high frequency of has been copied to numerous duplicates, and there must be a duplicate falling into the base band (within the operation bandwidth of PD1) by virtue of the large range of . Finally, the output electronic signal from PD1 is sent to electonic ADC1 synchronized with MLL1, where the signal after down-conversion sampling is quantified. Since the first optical sampling ADC works under a clock rate of , its sampling rate and instantaneous bandwidth are and , respectively. Therefore, there must be a frequency duplicate falling into the instantaneous bandwidth of the first optical sampling ADC, whose Fourier frequency can be calculated through fast Fourier transform (FFT) of the output data from electronic ADC1. Due to the period nature of bandpass sampling, the calculated Fourier frequency may represent the following input RF frequencywhere is an natural number to be determined, and represents the reminder of . Hence, all the potential input RF frequency can be calculated asThe operation principle of the other two optical sampling ADCs are exactly the same as that of the first one presented above. Therefore, all the potential input RF frequency can also be calculated from the other two optical sampling ADCs respectively as
where and are the repetition frequency of MLL2 and MLL3, respectively. and are the calculated Fourier frequency of the second and the third optical sampling ADCs, respectively. and are the potential input RF frequencies calculated from the second and the third optical sampling ADCs, respectively. and are natural numbers to be determined.In a proper frequency range, the three natural numbers , and can be determined and the final frequency measuring result can be calculated through the frequency recovery algorithm based on the following relation
It should be particularly pointed out that there should be an error redundancy when Eq. (9) is used to determine the natural numbers of , and , since the Fourier frequencies have limited resolutions due to both the finite data adopted for FFT calculation and the sampling rate drift of the photonic ADCs. The general method for achieving frequency recovery is using traversal algorithm through setting a proper error redundancy. However, the traversal algorithm is time-consuming, and the time consumption will increase exponentially with the number of the input frequencies. In order to improve the frequency recovery speed, a new algorithm for frequency recovery is proposed, whose flow chart is presented in Fig. 2. The procedure for frequency recovery can be described in detail as follows. Firstly, the data streams output from the three photonic ADCs are converted to the frequency domain through FFT, and the peaks above the threshold (i.e. the Fourier frequencies) are found. The threshold is set to be the higher one between the noise floor and the high-order harmonics, where the noise floor is determined by the quantization level, the sampling time jitter, and the frequency resolution of FFT (i.e., the sampling duration), and the high-order harmonics are dependent on the modulation index, the bias voltage drift, and the imperfect transmission curve of the MZMs. Secondly, after seeking out the Fourier frequencies, three matrixes which contain all the potential input frequencies are built according to Eqs. (6)-(8). Taking the first photonic ADC as an example, all the potential input frequencies recovered based on fF1 and f1 can be listed as a matrix of [fF1, f1 + fF1, 2f1 + fF1, … (k1-1)f1 + fF1, f1-fF1, 2f1-fF1, …k1f1-fF1], where k1 is a natural number equals to the frequency measurement range divided by f1. Hence, a vector N1, equal to [0:k1-1], can be created to simplify the expression of the matrix to be [N1f1 + fF1, (N1 + 1)f1-fF1] which is represented as matrix A1. Similarly, the matrixes of A2 and A3 that contain all the possible input frequencies for photonic ADC2 and photonic ADC3 can also be obtained, respectively. Finally, the input frequency can be recovered though seeking the intersection of the three matrixes A1, A2 and A3. It should be particularly pointed out that the intersection can only be obtained from A1, A2 and A3 in ideal conditions, i.e., without sampling rate drift and with infinitely-high frequency resolution. Nevertheless, in reality, the intersection of the three matrixes must be an empty set due to the existence of measurement errors introduced by both the sampling rate drift and the finite frequency resolution. Hence, before seeking the intersection of the three matrixes, the significant digit of the recovered input frequencies must be preset based on the measurement accuracy determined by both the sampling rate drift and the finite frequency resolution. To be more specific, the last bit of the significant digit of the matrixes should be the one that the Fourier frequencies are accurate to. For instance, if the Fourier frequencies can be accurate to MHz unit (i.e. the errors are lower than ± 0.5 MHz), the last bit of the significant digit is set to be MHz unit. Three more matrixes of B1, B2 and B3 are used to present the corresponding matrixes after setting the significant digit. The intersection of the matrixes B1, B2 and B3 must exist, whose locations can be found out in the matrixes B1, B2, and B3. Using the corresponding locations in the original matrixes A1, A2, and A3, the frequency measurement result can be calculated through averaging the three recovered frequencies in the original matrixes A1, A2, and A3.
In addition, the reason why three photonic ADCs should be used in the proposed scheme is presented based on the mathematical essence behind the bandpass sampling theory as follows. It can be clearly seen from Eq. (5) that the unambiguous frequency measurement range is the Nyquist bandwidth if a single photonic ADC is employed, which means that the only way to enlarge the frequency measurement range is increasing the sampling rate of the photonic ADC. Therefore, in order to reduce the pressure on the quantization step and the data handling capacity of DSP, multiple low-speed photonic ADCs with different sampling rates should be adopted to realize a large frequency measurement range, whose operation principle is analogous to, while not identical to, the process of seeking common multiple. For the scheme using multiple photonic ADCs with different sampling rates of f1, f2,…fq (q is the number of photonic ADCs), a linear equation group with q equations like Eq. (6) can be established for q + 1 variables of n1, n2, … nq and fs (fs is the replacement of fs1, fs2, fsq in the equation group). It should be noted that the input frequency can be recovered correctly (i.e., the equation group has unique solutions of n1, n2, … nq and fs), in a proper range, where the unambiguous frequency measurement range of fs depends on both the number and the sampling rate relationship of the photonic ADCs. For the scheme employing two photonic ADCs, the unique solution range of fs won’t grow greatly compared with the single photonic ADC scheme, due to the periodicity of the Fourier frequency in bandpass sampling theory. The frequency measurement range equals to the average of the two sampling rates since there must be a multiple solution at that position. For the scheme employing three photonic ADCs, the unique solution range of fs is approximately half of the smallest one in all the lowest common multiples (LCM) of any two sampling rates. Therefore, the frequency measurement range can be greatly enhanced through using three photonic ADCs, if the sampling rates of the three photonic ADCs are set to meet the condition that the LCM of any two sampling rates is larger than the expected frequency measurement range. Moreover, there is no doubt that the unique solution range of fs can be further enhanced through using more photonic ADCs. Nevertheless, the frequency measurement range in the scheme employing three photonic ADCs is able to reach the bandwidth limit of the commercially-available electro-optic modulators. Hence, there is no need to use more photonic ADCs to extend the frequency measurement range, if the system complexity and cost are taken into consideration. However, using more photonic ADCs can help reduce the probability of misjudgement in multi-tone signal measurement significantly. Therefore, whether more photonic ADCs are adopted or not depends on applications.
Furthermore, based on the principle described above, a large frequency measurement range can be guaranteed as long as none of the three sampling rates is another one’s integer multiple. Nevertheless, the frequency measurement range is determined by the LCMs of the sampling rates, it is doubtless that the measurement range of system with co-prime sampling rates will be larger than those with non-coprime sampling rates. Hence, the sampling rates employed in this paper are all choose to be co-prime.
3. Simulation and analysis
In this section, numerical simulation is implemented to evaluate the performance of the proposed microwave frequency measuring approach. The main parameters adopted in the simulation are listed as follows. The optical pulse width (full width at half maximum) of the three MLLs with an aperture jitter of 5 fs is set to be 1 ps, and the repetition frequencies are set as 1.005 GHz, 1.010 GHz and 1.015 GHz, respectively. Therefore, the LCMs of any two sampling rates are 203.01 GHz, 204.015 GHz and 205.03 GHz, which guarantee a frequency measurement range of 0-100 GHz. Since the polymer MZM (LX8900 from GIGOPTIX) have realized the bandwidth up to 110 GHz with a typical half-wave voltage of 6.5 V [14], the 3 dB bandwidth of the three MZMs utilized in the simulation is set to be 100 GHz. The three MZMs with the half-wave voltages of 6.5 V are biased at their quadrature points, and the modulation indexes are set to be 0.8 (i.e. modulation depth of 72%). For all of the PDs, the bandwidth is assumed to be 3 GHz. The electronic ADCs synchronizing with the corresponding MLLs have a quantization level of 8 bits, and 1024 sampling points in each ADC are used for FFT calculation.
Firstly, a single-tone microwave signal with a frequency of 30.0000 GHz is employed to be the input of the proposed microwave frequency measuring scheme. Figures 3(a)-(c) present the FFT spectra of the three optical sampling ADCs. The major peaks in Figs. 3(a)-(c) correspond to the calculated Fourier frequencies of the input microwave signal from the three optical sampling ADCs, which are located at 0.1501 GHz, 0.3003 GHz, and 0.4504 GHz, respectively. Based on the frequency recovery algorithm, the calculating values of , and are all equal to 30, which indicate the recovered frequencies from three optical sampling ADCs are 29.9999 GHz, 29.9997 GHz and 29.9996 GHz, respectively. Hence, the measured frequency can be obtained through averaging the three recovered frequencies, whose value is 29.9997 GHz.
Fig. 3 Simulation results for a single-tone input microwave signal. (a) FFT spectrum of photonic ADC1. (b) FFT spectrum of photonic ADC2. (c) FFT spectrum of photonic ADC3.
In the proposed scheme, the frequency measurement error is attributed to the limited spectrum resolution of FFT and the repetition frequency drift of the MLLs, where the spectrum resolution in inversely proportional to the sampling duration and the repetition frequency drift will be transferred to the frequency measurement result through magnification by tens or hundreds of times (depending on the nature number of , and ). In the simulation, the repetition frequency drift of the MLLs is not taken into consideration. The frequency measurement error is solely determined by the sampling sequence length of 1024 and the sampling rates of ~1 GS/s, which is ~1 MHz. Hence, the measurement accuracy in the simulation should be in the range of ± 0.5 MHz. Figure 4 exhibits the frequency measurement error for an input single-tone microwave signal with its frequency ranging from 0 to 100 GHz and a step of 2.5 GHz. It can be seen from Fig. 4 that the frequency measurement error is well confined in the range of ± 0.5 MHz. The frequency measurement resolution can be further improved through increasing the number of sampling points used for FFT calculation. However, the price paid for this resolution improvement is reducing the measuring speed due to the fact that the frequency resolution in the FFT calculation is inverse proportional to the recording time. Thus, a balance should be considered between measuring speed and measuring resolution in particular applications.
Fig. 4 Frequency measurement error of simulation for a single-tone input microwave signal with different frequencies.
Additionally, it can be seen from Figs. 3(a)-(c) that there are three minor peaks located at 0.4504 GHz, 0.1101 GHz and 0.3353 GHz, respectively, which correspond to a recovered frequency of 89.9998 GHz. This spurious frequency component originates from the nonlinear transfer function of the MZMs, and attributes to the third-order harmonic distortion of the MZMs biased at quadrature points. In a frequency measurement oriented to unknown input microwave signals, these spurious peaks are detrimental, since they may lead to a misjudgment. Fortunately, any combination of minor peaks and major ones will not lead to any misjudgment once the error tolerance is properly chosen in the frequency recovery calculation. In order to avoid misjudgment, spur-free dynamic range (SFDR) should be considered when the system is designed. SFDR is defined as the range of input microwave power that the fundamental frequency component is above while the distortions are under the noise floor in a 1 Hz bandwidth [17]. A two-tone input microwave signal with frequencies of 0.2 GHz and 100 GHz with equal amplitude is utilized to evaluate the SFDR over full-band of the system used in the simulation. Since the DC bias of the MZMs in the proposed scheme is stabilized at the quadrature point, the third-order intermodulation terms (IMD3) are the largest spurs, which impose a restriction on SFDR. As shown in Fig. 5, the output power of is lower than the output power of because of the frequency response characteristic of the MZMs, and the IMD3 of and are the largest spurs with equal power. The SFDR simulation result in Fig. 4 indicates a SFDR of 54 dB in 1 MHz bandwidth (i.e. 94 dB-Hz2/3). In this figure, the fundamental frequency component can be measured and the third-order harmonic is suppressed below the noise floor when the input microwave power is in the range of −48.6 dBm to 5.4 dBm, corresponding to a modulation depth of 0.1% to 28.1%. Therefore, misjudgement-free frequency measurement can be realized in the above-mentioned SFDR. In the simulation, the noise floor is determined by the quantization error (quantization-induced noise floor of approximately −70 dBm and thermal noise floor of −100 dBm), which can be reduced though using electronic ADCs with a higher quantization bits. Hence, the SFDR can be further improved. Besides, dual-output MZMs can also be employed in combination with arcsin algorithm to enhance the SFDR [18-19]. Additionally, it should be specially pointed out that the values of the modulation index in various systems need to be determined by the noise floor which is mainly dependent on the quantization level, the sampling jitter, and the frequency resolution of FFT.
Then, a multi-tone input microwave signal with frequencies of 2.0000 GHz, 50.0000 GHz, and 100.0000 GHz is measured, and the modulation depths are set to be 28% to suppress the third-order harmonics below the noise floor. Figures 6(a)-(c) show the FFT spectra of the three optical sampling ADCs. The correct combinations of the Fourier frequencies and the frequency measuring results obtained through the frequency recovery algorithm are shown in Table 1.
Fig. 6 Simulation results for a three-tone input microwave signal. (a) FFT spectrum of photonic ADC1. (b) FFT spectrum of photonic ADC2. (c) FFT spectrum of photonic ADC3.
Table 1. Frequency measuring results for a multi-tone input microwave signal
In addition, it should be noted that misjudgement may occur in some special cases of multi-tone signal frequency measurement. In these cases, the actual input frequency components can definitely be recovered, but spurious frequency components may also be given out. An example is presented in Table 2, where a multi-tone microwave signal with frequencies of 30.10 GHz and 70.60 GHz is adopted as the input. In the output, except for the input frequencies of 30.10 GHz and 70.60 GHz, a spurious frequency of 50.30 GHz is also provided. Fortunately, the occurrence probability of these misjudgment cases is extremely low, and it can be solved through using more photonic ADCs.
Table 2. Theoretical frequency measuring result of a special multi-tone microwave signal
4. Experimental results and discussion
A proof-of-concept experiment is carried out to demonstrate the proposed approach. The experimental setup is slightly different from the architecture shown in Fig. 1, where only one photonic ADC is implemented to obtain the spectrum of the signal. The MLL used for sampling is a home-made Erbium-doped passive mode-locked fiber laser whose repetition frequency can be tuned through a fiber-delay line in the cavity. In the experiment, the repetition frequency of the MLL is set to be 27.690 MHz, 27.710 MHz, and 27.730 MHz in turn, which corresponds to a measurement range up to 38 GHz. Additionally, the spectrum of the optically-sampled signal is measured by an electronic spectrum analyzer instead of an electronic ADC with FFT processing.
Due to the bandwidth limitation of the MZM used in the experiment, the maximum measurable frequency is constrained to be around 20 GHz. Figure 7 exhibits the measured spectra of a single-tone RF signal with a frequency of 20 GHz under the three sampling rates, where a threshold of −70 dBm is set to distinguish the effective and spurious peaks. In the experiment, the input RF signal power is 3 dBm and the half-wave voltage of the MZM is 8 V (@20 GHz), corresponding to a modulation index of 0.18. Additionally, the bandwidth and the resolution of the spectrum analyzer are set to be Nyquist bandwidth and 1 kHz, respectively. Based on the proposed algorithm, the peaks below the threshold are justified as noise and high-order harmonics while the ones above the threshold are recognized as effective signals. It can be found that the effective peaks in Figs. 7(a)-(c) are located at 7.812 MHz, 6.622 MHz, and 6.673 MHz, respectively, which are the Fourier frequencies of the input signal under the three sampling rates. Based on the proposed frequency recovery algorithm, the calculated values of , and are 722, 722, and 721, corresponding to recovered frequencies of 19.999992 GHz, 19.999998 GHz, and 20.000003 GHz, respectively. Therefore, the frequency is determined to be 19.999998 GHz by the average value of the three recovered frequencies, which demonstrates the effectiveness of the proposed scheme. It should be noted that the maximum measurement error of the three photonic ADCs is −8 kHz, which is bigger than the theoretical error of ± 0.5 kHz introduced by the finite spectrum resolution in the experiment. This measurement error can be attributed to the drift of the MLL’s repetition frequency, which is measured to be in the range of approximately ± 10 Hz in our experiment system. Hence, the measurement error induced by the sampling rate drift is approximately in the range of ± 7.2 kHz since the input frequency of 20 GHz is about 722 times of the sampling rates. Taking the measurement error induced by both the finite spectrum resolution and the sampling rate drift into consideration, the measurement error in the experiment should be in the range of approximately ± 8 kHz. Figure 8 presents the frequency measurement results and measurement errors for input single-tone microwave signal with different frequencies. It can be seen in Fig. 8 that the frequency measurement results fit well with the actual values, and the measurement errors are well confined in the range of ± 8 kHz determined by both the finite spectrum resolution and the sampling rate drift.
Fig. 7 Experimental measured spectra of a single-tone RF signal with a frequency of 20 GHz under sampling rate of (a) 27.690 MS/s, (b) 27.710 MS/s and (c) 27.730 MS/s.
Fig. 8 Experimental results of measured frequency and measurement error for a single-tone input microwave signal with different frequencies
For a further discussion, simulation is carried out based on the identical parameter settings to the experiment system. Firstly, MLLs without repetition frequency drift are employed to measure an input single-tone microwave signal with a frequency of 20 GHz, where the spectra under the three sampling rates are shown in Figs. 9(a)-(c), respectively. It can be found in Fig. 9 that the three Fourier frequencies are located at 7.8202 MHz, 6.6199 MHz and 6.6704 MHz, respectively, corresponding to a recovered frequency of 20.0000002 GHz. The measurement error of 0.2 kHz matches well with the 1 kHz frequency resolution since no sampling rate drift exists here. Additionally, it should be pointed out that the second order harmonics presented in the experimental spectra do not exist in the simulation ones. The second order harmonics in Fig. 8 can be attributed to the bias drift and the imperfect transmission curve of the MZM in reality, where they should doubtlessly vanish if a MZM with a perfect cosine-shape transmission curve is exactly biased at the quadrature point like the case in the simulation.
Fig. 9 Simulated spectra of a single-tone RF signal with a frequency of 20 GHz under a sampling rate of (a) 27.690 MS/s, (b) 27.710 MS/s and (c) 27.730 MS/s.
Then, the influence of the sampling rate drift on the measurement accuracy is discussed. For a passive mode-locked fiber laser, the sampling rate drift induced by the environment condition such as temperature variation and mechanical vibration is generally slow. In the proposed scheme, the sampling duration is set to be very short in order to achieve a fast measurement. Hence, the sampling rate in a single measurement can be regarded as a constant, i.e. the sampling rate drift is fixed in a single measurement. In addition, the sampling rate drift in different channels are random and independent. Hence, there are various combinations of sampling rate drift among the three channels. Table 3 presents the simulation results for an input microwave signal with a frequency of 20 GHz under several special sampling rate drift combinations, where the input frequencies are all recovered using sampling rates of 27.690 MHz, 27.710 MHz and 27.730 MHz. For a certain input frequency, the value of n1, n2 and n3 are fixed, which means that the influence of the sampling rate drift on the measurement error is fixed. In the simulation, the values of n1, n2 and n3 are approximately equal. Hence, the sampling rate drift induced error should be approximately -n1 times (i.e., −722-fold) of the average of the three sampling rate drift. Based on the above analysis, it can be found in Table 3 that the measurement error is well confined to be in the predicted range.
Table 3. Simulation results of a single-tone RF signal under different sampling rates drifts
Finally, it should be pointed out that there is a restrictive relation among the sampling rate, the sampling duration, and the frequency measurement range. For relatively low sampling rates (e.g., ~27MS/s in the experiment), in order to reach a large enough frequency measurement range, the difference between each sampling rate should be small enough to obtain large LCMs. In such a case, a longer sampling duration is required to enhance the resolution of the Fourier frequencies. Meanwhile, a rigorous stability of the sampling rates is essential to guarantee the measurement accuracy. To the contrary, for relatively high sampling rates (e.g., ~1GS/s in the simulation section), the requirement of the sampling rate difference, the recording time and the repetition frequency stability can be greatly relaxed. Therefore, the sampling rates of the photonic ADCs should be selected prudently based on the requirements of both the frequency measurement range and measuring time.
5. Conclusion
In conclusion, a microwave photonic approach to the microwave frequency measurement is demonstrated based on three low-speed optical sampling ADCs. Theoretical and simulation results show that, the unambiguous frequency measurement range is approximately half of the smallest one in all the LCMs of any two sampling rates. Therefore, through properly selecting the sampling rates of the three photonic ADCs, a broad frequency measurement range can be definitely realized. Simulation have been carried out to demonstrate frequency measurement of both single-tone and multi-tone microwave signals in a frequency range of 0-100 GHz, where three photonic ADCs with sampling rates of 1.005 GHz, 1.010 GHz and 1.015 GHz are employed. A proof-of-concept experiment is also carried out to demonstrate a frequency measurement with an accuracy of ± 8 kHz over a 0-20 GHz frequency range, where three photonic ADCs with sampling rates of 27.690 MS/s, 27.710 MS/s and 27.730 MS/s are employed. Larger frequency measurement range can be achieved by using an MZM with a larger bandwidth since the frequency measurement range is limited by the bandwidth of the MZM in the proposed scheme.
Funding
National Natural Science Foundation of China (NSFC) (Nos. 61575037, 614210021, 6130703, 61205109), the Innovation Funds of Collaboration Innovation Center of Electronic Materials and Devices (No. ICEM2015-2001), the National advanced Research Foundation of China (No. 9140A21030215HT05326), the National Key Laboratory Foundation of China (No. 9140C530202150C53012).
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