1. Introduction

Broadband microwave frequency measurement (FM) is vital in the electronic warfare applications, such as anti-stealth defense and radar warning receivers [1,2]. Photonic-assisted FM offers an approach with wide measurement range, which can be mainly divided into two types, frequency-to-power mapping (FTPM) [3–6] and frequency-to-time mapping (FTTM) [7–14]. FTPM is commonly based on amplitude comparison function (ACF) [3–6]. Although the FM range of FTPM can be wide, most of these methods can only characterize a single frequency component [3–5] and the measurement errors are generally about hundreds of megahertz. One of the effective solutions with multi-tone FM is using distributed FTPM to construct channelized ACF in different frequency band [6]. However, the multi-tone resolution is limited by the channel interval. FTTM is able to strike a balance between the wide FM range and multi-tone FM ability [7–14]. In Ref. [7,8], the signal under test (SUT) is frequency-shifted close to the reference carrier via a recirculating frequency shift (RFS) loop, to achieve wide FM range and multi-tone measurement ability. Nevertheless, the FM accuracy and multi-tone resolution are both limited by the frequency shift value of the RFS loop, usually hundreds of megahertz. In Ref. [9], the SUT is modulated on a linearly chirped optical pulse train, which is then compressed by the dispersion element and converted to electrical pulses. Short interception period of 27 ns and wide FM range of 0.6-42 GHz are achieved, but an electrical backend with a large bandwidth is needed to reach a low muti-tone resolution. Recently, frequency modulated continuous wave (FMCW) combined with electric-domain intermediate frequency (IF) envelop detector is utilized for FTTM-based FM, which is able to achieve low errors and muti-tone resolution at the same time [10–14]. However, the FM range of these methods is limited by the bandwidth of the FMCW. Although it can be extended through switching the order of FMCW sidebands [11,14], the low power of the higher-order sidebands makes it hard to realize further extension. Also, the electrical devices used for carrier-suppressed single sideband (CS-SSB) modulation restricts the implement of wide FM range, such as the 90-degree hybrid and the optical bandpass filter (OBPF) [10–14].

In this letter, we present an ultra-broadband multi-tone FM approach based on FMCW. The FM range is largely extended by extending the bandwidth of FMCW through a RFS loop, from 0.001 GHz-16 GHz to 0.001 GHz-437.5 GHz. It means the FM range will not be restricted by the electronic bottlenecks. The bandwidth-extended FMCW coherently beats with the CW light modulated by the SUT at the balanced photodetector (BPD). Then by selecting the beating component through a low-pass filter (LPF), pulses arise at the time when the instantaneous frequency of FMCW is equal to that of SUT, realizing FTTM. Besides, no reference signals are used, owing to the up- and down-chirps of FMCW. Experiments are carried out. Using an available microwave generator (MG) with a maximum output frequency of 43.5 GHz, a FM within 0.1 GHz-43.5 GHz is demonstrated with errors of less than $\pm$10 MHz for all frequencies. The multi-tone resolution of 60 MHz is achieved at the FMCW chirp rate of 3.1998 $\rm{GHz}/\mathrm{\mu}\rm{s}$, which agrees with the theoretical one. Discussion about the relationship between the multi-tone resolution and FMCW chirp rate is made, which indicates that the multi-tone resolution can be improved to 1 MHz by lowering the FMCW chirp rate.

2. Principle

Figure 1 shows the setup of the proposed multi-tone FM system and the time-frequency map of the signals at the corresponding points. A CW light is divided into two paths by an optical coupler (OC-1), one of the outputs is modulated by the up- and down-chirped FMCW through the Mach-Zehnder modulator (MZM-1), while another is modulated by the SUT through MZM-2. The two MZMs are biased at the null point to maximize the power of the 1st sidebands. Under the condition of small signal, the optical FMCW output from MZM-1 can be expressed as

 

Fig. 1. Setup of the proposed multi-tone FM system and the time-frequency map of the signals at the corresponding point. OC: optical coupler. AWG: arbitrary waveform generator. MZM: Mach-Zehnder modulator. OBPF: optical band-pass filter. EDFA: erbium-doped fiber amplifier. DP-MZM: dual-parallel MZM. PC: polarization controller. BPD: balanced photodetector. LPF: low-pass filter. OSC: oscilloscope.

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Then, ${E_{{\rm {RFS}}}}\left ( t \right )$ and ${E_{{\rm {SUT}}}}\left ( t \right )$ coherently beats at a pair of balanced photodetectors (BPD), which is followed by a narrow-bandwidth LPF. In the output of the LPF, pulses arise at the time when the instantaneous frequency of ${E_{{\rm {RFS}}}}\left ( t \right )$ is equal to that of ${E_{{\rm {SUT}}}}\left ( t \right )$. The lower sideband of ${E_{{\rm {SUT}}}}\left ( t \right )$ can be neglected, since it will only enhance the amplitude of pulses located in the time window $n = 0$, but not change the pulses arising time. Then, the pulses output by the LPF can be given by

It should be noted that in order to avoid interference from irrelative pulses, the BPD is used to reduce the amplitude of the pulses arising at the start and end of the time window $n=0$ and at the middle of the time window $n=1$, that are derived from the self-beating of the output of MZM-1.

3. Experiment

Experiments are carried out to verify the principle, repeatability and analyze the frequency resolution. The laser (Pure Photonics, PPCL-300) provides a 13.5-dBm CW light with a wavelength of 1550 nm, which is divided into two paths by OC-1. One of the outputs is launched into the MZM-1 (Fujitsu, FTM7939EK), and modulated by the FMCW from the port-1 of the arbitrary waveform generator (AWG, Keysight, M8195A). The FMCW has a frequency range of 0.001-16 GHz, pulse width of 10 $\mathrm{\mu}$s and period of 500 $\mathrm{\mu}$s. In the RFS loop, a 15.999-GHz single-tone signal comes from the port-2 of the AWG, and then passes through a 90-degree hybrid to generate two quadrature signals, which are ultimately launched into the DP-MZM (Fujitsu, FTM7961). The two sub-MZMs in the DP-MZM are biased at the null point, while the bias voltage of the main MZM is adjusted to introduce a 90-degree phase difference to the two sub-MZMs, realizing the CS-SSB modulation. The input and output optical spectrums of the DP-MZM are shown as the blue and red curves in Fig. 2(a). In the input of DP-MZM, the CW light is not completely suppressed and even has a power 8 dB higher than the FMCW sidebands. It is because the power of the FMCW (13 dBm) will spread across the bandwidth of 15.999 GHz, corresponding to a low modulation depth for each frequency. In the output of DP-MZM, the power of +1st sideband is 31 dB higher than that of the −1st sideband owing to a power amplifier (PA) placed before the 90-degree hybrid. A 2-km delay fiber is used to introduce a time delay of about 10 $\mathrm{\mu}$s. Adding the time delay introduced by the pigtails of other devices in the RFS loop, the total time delay of the RFS loop will be longer than 10 $\mathrm{\mu}$s. Therefore, the $n + 1$-times frequency-shifted FMCW can be added to the end of the $n$-times frequency-shifted one without interference. The center wavelength and span of the OBPF are set as 1548.65 nm and 3.5 nm. Figure 2(b) shows the input (blue curve) and output (red curve) optical spectrum of the RFS loop. As can be seen, the bandwidth of optical FMCW is 27-times extended to 3.5 nm (437.5 GHz), which indicates that the FM range is able to reach 0.001-437.5 GHz. It should be noted that the unevenness of the output spectrum is caused by the gain competition between the CW light and FMCW sidebands in EDFA. By utilizing an amplifier to the RF input of MZM-1, the CW light will be further suppressed and the gain competition will be lessened.

 

Fig. 2. Optical spectrums of the input (blue curves) and output (red curves) of (a) DP-MZM and (b) RFS loop.

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After multiple frequency shift in the RFS loop, the generated broadband FMCW has a decreasing signal-to-noise ratio (SNR) with increasing frequency. The optical SNR of the $m$-th frequency shift cycle (${\rm {OSN}}{{\rm {R}}_m}$) can be calculated as [15],

Another output of OC-1 is launched into MZM-2 to modulate SUT, which is provided by a MG (Rohde & Schwarz, SMF100A) with a maximum output frequency of 43.5 GHz. The outputs of MZM-2 and RFS loop coherently beats at the BPD with a bandwidth of 200 MHz, followed by a self-made 18.5-MHz LPF. The output waveforms of the LPF when the SUT frequency is 5 GHz and 12 GHz, 19 GHz and 26 GHz, 33 GHz and 40 GHz, are measured by an OSC (Tektronix, DSA 70804) with a sampling rate of 6.25 GSa/s, as shown in Fig. 3(a), (b) and (c), respectively. As can be seen, when frequency of SUT is within 0.001-16 GHz , two pairs of pulses will arise in the time window $n$ and $n + 1$, respectively. And the time interval of the first pair is used to calculate the frequency of SUT. In addition, the amplitude of the pulses located in the time window $n = 0$ is slightly higher than the pulses located in other time window, which is contributed by the beating between the lower sidebands of ${E_{{\rm {FMCW}}}}\left ( t \right )$ and ${E_{{\rm {SUT}}}}\left ( t \right )$. And there are three pulses with low amplitude, arising at start and end of the time window $n = 0$ and at the middle of the time window $n = 1$. They are from the beating of 1st sidebands of FMCW and the CW light that is not completely suppressed at MZM-2, which can be considered as the starting flags of FM. Besides, the direct current (DC) interference occurs due to the jitters of the fiber time delay in the zero-intermediate-frequency (zero-IF) architecture. Fortunately, the proposed scheme has a high tolerance to the DC interference. Since the DC interference only jitters the amplitudes of the pulses rather than their positions, the occurrence of the pulses peaks can be obtained accurately under DC interference.

 

Fig. 3. Output waveforms of LPF when SUT frequency is (a) 5 GHz and 12 GHz, (b) 19 GHz and 26 GHz, (c) 33 GHz and 40 GHz, (d) 5 GHz in dB unit with a power of 5 dBm, (e) 16 GHz and (f) 32 GHz.

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Figure 3(d) shows the output waveform of LPF in dB unit at a SUT frequency of 5 GHz and a power of 5 dBm, to illustrate the sensitivity and dynamic range (DR) of the proposed system. The noise floor and DR can be obtained as about −20 dBm and 30 dB, respectively, which means that the system can capture the SUT with a sensitivity as low as −25 dBm.

In particular, when the SUT frequency is close to $n{\times }$15.999+0.001 GHz, the output waveform of LPF will be a sine waveform, as shown in Fig. 3(d) and (e). Since the CW light is not completely suppressed, the residual part will experience frequency shift in the RFS loop. Thus, the frequency-shifted CW light will coherently beat with the upper sideband of ${E_{{\rm {SUT}}}}\left ( t \right )$, consequently producing the sine waveform with a frequency of $\left | {{f_x} - n \times 15.999\;{\rm {GHz}}} \right |$. Its stop time is the end of the time window $n$, and can be used to estimate SUT frequency as $n{\times }$15.999+0.001 GHz. The absolute value of the estimation errors for these frequencies will be lower than the bandwidth of LPF, i.e. 18.5 MHz.

Then, the frequency of MG sweeps from 0.1 GHz to 43.5 GHz. The measured time intervals of the first two pulses at each frequency are shown in Fig. 4(a). The wraps with a period of 15.999 GHz can be seen because each time window has a FM range of 15.999 GHz. The measured time interval can be substituted into Eq. (5) to calculate SUT frequency, in which $n$ can be determined by the pulses arising time window. Three repeated FMs are carried out and the results are shown in Fig. 4(b), (c) and (d). Warps are eliminated by the determined $n$, and the FM errors are within $\pm$10 MHz except frequencies of $n \times$15.999 GHz. The maximum mean and standard deviation of these FM errors are −0.3 MHz and 3.17 MHz, respectively. Therefore, the proposed scheme has a wide FM range and high accuracy.

 

Fig. 4. (a) Measured time interval of the first two pulses at the frequency 0.1-43.5 GHz. (b), (c), (d) are the results of three repeated FMs.

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The frequency resolution of the proposed scheme is analyzed, which is determined by the larger one of the two factors, bandwidth of LPF and $k \cdot {\tau _p}$, where ${\tau _p}$ is the full width at half maximum (FWHM) of the pulse. ${\tau _p}$ is measured at the SUT frequency of 10 GHz, as shown in Fig. 5(a). The two pulses located in the time window $n = 0$ are shown in Fig. 5(b) and (c), respectively, and have a same FWHM of 0.0198 $\mathrm{\mu}$s. Thus, the frequency resolution of the proposed scheme is $k \cdot {\tau _p}$=63.356 MHz, rather than LPF bandwidth of 18.5 MHz. Then, the frequencies of two pairs of dual-tone signals are measured. The first pair is 10 GHz and 10.1 GHz, as shown in Fig. 5(d)-(f). Four pulses can be clearly distinguished in the time window $n = 0$, the first and last of which are corresponding to the 10-GHz signal while another two are corresponding to the 10.1-GHz signal. And the two frequencies are measured as 10.0013 GHz and 10.1039 GHz. When the frequency interval of the two tones is reduced to 60 MHz, as shown in Fig. 5(g)-(i). The cross talks between the first two pulses are intensified, as are the cross talks between the last two pulses. However, the four peaks can still be found and the two frequencies are measured as 10.0033 GHz and 10.0578 GHz. Therefore, the frequency resolution of the proposed scheme can reach 60 MHz, which is slightly lower than the theoretical one. It is because the cross talk between the pulses has not reached the worst when the frequency interval of the two tones is just the theoretical frequency resolution.

 

Fig. 5. Analysis of frequency resolution. (a)-(b) Measurement of pulse FWHM. (d)-(f) Dual-tone FM of 10 GHz and 10.1 GHz. (g)-(i) Dual-tone FM of 10 GHz and 10.06 GHz.

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It should be noted that when the frequency interval of the dual-tone signal increases to a value close to $B$, the output waveform of LPF can be different. Assuming that the dual-tone signal has frequencies of $f_x$ and $f_x+B$, two cases need to be discussed. The first case is that $f_x$ is not close to an integer multiple of $B$. Eight pulses, distributing in three adjacent time windows, can be observed on the OSC. The last two pulses for the signal of $f_x$ and the first t wo pulses for the signal of $f_x+B$ arise in the middle time window. The two frequencies of the dual-tone SUT can be estimated from the pulses located in the first and last time windows, respectively. In another case, where $f_x$ is close to an integer multiple of $B$, superposition of sine waveforms can be observed on the OSC. For example, when $f_x=$16 GHz, superposition of the two sine waveforms shown in Fig. 3(d) and (e) can be observed. In this case, the frequencies of the dual-tone SUT can be distinguished owing to the special waveforms. Experiments are not demonstrated, since we do not have devices that can generate a dual-tone signal with a frequency interval of 16 GHz.

4. Discussion on frequency resolution

As mentioned above, the multi-tone resolution of the proposed FM scheme is determined by

Actually, the output of LPF is a FMCW. The output waveform of LPF acts as a pulse, when the bandwidth of LPF is narrower than the stop frequency of FMCW, i.e.

Thus, the frequency resolution can be given by

Accordingly, the minimum multi-tone resolution lies on the chirp rate, which can be lower by reducing the chirp rate of FMCW. As the chirp rate reduces by a factor of $m$, the minimum resolution is lowered by a factor of $\sqrt m$, as shown in Fig. 6 (at a chirp rate of $k = 3.1998 \;{\rm {GHz/\mu s}}$). It can be seen that when $m = 1$, the minimum resolution is about 67 MHz, which is approximately consistent with the experiment results. The small deviation is caused by the different definition of pulse width. In the discussion, the pulse width is defined as the 10% to 90% rise time plus 90% to 10% fall time, while in the experiment, the pulse width is defined as the FWHM. By lowering the FMCW chirp rate, the multi-tone resolution can be improved to 1 MHz.

 

Fig. 6. Reduction of the minimum frequency resolution as the chirp rate lowers by a factor of $m$ at a chirp rate of $k = 3.1998 \;{\rm {GHz/\mu s}}$.

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5. Conclusion

In conclusion, an ultra-broadband multi-tone FM approach based on FMCW is proposed. FTTM is constructed through coherently beating FMCW and SUT at the BPD. The FM range is largely extended by the RFS loop from 0.001 GHz-16 GHz to 0.001 GHz-437.5 GHz. The up- and down-chirped FMCW is used to achieve self-reference and keep high FM accuracy. Experiments are carried out to verify the principle, repeatability and analyze the multi-tone frequency resolution. Using available MG, a FM range of 0.1-43.5 GHz is demonstrated. Three repeated FMs indicate that errors are kept within 10 MHz with a maximum mean and standard deviation of −0.3 MHz and 3.17 MHz, respectively. The multi-tone resolution is analyzed to be about 60 MHz at the FMCW chirp rate of 3.1998 $\rm{GHz}/\mathrm{\mu}\rm{s}$, which is consistent with the theoretical one. The theoretical derivation indicates that lowering the chirp rate of FMCW can improve the multi-tone resolution to 1 MHz.