## Abstract

We present a photonic compressive receiver, where the frequency information of the captured signal is directly mapped to the time intervals between compressed pulses for multiple microwave frequency measurement. The theoretical measurement error, multiple-frequency resolution and effective measurement range are derived. The effects of dispersion deviation and the electrical bandwidth are also discussed. The theoretical results are verified by the measured pulse waveforms and frequency-time mapping relationship. A photonic compressive receiver with an effective measurement range of 42 GHz, a multiple-frequency resolution of 1.2 GHz, a measurement accuracy of 88 MHz and a signal interception period of 27 ns is experimentally obtained.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

In the electronic warfare (EW) applications, such as radar warning and electronic intelligence systems, it is important to realize the fast and accurate measurement of the frequencies contained in intercepted microwave signals [1]. Typical requirements for frequency measurement would include a wide operation bandwidth, high resolution, and near real-time response [2]. Instantaneous frequency measurement (IFM) receiver is usually applied to measure the frequency information before separately performing signal analysis and countermeasure [3]. However, modern electrical IFM systems are usually limited in the measurement range to around 18 GHz due to the limited bandwidth of the electronic components [4]. Moreover, they cannot measure multiple frequencies simultaneously [5]. Photonic measurement systems can overcome these limitations and offer advantages including large instantaneous bandwidth, electromagnetic immunity, potential for multi-frequency measurement, light weight and so on [6,7].

Generally, the photonic microwave frequency measurement system is realized by analyzing other easily measured parameters, such as optical power or time delay, to estimate the frequency of an unknown microwave signal [8]. The frequency-to-power mapping systems [2,9–12] which map the RF frequency to the output power, are the common schemes and usually offer advantages of low cost and simple structure. However, most of the frequency-to-power mapping systems [2,9–11] can only characterize a single frequency component, which considerably reduces their practicability in the spectrally cluttered environment. Systems based on frequency-to-time mapping to enable multiple-frequency measurement have been reported [4,8,13–16]. Most of these systems realize the frequency-to-time mapping through a frequency shifting recirculating delay line [4], a scanning optical filter [8], a tunable laser [13] or a frequency scanning signal [15] and observe different frequency points at different times by scanning. However, the signal interception periods of these approaches are limited by the number of circulation, the scanning speed of the optical filter or the sweeping rate of the laser, which cannot meet the requirement of intercepting signals with a pulse width less than 100 ns in EW applications [17]. In [14], an alternative frequency-to-time mapping system through dispersion effect is proposed, which can reach a signal interception period less than 100 ns. However, its resolution is unsatisfactory and it is unable to measure the signal below 15 GHz. The all-optical Fourier transform has been found during shaping the optical pulses by chromatic dispersion [18], and a structure based on all-optical Fourier transform has been presented to act as an ultrafast electrical spectrum analyzer [19]. However, the adopted optical temporal magnification, used to relax the bandwidth requirements of photodetector (PD) and oscilloscope, would introduce the limitation to the measurement range, nonlinear noise, and increase the system complexity. Furthermore, the theoretical measurement range, error, resolution and the effect of the dispersion deviation in this structure are also open issues.

In this paper, with the reference of the conventional compressive receiver [20], a photonic compressive receiver for multiple microwave frequency measurement is presented. In the proposed scheme, instead of using a temporal magnification to relax the bandwidth requirement of PD and oscilloscope [19], the ultrashort pulse is directly observed via an equivalent time sampling method [21]. By modeling the frequency-time mapping procedure, the theoretical measurement accuracy, multiple frequency resolution and measurement range are derived accordingly. The effect of dispersion deviation and the electrical bandwidth are also discussed. The proposed photonic compressive receiver can realize the multiple-frequency measurement with high accuracy and high resolution over an ultra-wide frequency range, and the ability to intercept the signal within 100 ns.

## 2. Modeling and analysis

Figure 1(a) illustrates the diagram of the proposed photonic compressive receiver. A linearly chirped optical pulse train is generated by a linearly chirped optical pulse generator, which is modulated by an intercepted microwave signal via an electro-optic modulator, usually a lithium niobate Mach-Zehnder modulator (MZM). The modulated optical pulse train is then compressed by dispersion effect and converted to electrical signals by a PD. The detected electrical signal is quantized by an electric analog-to-digital converter (EADC). Finally, the time intervals between the pulses corresponding to the modulation term of signal in each period are measured, which can be used to calculate the frequency components in the intercepted signal.

As shown in Fig. 1(b), the time domain expression of the linearly chirped optical pulse train generated by the pulse generator can be expressed as

According to Eq. (4), the equivalent procedure before PD is illustrated in Fig. 2, and the first step is the frequency-time mapping. Take a two-tone signal with frequency of $\omega _1$ and $\omega _2$ as the input signal for example, namely $v_I (t)=cos(\omega _1 t)+cos(\omega _2 t)$. The compressed signal in Eq. (4) can be modified as

The second step of the equivalent procedure is shaping, which shapes the impulse functions as a scaled spectrum envelope of $p_e (t)$. Finally, the shaped pulses are dispersion stretched through an equivalent dispersive medium with GVD of $\Delta \ddot {\Phi }$. Since $\Delta \ddot {\Phi }=-(u_0 \ddot {\Phi }+1) \ddot {\Phi }$, the dispersion stretching effect can be removed when the GVD of the dispersive medium is matched with the shift rate of liner chirped optical pulse, i.e. $\ddot {\Phi }=-1/u_0$. In this case, the width of the compressed pulse $\tau _0$ is equal to the pulse width of the transform-limited pulse, $P_E(t/\ddot {\Phi })$, whose bandwidth is determined by the frequency range of the liner chirped optical pulse, $f_r$. However, it’s difficult to strictly match the GVD of the dispersive medium and $u_0$. When the dispersion mismatch satisfies the conditions: $\left | \frac {t_1^2}{\Delta \ddot {\Phi }} \right |\ll 1$, where $t_1$ is pulse width of $P_E(t/\ddot {\Phi })$, the spectrum of the transform-limited pulse would be mapped to the time domain, and the width of the compressed pulse $\tau _0$ can be expressed as [22]

The compressed signal is then detected by a PD and sampled by an EADC. When the width of compressed pulse, $\tau _0$ is much larger than the temporal width of the impulse response accounting from the PD to the EADC, the spectrum of the compressed pulse is included in the passband accounting from the PD to the EADC, and hence the measured pulse width $\tau _1$ is approaching to $\tau _0$. It’s worth noting that, when the GVD of dispersive medium is matched well with the linearly chirped optical pulse, the width of the compressed pulse $\tau _0$ can become narrower than the temporal width of the impulse response accounting from the PD to the EADC. In this case, the measured pulse width $\tau _1$ will mainly be determined by the bandwidth accounting from the PD to the EADC since the spectrum of the compressed pulse exceeds the passband accounting from the PD to the EADC.Since the measured pulse width $\tau _1$ determines the minimum distinguishable interval between the adjacent compressed pulses, the time interval between each pair of compressed pulses should be larger than $\tau _1$ and the time interval difference between different pairs of compressed pulses should be larger than $2\tau _1$ to avoid the overlapping of the adjacent compressed pulses in each period. Considering the periodicity of the optical pulse train, the pulse interval between each pair of compressed pulses should be less than $(T_s - \tau _1)$ to avoid the overlapping of compressed pulses in adjacent periods. The feasible time interval range and time interval difference are shown in Fig. 3, which are corresponding to the theoretical measurement range and multi-frequency resolution, respectively.

In addition, the sampling rate of the EADC determines the measurement error of the pulse interval, which is corresponding to the theoretical frequency measurement error. According to the feasible time interval range, sampling interval, time interval difference, and Eq. (6), the theoretical measurement range, error and multi-frequency resolution can be respectively derived as

## 3. Experimental results

The experimental setup of the proposed photonic compressive receiver is illustrated in Fig. 4. A mode-locked laser (MLL) (Precision Photonics, FFL1560) is employed to generate an optical pulse train with a 36.5 MHz repetition rate, which is corresponding to the signal interception period of $\sim$27 ns. An optical filter (10dB passband:1563.08 $nm$-1564.68 $nm$) is used to control the spectrum width of the optical pulse, $\Delta \lambda$. The linearly chirped optical pulse train is generated by stretching the optical pulse train via a 120-km single mode fiber (SMF). An erbium-doped optical fiber amplifier (EDFA) is adopted to compensate the attenuation in dispersion process. The linearly chirped optical pulse train is modulated by the input RF signal from a microwave signal generator (Rohde & Schwarz, SMF 100A) via a 40 Gbps MZM biased at $V_\pi =6.2V$. The power level of microwave generator is set at 0 dBm to avoid the modulation of signal’s higher order terms. The modulated signal is compressed back to pulses via a dispersion compensating fiber (DCF, compensating 120-km SMF) and detected by a photodetector with a 50GHz bandwidth. The detected pulse train is then amplified by a 55 GHz electric amplifier (AMP) and finally measured by a 70 GHz sampling oscilloscope (Keysight, DCA-X 86100D).According to the equivalent time sampling method, 2 THz equivalent sampling rate is realized to capture the narrow pulse’s temporal shape and location by setting the equivalent sampling interval as 0.5 ps.

In above experimental set-up, the GVDs of the 120-km SMF and the DCF can be considered constant in the slight spectral range limited by the optical filter, and the influence of higher-order dispersion can be ignored. The nonlinear effect in DCF can also be ignored since the peak power of the linearly chirped optical pulse after temporal stretching is low enough. The nonlinear effect in the SMF will cause phase change of the linearly chirped optical pulse and affect the optical pulse compression in the DCF, which will result in the shape changing of the compressed pulse. However, in the case with well-matched SMF and DCF, the shape of measured pulse is mainly determined by the bandwidth from the PD to the EADC. Therefore, the influence of nonlinear effect in SMF can also be neglected. It’s worth noting the response time of the equivalent time sampling method should be $T_r=\frac {T_s^2}{\delta \tau }$ to sample the signal of a whole period. According to Eq.(8), the theoretical measurement error can be calculated as $\delta f=\frac {T_s^2}{4\pi T_r \left | \ddot {\Phi } \right |}$.

The measured single pulse waveform in the cases with different dispersion matching are illustrated in Figs. 5(a) and 5(b), which are normalized by the corresponding maximum value and minimum value. In the case of Fig. 5(a), the GVD of the 120 km SMF and DCF are $\sim 2052.2 \ ps/nm$ and $\sim -2105.9 \ ps/nm$, respectively. The measured 10 dB pulse width is about 80 ps, which is approach to the theoretical width of the compressed pulse according to Eq. (7). This is reasonable since the theoretical width of the compressed pulse, 85 ps, is large enough compared to the temporal width of the impulse response accounting from the PD to the EADC (about 20 ps for 50 GHz bandwidth ideal low pass filters). The SMF and DCF are adjusted to match better in Fig. 5(b), with the GVD of the SMF and DCF are $\sim 2118.9 \ ps/nm$ and $\sim -2113.0 \ ps/nm$, respectively. The measured 10 dB pulse width is about 20 ps which is much larger than the theoretical width of the compressed pulse, $\sim$9 ps according to Eq. (7). This is because the width of the compressed pulse, 9 ps, is narrower than the temporal width of the impulse response accounting from the PD to the EADC, and the measured pulse width is mainly determined by the bandwidth accounting from PD to EADC. The pulse tails in Fig. 5(b) is caused by the sharp roll-off response of the applied 50 GHz PD. Based on the pulse waveform shown in Fig. 5(b), considering the pulse width and pulse tails, the time interval between pulses should be larger than 20 ps and the corresponding lower limitation for frequency measurement and multi-frequency resolution are 0.6 GHz and 1.2 GHz, respectively.

The follow-up frequency measuring experiments are carried out in the case with the well matched SMF and DCF. Figure 6 indicates the measured waveform when the input signal is a single tone of 0.6 GHz, 4 GHz and 8 GHz, respectively. It can be see that a pair of pulses can be observed in every period for a single-tone input signal. The time interval between the pulse peaks has a positive correlation with the frequency of input signal. The amplitude attenuation with the increase of input frequency is mainly due to the limited MZM bandwidth and it does not affect the measurement accuracy. In other words, the frequency exceeding the 3dB bandwidth of MZM can also be measured as long as the corresponding pulses can be measured. In addition, the dispersion-induced power penalty will not appear in the proposed receiver, since the generation and detection of the compressed pulse are irrelevant to the beat between the sidebands and carrier. The maximum measurable frequency in this experimental setup is 42 GHz, which is limited by the output bandwidth of the microwave signal generator.

The input signal’s frequency is swept from 0.6 GHz to 42 GHz, five measurement results at each frequency are shown in Fig. 7(a). The red line is the theoretical frequency-time mapping line with the slope of $34.43 \ ps/GHz$, which is calculated by Eq. (6) and corresponding to $\ddot {\Phi }\approx -2113 \ ps/nm$. The measured results are in line with the theoretical results, and the measured time intervals are almost completely in proportion to the frequency of input signal. Figure 7(b) demonstrates the deviations between the measured result of each time and the theoretical value at each frequency point. We can see that the time deviations between the measured results and the theoretical value within the whole frequency measuring range are less than 3 ps, which is corresponding to the maximum frequency measurement error of $\sim$88 MHz. The measurement error is mainly caused by the sampling time jitter and the limited time resolution of the sampling oscilloscope.

To verify the proposed photonic compressive receiver’s capability of multiple-frequency measurement, two-tone microwave signals are applied as input signals. Figures 8(a) and 8(b) show the measured waveform for two-tone input signal of 3 GHz and 4.2 GHz, and 3 GHz and 10 GHz, respectively. We can see that each frequency in the input signals corresponds to a pair of pulses and the pulses peaks can be distinguished easily for both cases. The results indicate that the multiple frequency can be measured with a resolution of 1.2 GHz.

Table 1 shows the performance comparison of the existing microwave multi-frequency measurement systems. It can be seen that the proposed photonic compressive receiver has a relatively wide measurement range and a relatively small measurement error. The multiple frequency resolution can be improved by increasing the frequency shift rate of the linearly chirped optical pulse and the GVD of the dispersive medium, and/or elevating the bandwidth accounting from the PD to the EADC. The measurement range can also be extended through replacing the modulator with a wider one, which can reach 100 GHz. Although the measurement range will decrease with the increase of GVD, the theoretical measurement range can still reach 130 GHz for a measurement resolution of less than 0.2 GHz without considering the bandwidth limitation of MZM. Meanwhile, the avoidance of nonlinear effect such as stimulated Brillouin scattering and four-wave mixing makes the system more energy-efficient.

## 4. Conclution

In conclusion, we proposed a photonic compressive receiver for multiple microwave frequency measurement. The frequency-time mapping procedure is modeled, and the theoretical measurement error, multiple-frequency resolution and effective measurement range are derived accordingly. The effects of dispersion deviation and the bandwidth accounting from PD to EADC are also discussed. The theoretical results were verified by the measured pulse waveforms and frequency-time mapping relationship. A photonic compressive receiver with an effective measurement range of 42 GHz, a multiple-frequency resolution of 1.2 GHz, a measurement accuracy of 88 MHz and a signal interception period of 27 ns is experimentally obtained. The performance can be further raised up through employing a modulator with higher bandwidth, dispersive mediums with larger dispersion and/or an EADC with less time jitter.

## Appendix

This appendix presents the derivation of Eq. (4), which gives the compressed pulses in one period.

## Funding

National Natural Science Foundation of China (61535006, 61627817).

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