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Photonics-based simultaneous distance and velocity measurement of multiple targets utilizing dual-band symmetrical triangular linear frequency-modulated waveforms

Shaowen Peng, Shangyuan Li, Xiaoxiao Xue, Xuedi Xiao, and Xiaoping Zheng

Open Access Open Access

Abstract

The distance and velocity measurement can be obtained by the round-trip time and Doppler effect on the down-chirp and the up-chirp of the linear frequency-modulated waveform (LFMW), but false targets will appear in a multi-target situation, resulting in erroneous detection. Here, we report a photonics-assisted approach to realize unambiguous simultaneous distance and velocity measurement in multi-target situations utilizing a dual-band symmetrical triangular LFMW. Dual-band observation invariance is proposed, to effectively resolve the false targets. The de-chirped signals can be obtained from parallel de-chirping processing to the dual-band echoes. By measuring and calculating the beat frequencies of the de-chirped signals in the two frequency bands, the actual parameter measurements can be acquired according to the authenticity criterion. In the experiments, detections to three targets are performed, and the distance and velocities are acquired without false targets. The absolute measurement errors of the distance and the velocity are less than 9 mm and 0.16 m/s, respectively. These results show the feasibility of the proposed approach.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

The simultaneous distance and velocity measurement of the targets has been widely applied in military affairs and civilian activities, such as reconnaissance, automotive industry, navigation, and micro-motion detection [1–5]. At present, the distance and velocity measurement is mainly realized by radar [5,6] and lidar [7–9]. Both methods get the measurement based on the round-trip time and the Doppler effect. Thanks to the high frequency and large bandwidth of the transmitted light in lidar, the accuracy of the velocity and distance measurement is very high [8]. However, lidar cannot work when the weather is terrible, such as raining and fogging. While radar can work at all times and under all-weather conditions, and has a large detection zone compared to lidar [10]. Unfortunately, limited to the electronic bottlenecks, it is hard for the traditional radar to generate and process large-bandwidth and high-frequency radio signals [11,12], resulting in a low-accuracy measurement of distance and velocity.

During the past several decades, to overcome the electronic bottlenecks, microwave photonics (MWP) technology has been developed rapidly due to its excellent performance in high frequency, large bandwidth, and so on [13–15]. Many photonics-based approaches named frequency multiplication, photonics digital-to-analog converter, and optical shaping and frequency-to-time mapping, have been proposed to directly generate large-bandwidth and high-frequency signals [16–19]. Many schemes such as the mixer and the photonic analog-to-digital converter (ADC), have also been proposed to process these signals [20,21]. Based on these technologies, some photonics-based simultaneous distance and velocity measurement system have been presented [22,23]. In [22], Cheng et al., proposed an approach to realize a high-resolution distance and velocity measurement based on photonic linear frequency modulated waveform (LFMW) generation and detection. In the experiment, the distance of static targets was measured and the absolute error of distance measurement is less than 3 mm. In [23], Zhang, et al., realized the simultaneous measurement of distance and velocity using the multi-band LFMWs with opposite chirps. The measured relative errors for distance and velocity are less than 0.005% and 0.59%, respectively. Both approaches indicate the advantages of MWP technology in distance and velocity estimation. They obtained the measurement by using the Doppler effect on the down-chirp and the up-chirp LFMWs. However, in a multi-target situation, these detection methods cannot get the correct values of distance and velocity because of the existence of false targets [24].

In this paper, a photonics-assisted approach for realizing the simultaneous distance and velocity measurement of multiple targets is proposed and demonstrated. Dual-band symmetrical triangular LFMWs are simultaneously generated as the transmitted signals. The parallel de-chirping processing to the dual-band echoes is realized. By analyzing the spectrum of the dual-band de-chirped signals, the value of the distance and velocity with false targets are obtained in each frequencies band. Based on the dual-band observation invariance that the distance and velocity of the actual targets keep invariable while those parameters of false targets are changed under dual-band observations, the false targets can be resolved by processing the detection results from each frequency band according to the authenticity criterion. Then the simultaneous high-accuracy distance and velocity measurement of the actual targets can be obtained. In the experiment, we realized simultaneous distance and velocity measurement in a three-target situation using S-band and X-band symmetrical triangular LFMWs. The absolute error of distance and velocity measurement is obtained in single-target situation, and they are less than 9 mm and 0.16 m/s, respectively.

2. Principle

Figure 1(a) shows the schematic diagram of the proposed multi-target distance and velocity measurement system based on photonics. It mainly consists of five parts, which are band-pass filter (BPF), antennas, dual-band symmetrical triangular LFMW generator, parallel de-chirping processing module and spectrum analysis module, respectively. The generated LFMW can be expressed as

$${S_{bT}}(t )= {V_{bT}}\cos ({{\theta_{bT}}(t )} )= \left\{ {\begin{array}{{l}} {S_{bT}^ + (t )= {V_{bT}}\cos \left[ {2\pi \left( {({{f_{b1}} + {B_b}} )t + \frac{1}{2}{u_b}{t^2}} \right)} \right]}\\ {S_{bT}^ - (t )= {V_{bT}}\cos \left[ {2\pi \left( {({{f_{b1}} + {B_b}} )t - \frac{1}{2}{u_b}{t^2}} \right)} \right]} \end{array}} \right.\;\;\;\;\;\begin{array}{{c}} { - \frac{T}{2} < t \le 0}\\ {0 < t \le \frac{T}{2}} \end{array}$$
where $b({b = 1,2} )$ stands for the dual-band LFMWs respectively, $S_{bT}^ + (t )$ and $S_{bT}^ - (t )$ stand for the up-chirp segment and down-chirp segment respectively, ${\theta _{bT}}(t )$ is the phase, ${V_{bT}}$ is the amplitude, ${f_{b1}}$ is the starting frequency, ${B_b}$ is the bandwidth, T is the period and ${u_b}\textrm{ = 2}{B_b}/T$ is the chirp rate. The instantaneous frequency of the transmitted signal is shown as the solid curve in Fig. 1(b). This LFMW is split into two parts. One is sent to the receiver as a reference signal. The other is emitted to space by an antenna. When the signal hits the ${i^{th}}$ target, it will be reflected and partially received by the receiving antenna after a delay time, which can be expressed as
$${\tau _i}(t )= \frac{2}{c}({{R_i} - {v_i}t} )$$
where $i = 1,\;2,\;\ldots \;N$ is the serial number of the targets, N is the total number of the targets, ${R_i}$ is the initial distance, ${v_i}$ is the velocity of the target [25]. Then the echo signal can be expressed as
$${S_{bR}}(t )= \sum\limits_i^N {{V_{bR,i}}\cos ({{\theta_{bR,i}}(t )} )} = \sum\limits_i^N {{V_{bR,i}}\cos ({{\theta_{bT}}({t - {\tau_i}(t )} )} )}$$
where ${\theta _{bR,i}}(t )$ is the phase and ${V_{bR,i}}$ is the amplitude. The instantaneous frequency of the echo signal of the ${i^{th}}$ target is shown as the dashed curve in Fig. 1(b).

 figure: Fig. 1.

Fig. 1. The schematic diagram of the proposed multi-target distance and velocity measurement system. O/E: opto-electronic conversion.

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The received echo signals are separated into two branches after a BPF. Each branch contains the received signals in one frequency band. These signals and the reference signals are mixed in the parallel de-chirping processing module. The parallel de-chirping processing module consist of two similar photonic links. In each link, the received signal and the reference signal in the same frequency band are simultaneously modulated to a light. This electro-optical modulation can be realized by multiple modulators, for example, dual-drive Mach-Zehnder modulator (DMZM). The arms of the DMZM are driven by the received signal and the reference signal respectively. The DMZM is biased at the minimum transmission point. Only considering the first-order sidebands of the optical signal, the expression of the light at the output of the DMZM can be written as

$$\begin{array}{c} {E_b}(t )\propto {e^{j{w_c}t}}\left[ {\exp \left( {\frac{\pi }{{{V_\pi }}}{S_{bT}}(t )} \right) + \exp \left( {\frac{\pi }{{{V_\pi }}}{S_{bR}}(t )\textrm{ - }{\varphi_b}} \right)} \right]\\ \approx {e^{j{w_c}t}}[{{J_0}({{m_{bT}}} )+ 2j{J_1}({{m_{bT}}} )\cos ({{\theta_{bT}}(t )} )} ]+ \\ \;\;{e^{j{w_c}t - {\varphi _b}}}\prod\limits_i^N {[{{J_0}({{m_{bR,i}}} )+ 2j{J_1}({{m_{bR,i}}} )\cos ({{\theta_{bR,i}}(t )} )} ]} \end{array}$$
where ${w_c}$ is the frequency of the carried light, ${m_{bT}} = \pi {V_{bT}}/{V_\pi }$ and ${m_{bR,i}} = \pi {V_{bR,i}}/{V_\pi }$ are the modulation indices, ${V_\pi }$ is the half-wave voltage and ${\varphi _b}$ is the phase difference between the two arms, which is set to be $\pi$. After the opto-electronic conversion, the de-chirping processing is completed. Taking the main low-frequency signals as interests, the output signal can be expressed as
$$\begin{aligned} {i_b}(t )\propto &{E_b}(t )E_b^\ast (t )\\ \approx &4{J_1}({{m_{bT}}} )\sum\limits_i^N {\prod\limits_{{\scriptstyle u} = 1 \atop {\scriptstyle u} \ne i }^N {{J_0}({{m_{bR,u}}} ){J_\textrm{1}}({{m_{bR,i}}} )\cos ({{\theta_{bT}}(t )- {\theta_{bR,i}}(t )} )} } \\ & + 4{J_0}({{m_{bT}}} )\sum\limits_{p = 1}^N {\sum\limits_{{\scriptstyle q} = 1 \atop {\scriptstyle q} \ne p }^N {\prod\limits_{{\scriptstyle l} = 1 \atop {{\scriptstyle l} \ne p \atop {\scriptstyle l} \ne q }}^N {{J_0}({{m_{bR,l}}} ){J_\textrm{1}}({{m_{bR,p}}} ){J_\textrm{1}}({{m_{bR,q}}} )\cos ({{\theta_{bR,p}}(t )- {\theta_{bR,q}}(t )} )} } } \end{aligned}$$
Since the received signal is weak and the reference is strong, ${m_{bT}}$ is larger than ${m_{bR,i}}$. Then it can be obtained that the intensity of the first term of ${i_b}(t )$ is larger than the intensity of the second term of ${i_b}(t )$. Therefore, the second term of ${i_b}(t )$ can be removed by threshold detection [26] and Eq. (5) can be rewritten as
$${i_b}(t )\propto \sum\limits_i^N {\prod\limits_{{\scriptstyle u} = 1 \atop {\scriptstyle u} \ne i }^N {{J_0}({{m_{bR,u}}} ){J_\textrm{1}}({{m_{bR,i}}} )\cos ({{\theta_{bT}}(t )- {\theta_{bR,i}}(t )} )} }$$
The beat frequencies of the de-chirped signal in the up-chirp segment and the down-chirp segment can be obtained by spectrum analysis module. From Eq. (6), it can be expressed as
$$\left\{ {\begin{array}{{c}} {f_{b,i}^ +{=} {u_b}\frac{{2{R_i}}}{c} - {f_{b1}}\frac{{2{v_i}}}{c}}\\ {f_{b,i}^ -{=} {u_b}\frac{{2{R_i}}}{c} + {f_{b1}}\frac{{2{v_i}}}{c}} \end{array}} \right.$$
According to Eq. (7), the distance and velocity calculated from each frequency band can be derived as
$$\left\{ {\begin{array}{{l}} {{R_{b,i}} = \frac{c}{{4{u_b}}}({f_{b,i}^ +{+} f_{b,i}^ - } )}\\ {{v_{b,i}} = \frac{c}{{4{f_{b1}}}}({f_{b,i}^ -{-} f_{b,i}^ + } )} \end{array}} \right.$$
When there is only one target, the de-chirped signal would be a single beat frequency signal at the up-chirp and down-chirp segment, respectively. It is easy to combine them in each frequency band to get the distance and velocity by Eq. (8). However, when there are many targets, N beat frequency signals $\left\{ {\begin{array}{{cccc}} {f_{b,1}^ + }&{f_{b,2}^ + }&{\ldots }&{f_{b,N}^ + } \end{array}} \right\}$ and $\left\{ {\begin{array}{{cccc}} {f_{b,1}^ - }&{f_{b,2}^ - }&{\ldots }&{f_{b,N}^ - } \end{array}} \right\}$ will be founded at the up-chirp segment and down-chirp segment in each frequency band respectively. Considering all the combinations of the two set beat frequencies in each frequency band, there are ${N^2}$ groups, $({f_{b,p}^ + ,\;f_{b,q}^ - } )$, where $p,q = 1,2\ldots N$. From each group of the beat frequencies of $({f_{b,p}^ + ,\;f_{b,q}^ - } )$, a target with the distance and velocity of $({{R_{b,pq}},{v_{b,pq}}} )$ can be calculated. Therefore, we would get ${N^2}$ targets, ${N^2} - N$ targets of which are false targets. Fortunately, when the ratios of the starting frequency to the chirp rate in the two frequency bands are unequal, the distance and velocity of N actual targets are invariable while the distance and velocity of ${N^2} - N$ false targets are changed in the two frequency bands. Thus, we can compare the ${N^2}$ targets acquired from each frequency band to obtain the velocity and distance of the actual target. We define this theory as dual-band observation invariance. A specific analysis can be found below.

From Eq. (8), we can know that each beat frequency signal corresponds to an actual target. It can be assumed that the beat frequency of $f_{b,p}^ - ({p = 1,2\ldots N} )$ corresponds to the target with a distance of ${R_{b,p}}$ and a velocity of ${v_{b,p}}$, and the beat frequency of $f_{b,q}^ + ({q = 1,2\ldots N} )$ corresponds to the target with a distance of ${R_{b,q}}$ and a velocity of ${v_{b,q}}$. According to Eqs. (7) and (8), the calculated parameters from each group of the beat frequencies can be derived as

$$\left\{ {\begin{array}{{l}} {{R_{b,pq}} = \frac{1}{2}({{R_{b,p}} + {R_{b,q}}} )+ \frac{{{f_{b1}}}}{{2{u_b}}}({{v_{b,p}} - {v_{b,q}}} )}\\ {{v_{b,pq}} = \frac{1}{2}({{v_{b,p}} + {v_{b,q}}} )+ \frac{{{u_b}}}{{2{f_{b1}}}}({{R_{b,p}} - {R_{b,q}}} )} \end{array}} \right.$$
From Eq. (9), it can be clearly seen that the distance of ${R_{b,pq}}$ and the velocity of ${v_{b,pq}}$ are the parameters of actual targets when $p = q$, and their values are completely independent of the ratios of the starting frequency to the chirp rate ${f_{b1}}/{u_b}$. While the distance of ${R_{b,pq}}$ and the velocity of ${v_{b,pq}}$ are the parameters of false targets when $p \ne q$, and their values are dependent of the ratios of the starting frequency to the chirp rate ${f_{b1}}/{u_b}$. Hence, the actual distance and velocity can be calculated out by comparing ${N^2}$ the targets obtained from the two frequency bands with different ratios of the starting frequency to the chirp rate. Considering the measurement error, the authenticity criterion to obtain the actual target can be designed as following
$$\left\{ {\begin{array}{{c}} {|{{R_{1,pq}} - {R_{2,pq}}} |\le \Delta {r_1} + \Delta {r_2}}\\ {|{{v_{1,pq}} - {v_{2,pq}}} |\le \Delta {v_1} + \Delta {v_2}} \end{array}} \right.$$
where $\Delta {r_1} = c/2{B_1}$, $\Delta {r_2} = c/2{B_2}$ are the resolution of the distance in the two frequency bands, respectively, $\Delta {v_1} = c/T{f_{11}}$, $\Delta {v_2} = c/T{f_{21}}$ are the resolution of the velocity in the two frequency bands, respectively. If the calculated distance and velocity of the targets in the two frequency bands satisfied the authenticity criterion, then they can be considered as measured values of the actual targets.

3. Experiment results and discussion

To evaluate the performance of the proposed approach for distance and velocity measurement of multiple targets, an experiment is conducted based on the setup in Fig. 2. The electro-optical modulations of the dual-band signals are realized by a polarization-multiplexing dual-drive Mach-Zehnder modulator (PM-DMZM) (FTM7980EDA/301) simultaneously. A light with a power of 13 dBm generated from a laser diode (Agilent N7714A) is injected into the PM-DMZM. The transmitted signal and the echo signal are generated by the arbitrary waveform generator (AWG) (Tektronix AWG70002A) according to Eqs. (1) and (3). These signals are dual-band LFMWs at S-band and X-band. Both LFMWs have a time duration of 10 ms and a bandwidth of 1 GHz. The carried frequency of S-band LFMW is 3 GHz and the carried frequency of X-band LFMW is 11 GHz. The frequency and bandwidth of the signals are limited by the AWG, which can be improved by other photonic techniques for higher accuracy. The waveform, spectrum and the instantaneous frequency of the transmitted signal are shown in Figs. 3(a), 3(b), and 3(c), respectively. Both transmitted signal and the echo signal are amplified, then filtered by a multichannel filter with the pass-bands of 2-4 GHz and 8-12 GHz. Subsequently, the filtered LFMWs at the two frequency bands are sent to drive the PM-DMZM. The output light from the PM-DMZM is split into two parts by the polarization beam splitter (PBS), each of which is injected into a PD with a bandwidth of 3 GHz to realize the de-chirping process. After filtered by a low-pass filter whose cut-off frequency is 50 MHz, the de-chirped signals are sampled by a digital storage oscilloscope (DSO) (Agilent DSO81204B). The de-chirped signal can be divided into two segments in the time domain, which including the beat frequencies at the up-chirped and down-chirped segments, respectively. After FFT is performed to two separated signals respectively, the spectra of the beat frequencies at the up-chirped and down-chirped segments can be obtained [27].

 figure: Fig. 2.

Fig. 2. Experiment setup of the proposed system. LD: laser diode; PM-DMZM: polarization-multiplexing dual-drive Mach-Zehnder modulator; BPF: band-pass filter; PBS: polarization beam splitter; PD: photodetector; HA: horn antenna; ADC: analog to digital converter. Amp: amplifier; LPF: low-pass filter; AWG: arbitrary waveform generator; DSO: digital storage oscilloscope.

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 figure: Fig. 3.

Fig. 3. The transmitted dual-band LFMW generated by the AWG. (a) The waveform of the signal. (b) The spectrum of the signal, (c) The instantaneous frequency of the signal.

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The AWG is used to generate echo signals to emulate the echo waves from targets by changing the distance ${R_i}$ and the velocity ${v_i}$ according to Eq. (3). In actual applications, distortion will occur in the echo waves from targets due to the electromagnetic scattering characteristic of the targets, transport characteristics of the environment. This will broaden spectrum of the de-chirped signal, causing inaccurate frequency measurement when the distortion is serious, and in turn reduces the accuracy of the ranging and velocity measurement. At present, this effect can be eliminated to a certain extend by some compensation approaches [28]. Besides, this effect does not affect the effectiveness of the presented method to remove false targets and realize simultaneous distance and velocity measurement in multi-target detection. So, these influences are ignored in our experiments.

3.1 Single-target detection

Firstly, in order to explore the simultaneous distance and velocity detection capabilities and performance of the system, two sets of experimental scenarios are designed. In one of the scenarios, the assumed distance is fixed at 9 km and the assumed velocity varies from 10 m/s to 50 m/s. The measured results and absolute errors by S-band data are presented in Figs. 4(a) and 4(b), respectively. The measured results and absolute errors by X-band data are presented in Figs. 4(c) and 4(d), respectively. In another scenario, the assumed velocity is fixed at 30 m/s and the assumed distance varies from 3 km to 15 km. The measured results and absolute errors by S-band data are presented in Figs. 5(a) and 5(b), respectively. The measured results and absolute errors by X-band data are presented in Figs. 5(c) and 5(d), respectively. They show that the absolute error of the distance is less than 9 mm whether detected by S-band or X-band. The error of the velocity is less than 0.7 m/s detected by S-band and less than 0.16 m/s detected by X-band. Because the accuracy of the velocity at X-band is higher than S-band, the detection results by the X-band data are chosen as the measured results.

 figure: Fig. 4.

Fig. 4. (a) The measured results of distance and velocity by S-band data. (b) The absolute error of distance and velocity measured by S-band data. (c) The measured results of distance and velocity by X-band data. (d) The absolute error of distance and velocity measured by X-band data.

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 figure: Fig. 5.

Fig. 5. (a) The measured results of distance and velocity by S-band data. (b) The absolute error of distance and velocity measured by S-band data. (c) The measured results of distance and velocity by X-band data. (d) The absolute error of distance and velocity measured by X-band data.

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3.2 Multi-target detection

Then, a complex experimental scenario is set to verify that the approach can detect multiple targets. Three targets are assumed in the scenarios. Their distance and velocity are (3000 m, 20 m/s), (6000 m, 30 m/s) and (9000 m, 40 m/s), respectively. Figures 6(a)–6(d) show the spectrum of the S-band and X-band de-chirped signals at the up-chirp segment and down-chirp segment respectively. Three beat frequencies of the de-chirped signal can be obtained at the up-chirp segment or down-chirp segment from each frequency band. Combining the beat frequencies at the up-chirp and the down-chirp segment of the S-band signal, nine targets with different distance and velocity were obtained. They are shown in Fig. 7, marked by blue triangles. Similarly, combining the beat frequencies at the up-chirp and the down-chirp segment of the X-band signal, nine targets with different distance and velocity were also obtained. They are shown in Fig. 7, marked as red plus signs. It is clearly seen that many false targets appeared there, whether detected by S-band or X-band. Comparing the values of the velocity and distance of the nine targets between the two frequency bands, we find there are three targets with similar values of the distance and velocity, which satisfied the authenticity criterion Eq. (10). These three targets are shown in Fig. 7, marked as orange circles. Based on the theory that the distance and velocity of the actual targets keep invariable while those parameters of false targets are changed under dual-band observations, it can be concluded that these three targets are actual targets. The detailed parameters of the three targets measured by S-band and X-band are presented in Table 1. According to the single-target detection results, the accuracy of the velocity at X-band is higher than S-band, thus the detection results by the X-band data are chosen as the measured results. Therefore, the measured distance and velocity of the actual targets are (3000.001 m, 20.01 m/s), (6000.001 m, 30.02 m/s) and (8999.999 m, 39.98 m/s), respectively, which is very close to the real values. It shows the proposed approach is effective to realize the simultaneous distance and velocity of multiple targets.

 figure: Fig. 6.

Fig. 6. (a) The spectrum of the S-band de-chirped signal at the up-chirp segment. (b) The spectrum of the S-band de-chirped signal at the down-chirp segment. (c) The spectrum of the X-band de-chirped signal at the up-chirp segment. (d) The spectrum of the X-band de-chirped signal at the down-chirp segment.

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 figure: Fig. 7.

Fig. 7. The measured results of the distance and velocity of multiple targets.

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Tables Icon

Table 1. The detection results of the three targets by S-band and X-band data

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4. Conclusion

We have proposed and experimentally demonstrated a method to realize simultaneous distance and velocity measurement of multiple targets utilizing dual-band symmetrical triangular LFMW. The dual-band observation invariance is presented to resolve the false targets. Combining the beat frequencies obtained from the de-chirped signals at the up-chirp and down-chirp segments by photonics-based parallel de-chirping processing, two sets of distance and velocity measurement results are obtained by the dual-band signals. Further, according to the authenticity criterion, the false targets are resolved and the actual targets are found out by processing the dual-band results. In the experiments, the absolute measurement errors of distance and velocity are less than 9 mm and 0.16 m/s, respectively, and the unambiguous distance and velocity of actual targets are obtained in a three-target scenario. This approach is suitable for most scenarios. Even for targets at the same distance, same velocity but different azimuths, this approach can achieve simultaneous distance and velocity measurement combined with other detection technologies, such as beam scanning [29].

Funding

National Natural Science Foundation of China (61690191); National Key Research and Development Program of China (2018YFB2201703, 2019YFB2203301); Natural Science Foundation of Beijing Municipality (4172029).

Disclosures

The authors declare no conflicts of interest.

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Figures (7)
Fig. 1.
Fig. 1. The schematic diagram of the proposed multi-target distance and velocity measurement system. O/E: opto-electronic conversion.

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Fig. 2.
Fig. 2. Experiment setup of the proposed system. LD: laser diode; PM-DMZM: polarization-multiplexing dual-drive Mach-Zehnder modulator; BPF: band-pass filter; PBS: polarization beam splitter; PD: photodetector; HA: horn antenna; ADC: analog to digital converter. Amp: amplifier; LPF: low-pass filter; AWG: arbitrary waveform generator; DSO: digital storage oscilloscope.

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Fig. 3.
Fig. 3. The transmitted dual-band LFMW generated by the AWG. (a) The waveform of the signal. (b) The spectrum of the signal, (c) The instantaneous frequency of the signal.

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Fig. 4.
Fig. 4. (a) The measured results of distance and velocity by S-band data. (b) The absolute error of distance and velocity measured by S-band data. (c) The measured results of distance and velocity by X-band data. (d) The absolute error of distance and velocity measured by X-band data.

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Fig. 5.
Fig. 5. (a) The measured results of distance and velocity by S-band data. (b) The absolute error of distance and velocity measured by S-band data. (c) The measured results of distance and velocity by X-band data. (d) The absolute error of distance and velocity measured by X-band data.

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Fig. 6.
Fig. 6. (a) The spectrum of the S-band de-chirped signal at the up-chirp segment. (b) The spectrum of the S-band de-chirped signal at the down-chirp segment. (c) The spectrum of the X-band de-chirped signal at the up-chirp segment. (d) The spectrum of the X-band de-chirped signal at the down-chirp segment.

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Fig. 7.
Fig. 7. The measured results of the distance and velocity of multiple targets.

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Tables (1)

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Table 1. The detection results of the three targets by S-band and X-band data

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Equations (10)

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S b T ( t ) = V b T cos ⁡ ( θ b T ( t ) ) = { S b T + ( t ) = V b T cos ⁡ [ 2 π ( ( f b 1 + B b ) t + 1 2 u b t 2 ) ] S b T − ( t ) = V b T cos ⁡ [ 2 π ( ( f b 1 + B b ) t − 1 2 u b t 2 ) ] − T 2 < t ≤ 0 0 < t ≤ T 2
τ i ( t ) = 2 c ( R i − v i t )
S b R ( t ) = ∑ i N V b R , i cos ⁡ ( θ b R , i ( t ) ) = ∑ i N V b R , i cos ⁡ ( θ b T ( t − τ i ( t ) ) )
E b ( t ) ∝ e j w c t [ exp ⁡ ( π V π S b T ( t ) ) + exp ⁡ ( π V π S b R ( t )  -  φ b ) ] ≈ e j w c t [ J 0 ( m b T ) + 2 j J 1 ( m b T ) cos ⁡ ( θ b T ( t ) ) ] + e j w c t − φ b ∏ i N [ J 0 ( m b R , i ) + 2 j J 1 ( m b R , i ) cos ⁡ ( θ b R , i ( t ) ) ]
i b ( t ) ∝ E b ( t ) E b ∗ ( t ) ≈ 4 J 1 ( m b T ) ∑ i N ∏ u = 1 u ≠ i N J 0 ( m b R , u ) J 1 ( m b R , i ) cos ⁡ ( θ b T ( t ) − θ b R , i ( t ) ) + 4 J 0 ( m b T ) ∑ p = 1 N ∑ q = 1 q ≠ p N ∏ l = 1 l ≠ p l ≠ q N J 0 ( m b R , l ) J 1 ( m b R , p ) J 1 ( m b R , q ) cos ⁡ ( θ b R , p ( t ) − θ b R , q ( t ) )
i b ( t ) ∝ ∑ i N ∏ u = 1 u ≠ i N J 0 ( m b R , u ) J 1 ( m b R , i ) cos ⁡ ( θ b T ( t ) − θ b R , i ( t ) )
{ f b , i + = u b 2 R i c − f b 1 2 v i c f b , i − = u b 2 R i c + f b 1 2 v i c
{ R b , i = c 4 u b ( f b , i + + f b , i − ) v b , i = c 4 f b 1 ( f b , i − − f b , i + )
{ R b , p q = 1 2 ( R b , p + R b , q ) + f b 1 2 u b ( v b , p − v b , q ) v b , p q = 1 2 ( v b , p + v b , q ) + u b 2 f b 1 ( R b , p − R b , q )
{ | R 1 , p q − R 2 , p q | ≤ Δ r 1 + Δ r 2 | v 1 , p q − v 2 , p q | ≤ Δ v 1 + Δ v 2
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