# 2-D Coherent Integration Processing and Detecting of Aircrafts Using GNSS-Based Passive Radar

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Air Target Echo Model in GNSS-Based Passive Radar

#### 2.1. GNSS-Based Passive Radar Geometry and Doppler Characteristic Analysis

**P**

_{tar}(X

_{A},Y

_{A},Z

_{A}) and moves at a constant velocity

**V**

_{tar}(V

_{x}, V

_{y}, 0) and in a constant level. The GNSS transmitter is located at

**P**

_{Tran}(X

_{G},Y

_{G},Z

_{G}) with a velocity vector

**V**

_{Tran}(V

_{Tx}, V

_{Ty}, V

_{Tz}).

_{R}(η) and transmitter slant range R

_{T}(η), which can be rewritten by Taylor series expansion.

_{ref}, λ, f

_{d}, and f

_{r}denote the illumination time, reference range, signal wavelength, target Doppler centroid, and Doppler modulated rate, and f

_{d}and f

_{r}are derived as follows [36]:

_{d}

_{0}and f

_{r}

_{0}represent the Doppler centroid frequency and Doppler frequency modulated rate of the stationary target, and f

_{d}

_{,v}and f

_{r}

_{,v}are the corresponding variations caused by target motion. Besides, in GNSS-based passive radar, the reference signal, and its Doppler are also important for signal synchronization and range compression. As both the transmitter and the receiver co-ordinates are known, it is easy to remove the Doppler of reference signal from the synchronization outputs [14]. Therefore, to simplify the subsequent derivation, the Doppler of reference signal are not considered in this manuscript.

_{d}

_{,v}and f

_{r}

_{,v}will cause Doppler spectrum shift and image defocusing, which is a serious problem for long duration coherent integration processing and target detection. Assuming the air target is moving in the XOY plane with a velocity 220 m/s, Figure 2 shows the variations of f

_{d}

_{,v}and f

_{r}

_{,v}, using the parameters listed in Table 1. As shown in Figure 2, both f

_{d}

_{,v}and f

_{r}

_{,v}are greatly influenced by passive radar geometry and target motion direction. f

_{d}

_{,v}is mainly determined by the equivalent range velocity, and f

_{r}

_{,v}is mainly influenced by the equivalent azimuth velocity [36]. Normally, as depicted in Figure 2a, the range of f

_{d}

_{,v}is from −1500 Hz to 1500 Hz when the target velocity is smaller than 220 m/s (approximately 800 km/h). However, the pulse repetition frequency (PRF), f

_{p}, of the GNSS signal is 1000 Hz. As a result, the Doppler ambiguity phenomenon should be considered in the signal processing; otherwise, the blind speed will be inevitably detected. On the other side, f

_{r}

_{,v}is about 2.6 ± 0.2 Hz/s on this condition, which is relatively small because of the air target is far away from the receiver, and the corresponding range-curvature term is smaller than one range cell normally.

#### 2.2. Air Target Echo Signal Model of GNSS-Based Passive Radar

_{T}is the signal amplitude, Q(·) is the modulation code, and T

_{p}, c, and λ denote signal period, speed of light, and signal wavelength, respectively. Then, set η = nT

_{p}+ τ, t

_{n}= nT

_{p}, and the range history can be written as:

_{n}is ignored for simplicity. As the transmitted code in each GNSS satellite is high-rate pseudo random noise (PRN) sequence, it has particularly excellent auto-correlation and cross-correlation properties, and these cross-correlation values are so small that they usually can be ignored. As a result, corresponding PRN code could not only be used as the matched filter but also be employed for separating the echo signal from different visible GNSS satellites. Hence, to simplify the subsequent derivation, one specific visible GNSS satellite is chosen as the transmitter, and the echo signal can be written as:

_{ref}is the target reference range in center time, and the range-curvature in the envelop term is ignored, as it is much smaller than one range cell.

## 3. The Proposed 2-D Coherent Integration Processing and Target Detection Algorithm

#### 3.1. The Principle of RFT

_{0}t + r

_{0}, where r

_{0}is the initial target position and v

_{0}is target radial velocity. To coherently integrate the moving target’s azimuth echoes pulse by pulse, the traditional RFT for radar signal is by jointly searching range and velocity (r, v), which is [18]:

_{v}= 2v/λ, the range history can be rewritten by R(t) = λf

_{v}

_{0}t/2 + r

_{0}, where f

_{v}

_{0}is the target Doppler frequency. So, the RFT in the range-Doppler plane is:

_{v}are range and Doppler searching steps, r

_{min}and f

_{min}are the minimum target range and Doppler searching value, and N

_{a}, N

_{r}, and N

_{f}are the sample number of azimuth, range, and Doppler domains, respectively. As shown in (7)–(9), the RFT actually performs echo signal integration along a parameterized line defined by (r, v) or (r, f

_{v}). When the parameterized line (r

_{0}, v

_{0}) or (r

_{0}, f

_{v}

_{0}), coincides with the target range history, all echoes distributed on the line are integrated to a peak, and full coherent-integration gain processing is achieved.

#### 3.2. Air Target Detection Based on 2-D Coherent Integration

#### 3.2.1. Signal Preprocessing

_{f}(·) is the FFT form of the GNSS modulation code,$A=\sigma {A}_{T}\text{\hspace{0.17em}}\mathrm{exp}\left\{j\frac{2\pi {R}_{ref}}{\lambda}\right\}$. Then, the phase term and range-walk caused by f

_{d}

_{0}and f

_{r}

_{0}are compensated with the 2-D filter ${H}_{p}\left(t,{f}_{\tau}\right)$.

_{d}

_{0}and f

_{r}

_{0}are unknown, because the target position is unknown in advance. However, as the satellites orbit is about 20,000 km, the changing of slant range R

_{T}in different range cell is very small compared with the R

_{T}value itself. So, the difference of f

_{d}

_{0}and f

_{r}

_{0}in each position of the whole experiment scenario is quite small, and any range cell’s f

_{d}

_{0}and f

_{r}

_{0}is enough to guarantee the accuracy of 2-D filter in (11). In this situation, it is recommended to use the f

_{d}

_{0}and f

_{r}

_{0}in the scene center to construct the 2-D filter.

#### 3.2.2. Azimuth Coherent Integration

_{r}

_{,v}, and residual range-walk term caused by the unknown, f

_{d}

_{,v}. For QPE estimation and compensation, many methods have been proposed in SAR imaging, and the simple and easy way is based on amplitude auto-focus method [37]. Besides, the residual range-walk can be removed by Doppler searching. Therefore, the azimuth coherent integration processing can be performed by QPE compensation, range-walk removal, and RFT, which is:

- The first phase term in the first line of Equation (13) is QPE compensation, and ${\widehat{f}}_{r}$ is the estimated Doppler rate parameter. As shown in Figure 2b, the variation range of Doppler rate is very small, and the maximum variation value is smaller than 0.6 Hz/s. Hence, we can use parallelized step search to estimate the Doppler rate (as shown in Figure 4), and this method has a high searching efficiency because of the small variation range of f
_{r}_{,v}. Normally, the estimated Doppler rate can be expressed as:$${\widehat{f}}_{r}^{k}={\widehat{f}}_{r}^{0}+k\Delta {\widehat{f}}_{r},\text{\hspace{1em}}k=0,1,2,\dots ,{N}_{k}$$ - The second phase term in the first line of Equation (13) is residual range-walk removal, and f
_{v}is the Doppler searching parameter. As shown in Equation (9), non-integer range cell is inevitable during the range history line searching, so range interpolation processing is necessary for traditional RFT. However, in Equation (13), when f_{v}matches the target’s Doppler frequency, f_{d}_{,v}, all range-walk term is removed, and the echo signal will distribute along the line perpendicular to the range direction. Therefore, the non-interpolation processing is needed in the proposed algorithm, and the following azimuth integral processing is greatly simplified. - The last phase term in the first line of Equation (13) is RFT processing, or called azimuth integral. As the range-walk term is removed, the echo signal is distributed along the azimuth direction. So, the RFT operation is similar to azimuth FFT operation.

_{d}

_{,v}, the estimated Doppler rate equals f

_{r}

_{,v}, and the azimuth peak will occur at the target Doppler frequency point.

_{A}N

_{R}N

_{V}), where N

_{A}, N

_{R}, and N

_{V}denote azimuth sample number, range sample number, and Doppler searching number, respectively. Therefore, the azimuth coherent integration processing will take a lot of memory and increase the computational complexity. To improve the algorithm’s efficiency, a fast implementation of azimuth coherent integration based on the chirp-Z transform (CZT) [38] is presented in the following.

_{v}denote the ambiguous Doppler index and non-ambiguous Doppler searching number. Based on this, Equation (13) can be rewritten as:

_{m}(·) and IDFT

_{m}(·) denote the Discrete Fourier Transform and Inverse Discrete Fourier Transform, respectively.

_{R}N

_{V}), which is significantly lower than the 3-D processing. More details about computation complexity analysis can be seen in Chapter 4.2.4. Based on (20), the processing flow of azimuth coherent integration is shown in Figure 5.

#### 3.2.3. Range Compression

_{d}

_{,v}. This will introduce a time-varying phase in the range direction and change the response of range compression. The traditional cross-ambiguity function could be used for range compression with Doppler and range 2-D joint-searching in each signal period [20,21,22]. But it will very time consuming because of the extremely long azimuth coherent time. Therefore, a special matched filter including Doppler frequency is necessary.

^{C}is the gain of range compression, normally Q

^{C}equals B

_{w}/f

_{p}, and B

_{w}is the signal bandwidth. D(·) is the normalized envelope of the auto-correlation result of GNSS transmitted signal, and D(0) = 1. $\delta \left({f}_{v}-{f}_{d,v}\right)$ is the factor caused by mismatched Doppler frequency in range compression, and $\delta \left(0\right)=1$, which means the range compression is matched filtering when ${f}_{v}={f}_{d,v}$. Here, as shown in Figure 4, we use parallelized search to estimate the Doppler rate. Then, when the error of estimated Doppler rate $\Delta {f}_{r}$ is zero, the signal ${S}_{o}\left({f}_{d,v},{R}_{ref}\right)$ is:

_{A}, equals coherent time multiplied by the signal repetition rate, and N

_{A}is the azimuth coherent integration. Considering the geometry of GNSS-based passive radar, each moving target’s range Doppler parameters are totally different. Therefore, even in multi-targets case, all targets are focused at their own Doppler parameter position.

#### 3.2.4. Target Detection

^{−6}in the additive white Gaussian noise background, the minimum signal-to-noise ratio (SNR) is 12.8 dB [18]. In TBD, the particle filter (PF) is commonly used, and it is capable of detecting a moving target with SNR as low as 7 dB [35]. In this paper, the CFAR method is used in the target detection part.

## 4. Experiments and Discussion

#### 4.1. Experiments and Results

#### 4.1.1. Comparing Results Using the Proposed Algorithm and Traditional RFT

#### 4.1.2. Multi-Static Experiments and Aircrafts Motion Parameters Estimation

#### 4.2. Discussion

#### 4.2.1. Power Budget and Coherent Time

_{in}of echo signal is:

_{E}is the GNSS signal power near the Earth’s surface, and it equals −154 dB in GPS L5 signal case [40]; G

_{R}is the antenna gain of receiver, and it is set to 35 dB as listed in Table 1; σ is target RCS, for a medium-sized aircraft (like Boeing 737–400), its RCS exceeds 20 dB in L-band, and in some azimuth angle, the RCS is larger than 26 dB [41]; R

_{R}is the range between target and receiver, k is the Boltzmann constant, T

_{s}is system Kelvin temperature, B

_{w}is the signal bandwidth, and F

_{n}L is system noise coefficient. For GNSS-based passive radar with the GPS-L5 signal, SNR

_{in}is roughly smaller than −70 dB. Hence, the 2-D coherent integration gain is necessary to obtain a sufficiently high SNR for target detection.

_{o}of processing result is:

_{c}is the coherent time, T

_{p}B

_{w}and f

_{p}T

_{c}are the range and azimuth processing gain, respectively. Normally, to guarantee the effectiveness of target detection, the minimum SNR (SNR

_{min}) of the processing results should be obtained. So, the coherent time, T

_{c}, needs to be longer than the lower bound, which is:

_{w}in the quasi-monostatic case, and the range-curvature term is $\frac{\lambda}{2}\left({f}_{r0}+{f}_{r,v}\right)\cdot {\left(\frac{{T}_{c}}{2}\right)}^{2}$. So, this assumption determines the upper bound of coherent time T

_{c}, which is:

_{min}equal to 12.8 dB. Then, with coherent time T

_{c}equal to 5 s, the result of power-budget analysis is shown in Figure 8b. As shown, the detection range of aircraft using GNSS-based passive radar is larger than 80 km, and it will be increased twofold in the RCS enhancement area. Furthermore, with the advanced target detection method, like TBD, the detection range can easily reach 200 km, as shown in Figure 8b.

#### 4.2.2. Searching Step of Doppler Rate

_{d}

_{,v}, R

_{ref}) is:

_{c}is 5 s), the energy loss (less than 0.2 dB) could be ignored when $\Delta {\widehat{f}}_{r}$ is 0.04 Hz/s. Besides, the variation of f

_{r}

_{,v}is smaller than 0.4 Hz/s, as shown in Figure 2b, and as a result, the maximum searching number is only 10, which is absolutely acceptable.

#### 4.2.3. Doppler Ambiguity

_{d}

_{,v}varies from −1500 Hz to 1500 Hz. Therefore, the range of searching Doppler frequency is [−1500, 1500] Hz, and the Doppler ambiguity is inevitable because f

_{p}of the GNSS signal is 1000 Hz. Let the ambiguous Doppler frequency be:

_{A}= 5000, BDSR(±1) = −17.06 dB, and BDSR(±2) = −27.40 dB. As the BDSR is very low, the blind Doppler sidelobe will submerge in the noisy background. Therefore, both the range/Doppler profile and BDSR results demonstrate that the proposed algorithm has a good performance even in the Doppler ambiguity situation, and the problem of blind speed is also overcome.

#### 4.2.4. Computation Analysis about the Azimuth Coherent Processing

_{2}(N) FLOP. Therefore, based on (13) and (17), the azimuth processing computation load is:

_{1}/C

_{2}. Normally, we let N

_{V}= N

_{A}, and with the condition of the same parameters as listed in Table 1, the computational efficiency γ is:

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. Delay-Doppler Ambiguity Function of GNSS and LFM Signal

_{D}denote the range, time delay, and Doppler frequency, respectively.

_{D}= 0), and even a small Doppler shift will lead to large SNR loss. Therefore, we can define that the GNSS signal is Doppler-intolerant. For airplane target detection, the range of target Doppler frequency is from −1500 Hz to 1500 Hz (shown in Figure 2), and the big Doppler shift is obviously not negligible, which means the Doppler frequency should be considered during the range compression processing.

**Figure A1.**Delay-Doppler ambiguity function of LFM and GPS-L5 signal. (

**a**) Three-dimensional results of LFM (

**b**) Doppler frequency profile of LFM (

**c**) Three-dimensional results of GPS-L5 (

**d**) Doppler frequency profile of GPS-L5.

## Appendix B

_{r}

_{,v}, and the estimated value, ${\widehat{f}}_{r}$.

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**Figure 1.**Air target detection geometric configuration with global navigation satellite system (GNSS) illuminators.

**Figure 2.**Variations of f

_{d}

_{,v}and f

_{r}

_{,v}caused by air target motion with 220 m/s in different geometric configurations (the X-axis is the angle between the target velocity and the Y-axis). (

**a**) Variation of f

_{d}

_{,v}(

**b**) Variation of f

_{r}

_{,v}.

**Figure 6.**Coherent integration processing results of target-1 with SVN #7 transmitter: (

**a**) Range-Doppler results using the proposed algorithm; (

**b**) range-Doppler results using traditional Radon Fourier Transform (RFT); (

**c**) range profiles of the results; (

**d**) Doppler frequency profiles of the results.

**Figure 7.**Coherent integration processing results for all targets by the proposed algorithm with different transmitter. (

**a**) Results with SVN #2; (

**b**) Results with SVN #7; (

**c**) Results with SVN #12.

**Figure 8.**Available coherent time and power budget for aircraft detection using GNSS-based passive radar. (

**a**) Available coherent time. (

**b**) Power budget analysis.

**Figure 10.**The range and Doppler profile when the Doppler ambiguity exists. (

**a**) Range profile (

**b**) Doppler profile.

Parameters | Values | Parameters | Values |
---|---|---|---|

Wavelength | 0.255 m | SVN #2 position | (1.0234, −1.5530,1.2411) × 10^{4} km |

Sampling rate | 42.0 MHz | SVN #7 position | (0.9767, 1.2291, 1.4880) × 10^{4} km |

$\Delta {f}_{v}$ | 0.2 Hz | SVN #12 position | (−1.8017, 1.6150, 0.4690) × 10^{4} km |

$\Delta {\widehat{f}}_{r}$ | 0.04 Hz/s | SVN #2 velocity | (187.9, −2113.8, −1799.3) m/s |

N_{V} | 15,000 | SVN #7 velocity | (−2398.5, −632.5, 1426.5) m/s |

f_{p} | 1000 Hz | SVN #12 velocity | (−2097.3, −729.0, −2321.4) m/s |

Experiment scenario Center | (60, 40, 10) km | Target RCS [41] | 20 dB |

Antenna gain (Receiver) | 35 dB | Coherent time | 5 s |

Target 1 position | (70, 20, 10) km | Target 1 velocity | (150, 120) m/s |

Target 2 position | (65, 30, 10) km | Target 2 velocity | (−135, 140) m/s |

Target 3 position | (50, 50, 10) km | Target 3 velocity | (100, −170) m/s |

Target-1 | Target-2 | Target-3 | |||
---|---|---|---|---|---|

SVN #2 | f_{d}_{,v} (Hz) | True | 748.87 | 375.28 | −835.72 |

Estimated | 749.20 | 375.40 | −835.80 | ||

Variation from reference range (km) | True | −17.79 | −9.76 | 10.13 | |

Estimated | −17.78 | −9.76 | 10.13 | ||

SVN #7 | f_{d}_{,v} (Hz) | True | 156.66 | −322.39 | 9.49 |

Estimated | 156.40 | −322.20 | 9.60 | ||

Variation from reference range (m) | True | 7.55 | 2.92 | −2.56 | |

Estimated | 7.55 | 2.92 | −2.56 | ||

SVN #12 | f_{d}_{,v} (Hz) | True | 812.52 | −995.59 | 531.07 |

Estimated | 812.60 | −998.60 | 531.20 | ||

Variation from reference range (m) | True | 21.08 | 9.68 | −15.25 | |

Estimated | 21.08 | 9.68 | −15.24 |

Target-1 | Target-2 | Target-3 | ||
---|---|---|---|---|

Target Location (X_{A},Y_{A},Z_{A}) (km) | True | (70, 20, 10) | (65, 30, 10) | (50, 50, 10) |

Estimated | (70.02, 19.99, 10.00) | (64.98, 30.01, 10.01) | (50.01, 50.02, 10.00) | |

Target Velocity (V_{x},V_{y},0) (m/s) | True | (150, 120, 0) | (−135, 140, 0) | (100, −170, 0) |

Estimated | (150.2, 119.9, 0) | (−135.0, 140.1, 0.0) | (99.9, −170.1, 0) |

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Zeng, H.-C.; Chen, J.; Wang, P.-B.; Yang, W.; Liu, W. 2-D Coherent Integration Processing and Detecting of Aircrafts Using GNSS-Based Passive Radar. *Remote Sens.* **2018**, *10*, 1164.
https://doi.org/10.3390/rs10071164

**AMA Style**

Zeng H-C, Chen J, Wang P-B, Yang W, Liu W. 2-D Coherent Integration Processing and Detecting of Aircrafts Using GNSS-Based Passive Radar. *Remote Sensing*. 2018; 10(7):1164.
https://doi.org/10.3390/rs10071164

**Chicago/Turabian Style**

Zeng, Hong-Cheng, Jie Chen, Peng-Bo Wang, Wei Yang, and Wei Liu. 2018. "2-D Coherent Integration Processing and Detecting of Aircrafts Using GNSS-Based Passive Radar" *Remote Sensing* 10, no. 7: 1164.
https://doi.org/10.3390/rs10071164