Aerospace Electronic and Defense Systems: Multistatic UAV SAR Joint Synchronization Based on Multiple Direct Wave Pulses Exchange | IEEE Journals & Magazine

$Fig. 1. - Signal link and timing configuration of MUAV-SAR with four separate stations. (a) Six direct wave links for four stations and echo links where only the two transceiver combinations of $T_{1}R_{1}$ and $T_{1}R_{2}$ ( $T$ denotes transmitter and $R$ means receiver) are shown. (b) Timing configuration of the bistatic direct wave link (top) and the monostatic and bistatic echo links (bottom).$

Signal link and timing configuration of MUAV-SAR with four separate stations.

(a) Six direct wave links for four stations and echo links where only the two transceiver combinations of

T1R1 and T1R2 (T denotes transmitter and R means receiver) are shown.
(b) Timing configuration of the bistatic direct wave link (top) and the monostatic and bistatic echo links (bottom).

Multistatic UAV SAR Joint Synchronization Based on Multiple Direct Wave Pulses Exchange | IEEE Journals & Magazine | IEEE Xplore

Abstract:

Multistatic unmanned aerial vehicle synthetic aperture radar (MUAV-SAR) 3-D imaging system suffers from the time and phase synchronization errors among multiple stations. The classical two-way direct wave pulse exchange synchronization method introduces the π -ambiguity phase error and causes limited time–phase synchronization accuracy with multiple system nodes, leading to 3-D image defocusing. An MUAV-SAR joint synchronization method based on multiple direct wave pulses exchange is proposed to solve the π -ambiguity problem robustly and improve the synchronization accuracy significantly.

First, the π -ambiguity phase error is estimated through the comparative calculation of delay-phase information extracted from direct wave pulses and the high estimated success probability (99.73%) of the π -ambiguity can be achieved through the noise smoothing.

Second, the synchronization accuracy is improved, that is, the time and phase errors are reduced to about (2/N)1/2 of the existing method by utilizing N stations information fusion to jointly process redundant information of direct wave pulses from multiple synchronization links.

Finally, a four-station UAV SAR real data experiment verifies the effectiveness of the proposed approach.

Published in: IEEE Geoscience and Remote Sensing Letters ( Volume: 21)

Article Sequence Number: 4005705

Date of Publication: 26 February 2024

ISSN Information:

DOI: 10.1109/LGRS.2024.3369684

Funding Agency:

SECTION I.

Introduction

Multistatic unmanned aerial vehicle synthetic aperture radar (MUAV-SAR) is a novel imaging system coherently networked through multiple spatially separated SARs. MUAV-SAR can obtain 3-D radar images by only single pass by forming an elevation aperture through multistatic stations with the advantages of high time efficiency and high resolution [1]. Since the SARs are placed on different UAV platforms independently, the distributed frequency sources will lead to time and phase synchronization errors, which will cause 3-D image defocusing. Therefore, MUAV-SAR requires high-precision time and phase synchronization among multiple stations.

Currently, there are some widely used bistatic SAR synchronization methods [2], which can be roughly divided into three types: the use of high-stability frequency sources [3], the use of global navigation satellite system (GNSS) signals to tame the frequency sources [4], [5], and the establishment of direct wave links for synchronization [6]. Among them, the two-way direct wave link synchronization method is usually adopted due to its relatively highest synchronization accuracy, which has been applied in the TanDEM-X [7] and LuTan-1 mission [8]. However, there are problems in directly applying the existing two-way direct wave link synchronization method to MUAV-SAR. On the one hand, due to the periodicity of the phase, π -ambiguity will be introduced in the phase synchronization process. On the other hand, the synchronization accuracy is limited due to the degraded signal-to-noise ratio (SNR) of direct wave pulses, while the number of system nodes increases, and the residual synchronization errors will lead to 3-D image defocusing. Wang et al. [1] proposed a multistatic synchronization method, which relies on the isolated strong points to solve π -ambiguity and multistatic synchronization phase error, and thus, the high probability of successful solution cannot be guaranteed. Also, the transmission of multiple stations in the time-division system will also lead to a decrease in duty cycle and poorer synchronization accuracy. Therefore, the high robustness and accuracy MUAV-SAR synchronization is still a challenge. In addition, the time and phase error model proposed in [1, Sec. III-A] is beneficial to the analysis of synchronization method, and the model is summarized in the following to demonstrate the problem of bistatic two-way synchronization based on the direct wave pulse exchange.

In this letter, an MUAV-SAR joint synchronization method based on multiple direct wave pulse exchange is proposed. This method first extracts the peak delay and phase of direct wave pulses, then uses the peak envelope phase comparison of the direct wave to solve the π -ambiguity problem of the traditional two-way link synchronization method, and finally utilizes the multistatic synchronization error joint solution based on multiple links error transfer modeling and least-squares method to improve the synchronization accuracy.

This letter is organized as follows. In Section II, an MUAV-SAR synchronization error model proposed in [1] is summarized, and the problem existing in the traditional synchronization method in [6] is analyzed. The proposed method is described and its synchronization accuracy is analyzed in Section III. Section IV presents a numerical simulation and the real MUAV-SAR experiment to verify the proposed method. The study is summarized in Section V.

SECTION II.

Traditional Synchronization Scheme

A. Synchronization Error Model

The signal link configuration of the MUAV-SAR system is shown in Fig. 1(a), in which the direct wave signal among the stations is used to exchange the synchronization information, and the MUAV-SAR system illuminates the scene and receives the echo forming the echo link. As shown in Fig. 1(b), the timing is configured based on time-division waveform. The symbol with or without subscript S is used to represent the echo link and the direct wave link, respectively. Tp denotes the duration of the direct wave pulse, τij denotes the time delay of the direct wave from station i to station j , and τsys denotes the exchange interval set by the system. Similarly, τji,τii,S,τij,S,τji,S,τjj,S , and τsys,S can be defined.

Fig. 1. Signal link and timing configuration of MUAV-SAR with four separate stations.
(a) Six direct wave links for four stations and echo links where only the two transceiver combinations of

T1R1 and T1R2 (T denotes transmitter and R means receiver) are shown.
(b) Timing configuration of the bistatic direct wave link (top) and the monostatic and bistatic echo links (bottom).

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Due to differences in frequency sources, there are inevitably time and phase inconsistencies among these stations, which are called synchronization errors.

1) Time Error Model:

The envelope of the signal transmitted by station i and received by station j is

s T i j (t j; t 0) = w (t j + Δ t i j - τ i j - t 0) (1)

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where w(⋅) is the envelope of the transmitted signal, t0 denotes the generation time of the signal at station i , Δtij=ti−tj denotes the visual clock difference between stations i and j , and τij is the propagation delay of the signal from station i to station j through space; for the direct wave, it is directly from station i to station j , and for scene echo, it is the time delay of the whole process from station i to scene and then received by station j .

In the same way, the envelope of the signal transmitted by station j and received by station i is

s T j i (t i; t 0) = w (t i - Δ t i j - τ j i - t 0) . (2)

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2) Phase Error Model:

The phase error is caused by the accumulation of frequency source errors, the initial phase difference, and phase noise.

The phase of the signal transmitted by station i and received and demodulated by station j is

φ dm_i j (t j; t 0) \approx + 2 π Δ f i j (t j - t 0) + 2 π f 0 Δ t i j - 2 π f 0 τ i j + φ i, ini - φ j, ini + n i (t j) - n j (t j) (3)

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where f0 is the nominal carrier frequency of all stations, Δfij=Δfi−Δfj is the carrier frequency error between stations i and j , φi,ini is the initial phase of the carrier, and ni(tj) is the phase noise.

Similarly, the phase of the signal transmitted by station j and received and demodulated by station i is [1]

φ dm_j i (t i; t 0) \approx - 2 π Δ f i j (t i - t 0) - 2 π f 0 Δ t i j - 2 π f 0 τ j i + φ j, ini - φ i, ini + n j (t i) - n i (t i) . (4)

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The time error and phase error in the signal will cause the offset and defocus of the 3-D imaging results, which needs to be estimated.

B. Bistatic Two-Way Synchronization

The traditional bistatic two-way synchronization relies on increasing system complexity, that is, adding the direct wave links in exchange for high precision.

The first step is to extract the peak delay and phase of the direct wave pulses. The echo transmitted by station i and received by station j can be expressed as

s C i j (t j; t 0) = s T i j (t j; t 0) \cdot exp [j φ dm_i j (t j; t 0)] + n r (t j) (5)

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where nr(⋅) represents the receiver noise.

The bistatic time error between stations i and j can be estimated by

Δ t^i j = t P , j i - t P , i j 2 (6)

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where tP,ij is the apparent time of the direct wave transmitted by station i and received by station j at station j as tP,ij=−Δtij+τij+t0+εt,ij . tP,ji is the apparent time of the direct wave transmitted by station j and received by station i at station i as tP,ji=Δtij+τji+t0+εt,ji according to (1) and (2). εt,ij and εt,ji are the extraction errors caused by receiver thermal noise.

After time synchronization, the signal envelope received by station i and transmitting station j is

s T C i j (t j; t 0) = sin c (1 ρ t (t j - τ i j - t 0 - t ε)) (7)

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where ρt is the range resolution. tε=(εt,ij−εt,ji)/2 is the combination of peak delay extraction errors, which is affected by the SNR of the direct wave.

The bistatic phase error can be estimated by

Δ φ^i j = φ dm _ j i ( t i ; t 0 ) - φ dm _ i j ( t j ; t 0 ) 2 + φ ε (8)

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where the phases φdm_ij and φdm_ji defined in (3) and (4) are achieved at the peaks of the direct waves after the range compression at stations i and j , respectively. φε=(εφ,ij−εφ,ji)/2 is the combination of peak delay extraction errors, which is affected by the SNR of the direct wave. Note that the “divide by two” step in formula (8), which changes the period of the phase from 2π to π and may result in a π error between the extracted value of the phase and the true value. We use kij to represent the π -ambiguity number, where kij can take values of 0 and 1.

Thus, the phase of the signal transmitted by station i and received by station j after the bistatic phase synchronization yields

φ c, i j = - 2 π f 0 (τ c, i j + t ε) + k i j π - φ ε . (9)

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From formula (9), we know that kijπ may cause the phase compensation result to differ by π from the true value. Thus π -ambiguity should be estimated and compensated.

SECTION III.

Proposed Method

The proposed method is mainly divided into two steps. First, the peak delay and phase of the direct wave during the synchronization period are extracted and the π -ambiguity in each group of bistatic synchronization is solved. Then, calculate the time and phase errors between the other two stations based on the reference time and phase errors, and finally compensate the calculation results in the scene echo.

A. Delay-Phase Joint Solution to Time-Invariant Phase Error (π -Ambiguity)

Equations (7) and (9) can be calculated by making the difference between the peak phase after compensation and the phase corresponding to the peak delay after compensation to solve π -ambiguity

k i j, m = φ c , i j + 2 π f 0 τ c , i j π + φ ε + 2 π f 0 t ε π (10)

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where subscript m represents the slow time sequence number. In general, φε and tε are random variables that follow a Gaussian distribution with mean of zeros and a variance of σφ and στ , respectively. The linear combination of Gaussian random variables still follows a Gaussian distribution. Thus, we assume that kij follows a Gaussian distribution with a mean of μk=0 and 1 and a variance of σk . Since π -ambiguity only exists in two cases of 0 and π , the tolerable upper limit of error of π -ambiguity solution should be less than 0.5π .

In practice, data accumulated over a period of time can be used for statistical analysis to reduce the impact of noise, that is,

k^i j \approx k i j, M = 1 M \sum m = 1 M k i j, m . (11)

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As shown in formula (11), we use the expectation (mean) of random variables as an estimate of the true value. Assuming that the accumulation point number is M , the variance of kij,M will decrease to (σk/(M)1/2) . According to the 3σ principle, we believe

| k i j, M - μ k | < 3 σ k M - - \sqrt = 3 σ k, M < 1 2 . (12)

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Therefore, by setting an appropriate σk,M , the data mean kij,M can be regarded as an estimation of μk , and a relatively high probability (99.73%) of successful solution can be guaranteed.

B. Multistatic Joint Solution to Time-Varying Delay and Phase Error

As shown in Fig. 2, there are six direct wave links for four stations. The time and phase errors in direct wave links have the linear constraint relationship, for example, the synchronization error between stations 2 and 4 can be represented by the error between stations 1 and 4 and stations 1 and 2, i.e., ΔX24=ΔX14−ΔX12 . To expand, assuming that there are N stations, there are C2N (C2N represents the combinatorial number of two elements taken from N different elements calculated as N(N−1)/2 ) dependent direct wave links. Generally, the degrees of freedom of C2N direct wave links are bigger than that of N independent clocks. Therefore, the joint synchronization is beneficial to improve multistatic time and phase error estimation accuracy.

Fig. 2.

Schematic of synchronization time and phase errors linear relationship for multiple transceiver combination.

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First, the time synchronization error measurement or phase synchronization error measurement Δx^ij on each synchronization link is obtained by (6) and (8), where i=1,2,…,N and j=i+1,i+2,…,N . Then, take station 1 as a reference, set the time synchronization error or phase synchronization error to be solved as X=[Δx12,Δx13,…,Δx1N]T , and finally use the linear relationship of time synchronization error or phase synchronization error between stations Δxij=Δx1j−Δx1i ; we can write a system of linear equations

A X = Δ B (13)

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where ΔB=[Δx^12,Δx^13,…,Δx^1N,Δx^23,…,Δx^(N−1)N]T is the measurement matrix of time error or phase error, which is affected by factors such as SNR and has measurement noise. A is the coefficient matrix of the linear equation

A = [I A d] C 2 N \times (N - 1) (14)

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where I is the (N−1) -order identity matrix and Ad represents the linear relationship between errors

A d = ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ - 1 - 1 ⋮ 0 10 ⋮ 0 01 ⋮ 0 \dots \dots \dots 00 ⋮ - 1 00 ⋮ 1 ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ C 2 N - 1 \times (N - 1) . (15)

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The ordinary least-squares solution of the linear equation system (13) is obtained as

X^ls = arg min X ∥ A X - Δ B ∥ 22 . (16)

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Observing the structure of the matrix A , we can see that the matrix A has full rank, so ATA is nonsingular, and the linear equation system (13) has a unique solution

X ls = (A T A) - 1 A T Δ B . (17)

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Finally, linearly combine the time synchronization error Xls with station 1 as the reference, and then, the joint calculation result of the time or phase synchronization error that needs to be compensated by each link echo can be obtained

Y syn = A \cdot X ls . (18)

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C. Accuracy Comparison of Synchronization Error

When the synchronization rate is high, the synchronization accuracy is only related to the SNR of the direct wave.

The relationship between the peak time delay and phase extraction accuracy and SNR after pulse compression is

σ τ = 3 - \sqrt π B 2SNR - - - - - \sqrt, σ φ = 1 2SNR - - - - - \sqrt (19)

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where στ is the standard deviation (STD) of the peak delay estimate, σφ is the STD of the peak phase estimate, B is the signal bandwidth, and SNR is the signal-to-noise ratio after the direct wave pulse compression.

1) Requirements of Time and Phase Extraction Accuracy for π -Ambiguity Estimation:

According to formula (10), we have the expression of variance of π -ambiguity solution, that is,

σ k = 1 2 - \sqrt π σ 2 φ + (2 π f 0 σ τ) 2 - - - - - - - - - - - - \sqrt (20)

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where the factor (2)1/2 in the denominator is due to the “divide by two” operation in the bistatic two-way synchronization.

From this, we can obtain the success probability of solving π -ambiguity, that is,

f (σ k) = \int μ k + 1 2 μ k - 1 2 1 2 π - - \sqrt σ k exp [- ( x - μ k ) 2 2 σ 2 k] d x . (21)

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It is worth mentioning that combining formulas (12), (19), and (20), a typical value of M can be obtained to guarantee the high success probability of solving π -ambiguity, which is set to 100 for current system.

2) Synchronization Accuracy Analysis:

The estimated value of the synchronization error after multistatic joint is shown in formula (17), where the measured value of the synchronization error ΔB can use the sum of the true value ΔBtrue and the measurement error ΔBerr represents

X ls = (A T A) - 1 A T (Δ B true + Δ B err) . (22)

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The measurement error is mainly caused by the thermal noise of the receiver, so it can be reasonably assumed that the measurement errors of the direct waves of each link are independent of each other, and the mean value is 0 and the variance is σ2 , so the residual error STD after the joint solution is

σ X ls_i = \sum (\partial X ls _ i \partial Δ B err i σ Δ B err i) 2 - - - - - - - - - - - - - - - - - - \sqrt = 2 N - - - \sqrt σ (23)

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where i represents the i th element in the solution vector.

It can be seen from formula (23) that after the joint solution, the residual error STD is reduced to (2/N)1/2 before the joint solution, where N is the number of radars in the system. The comparison results of the synchronization accuracy between the proposed joint solution synchronization method and the traditional method in [6] are shown in Fig. 3. The synchronization accuracy of the proposed method is improved as the number of stations increases, while the traditional method does not change.

Fig. 3.

Comparison of the proposed method for four (red star), 16 (yellow dot), and 128 (purple dotted) stations and the traditional method in [6] for 416128 stations (blue circle) in terms of (a) time and (b) phase synchronization accuracy. The synchronization accuracy is evaluated by the STD of synchronization errors. The plots show the variation curve of synchronization error with SNR.

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SECTION IV.

Experiment

In this section, a computer simulation and a real data experiment are performed based on the parameters of MUAV-SAR data, which are shown in Table I. The four-station MUAV-SAR illuminates the targets array and collects scene echoes and direct waves at the same time.

TABLE I Radar Parameters

First, the targets array simulation is conducted. The configuration of the targets array is shown in Fig. 4 and the inconsistency in the clocks is simulated based on [6]. The π -ambiguity statistical results are shown in Fig. 5. The solutions of the π -ambiguity are (0,0,π,π,0,π) for (T1R2,T1R3,T1R4,T2R3,T2R4,T3R4) , which is consistent with the real result, where “TiRj ” indicates that station i transmits the signal and station j receives the echo.

Fig. 4.

3-D configuration (top) of targets array in simulation with the views (bottom) along range (middle), azimuth (right), and elevation (left). The targets are distributed on the equal range-Doppler (RD) curves in elevation and overlapped on the RD plane.

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$Fig. 5. - $\pi $ -ambiguity judgment results $\hat {k}_{ij,m}\pi $ and the statistical histogram of $k_{ij,m}\pi $ for six transceiver combination. The background color of each subfigure represents the judgment result. The black solids show the fitting curve of the statistical histogram. The red solids show the mean of $k_{ij,m}\pi $ . The pink boxes show the judgment threshold for the mean of $k_{ij,m}\pi $ , and the judgment result is determined as the middle value ( $T_{1}R_{2}, T_{1}R_{3}, $ and $T_{2}R_{4}$ for 0 and the others for $\pi $ ) of the coordinate axis (the red solid in boxes) or the contrast value (the red solid out of boxes).$

Fig. 5.

π -ambiguity judgment results k^ij,mπ and the statistical histogram of kij,mπ for six transceiver combination. The background color of each subfigure represents the judgment result. The black solids show the fitting curve of the statistical histogram. The red solids show the mean of kij,mπ . The pink boxes show the judgment threshold for the mean of kij,mπ , and the judgment result is determined as the middle value (T1R2,T1R3, and T2R4 for 0 and the others for π ) of the coordinate axis (the red solid in boxes) or the contrast value (the red solid out of boxes).

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Under the condition that the SNR of the direct wave pulses after range compression is 30 dB, the synchronization accuracy of the proposed method and the traditional two-way link method in [6] is compared, as shown in Table II. After synchronization with the proposed method, the time and phase errors are reduced to about 1/(2)1/2 , which is consistent with the result given by formula (23), and it shows that the proposed method offers superior synchronization accuracy.

TABLE II Comparison Results of Synchronization Accuracy With the Traditional Method in [6] and the Proposed Method

The 3-D imaging results are obtained based on the 3-D backprojection (BP) method. Also, the results after synchronization utilizing the traditional method in [6], the method in [1], and the proposed method are shown in Fig. 6, whose image entropy is 16.5481, 16.2084, and 15.9105 respectively. The images indicate that the method in [1] is unable to adapt to scenarios with nonisolated strong targets, while the proposed method is robust and has good adaptability to diverse scenes.

Fig. 6.

Profiles of 3-D imaging results and the view along the azimuth direction after synchronization with (a) and (e) traditional method in [6], (b) and (f) method in [1], (c) and (g) proposed method, and (d) and (h) targets array configuration.

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In September 2021, a single-pass MUAV-SAR 3-D imaging experiment was carried out in Pinggu, Beijing. The overall 3-D imaging result of the imaging scene is shown in Fig. 7. The six-story building complex is clearly visible in the picture, and the corner reflector is well-focused processed whose Rayleigh resolution is 2.7×1.1×4.4 m (range × azimuth × elevation). The height difference between the building and the bungalow is about 20 m, which is consistent with the actual situation. The real data of MUAV-SAR verify the effectiveness of the proposed synchronization method.

Fig. 7.

3-D imaging real data processing results of the field test scene. The left image shows the (2-D elevation coloring map of the observation area, and the right plots show the 3-D imaging results of the corner reflector (top) using the traditional method in [6] (left) and the proposed method (right), and the 3-D enlarged view (left) and the optical view (right) of the six-story building complex (bottom).

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SECTION V.

Conclusion

In this letter, aiming at the problems of the occurrence of π -ambiguity phase error and insufficient synchronization accuracy of the traditional two-way direct wave pulse exchange synchronization method for MUAV-SAR system, a multistatic joint UAV-SAR synchronization method is proposed. First, compare the peak delay and phase of the direct wave to solve π -ambiguity and achieve the probability of successful solution to 99.73% through the azimuth accumulation (noise smoothing). Then, the synchronization accuracy improves by establishing error transfer equations for all links and jointly solving synchronization errors through least squares. Eventually, the proposed method was applied to the four-station UAV SAR real data processing to obtain an elevation resolution of 2.5 m. Future work will focus on the joint solution of synchronization and motion errors.

Aerospace Electronic and Defense Systems

Thursday, March 7, 2024

Multistatic UAV SAR Joint Synchronization Based on Multiple Direct Wave Pulses Exchange | IEEE Journals & Magazine | IEEE Xplore

Multistatic UAV SAR Joint Synchronization Based on Multiple Direct Wave Pulses Exchange | IEEE Journals & Magazine | IEEE Xplore

Abstract:

Introduction

Traditional Synchronization Scheme

A. Synchronization Error Model

1) Time Error Model:

2) Phase Error Model:

B. Bistatic Two-Way Synchronization

Proposed Method

A. Delay-Phase Joint Solution to Time-Invariant Phase Error (π -Ambiguity)

B. Multistatic Joint Solution to Time-Varying Delay and Phase Error

C. Accuracy Comparison of Synchronization Error

1) Requirements of Time and Phase Extraction Accuracy for π -Ambiguity Estimation:

2) Synchronization Accuracy Analysis:

Experiment

Conclusion

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