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.
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
sTij(tj;t0)=w(tj+Δtij−τij−t0)(1)
View Source 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
sTji(ti;t0)=w(ti−Δtij−τji−t0).(2)
View Source2) 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
φdm_ij(tj;t0)≈+2πΔfij(tj−t0)+2πf0Δtij−2πf0τij+φi,ini−φj,ini+ni(tj)−nj(tj)(3)
View Source 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]
φdm_ji(ti;t0)≈−2πΔfij(ti−t0)−2πf0Δtij−2πf0τji+φj,ini−φi,ini+nj(ti)−ni(ti).(4)
View SourceThe
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
sCij(tj;t0)=sTij(tj;t0)⋅exp[jφdm_ij(tj;t0)]+nr(tj)(5)
View Source where
nr(⋅)
represents the receiver noise.
The bistatic time error between stations i
and j
can be estimated by
Δt^ij=tP,ji−tP,ij2(6)
View Source 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
sTCij(tj;t0)=sinc(1ρt(tj−τij−t0−tε))(7)
View Source 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
Δφ^ij=φdm_ji(ti;t0)−φdm_ij(tj;t0)2+φε(8)
View Source 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
φc,ij=−2πf0(τc,ij+tε)+kijπ−φε.(9)
View SourceFrom 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
kij,m=φc,ij+2πf0τc,ijπ+φε+2πf0tεπ(10)
View Source 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^ij≈kij,M=1M∑m=1Mkij,m.(11)
View SourceAs 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
|kij,M−μk|<3σkM−−√=3σk,M<12.(12)
View SourceTherefore, 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.
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
AX=ΔB(13)
View Source 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=[IAd]C2N×(N−1)(14)
View Source where
I
is the
(N−1)
-order identity matrix and
Ad
represents the linear relationship between errors
Ad=⎡⎣⎢⎢⎢⎢−1−1⋮010⋮001⋮0⋯⋯⋯00⋮−100⋮1⎤⎦⎥⎥⎥⎥C2N−1×(N−1).(15)
View SourceThe ordinary least-squares solution of the linear equation system (13) is obtained as
X^ls=argminX∥AX−ΔB∥22.(16)
View SourceObserving 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
Xls=(ATA)−1ATΔB.(17)
View SourceFinally, 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
Ysyn=A⋅Xls.(18)
View SourceC. 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
στ=3–√πB2SNR−−−−−√,σφ=12SNR−−−−−√(19)
View Source 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,
σk=12–√πσ2φ+(2πf0στ)2−−−−−−−−−−−−√(20)
View Source 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(σk)=∫μk+12μk−1212π−−√σkexp[−(x−μk)22σ2k]dx.(21)
View SourceIt 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
Xls=(ATA)−1AT(ΔBtrue+ΔBerr).(22)
View SourceThe
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
σXls_i=∑(∂Xls_i∂ΔBerriσΔBerri)2−−−−−−−−−−−−−−−−−−√=2N−−−√σ(23)
View Source 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.
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.
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.
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.
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.
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.
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.
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