A Novel Interpolation Method for TDOA and FDOA Estimation based on Second-order Cone Programming

Full text

The 19th International Radar Symposium IRS 2018, June 20-22, 2018, Bonn, Germany
978-3-7369-9545-1 ©2018 DGON
1
A Novel Interpolation Method for TDOA and FDOA Estimation
based on Second-order Cone Programming
Zhixin Liu
*
, Dexiu Hu
**
, Yongjun Zhao
***
, Yongsheng Zhao
****
National Digital Switching System Engineering and Technological Research Center (NDSC)
Zhengzhou, CHINA
*
email: liuzhixin54@sina.com,
**
email: paper_hdx@126.com,
***
email: zhaoyjzz@163.com,
****
email: ethanchioa@aliyun.com
Abstract: Locating an emitting source based on time difference of arrival (TDOA) and
frequency difference of arrival (FDOA) measurements from passive sensors has widely
been used in radar and sensor networks. The localization accuracy is directly influenced
by the TDOA and FDOA estimation accuracy. While considerable research works have
been performed on algorithm development, limited accuracy has been paid in TDOA and
FDOA estimation. Therefore, a novel joint TDOA and FDOA estimation interpolation
method with sub-sample accuracy based on second-order cone programming (SOCP) is
presented in this paper. It combines the advantages of "soft" constraint and "hard"
constraint to build optimization models, and obtains the interpolation surface through the
SOCP, which can change the discrete ambiguity surface into continuous surface. The
estimation accuracy of TDOA and FDOA is no longer forced to lie on sampling interval
and sampling time. Simulation results indicate that the proposed method is effective and
outperforms the other traditional interpolation algorithms regarding the estimation
accuracy.
1. Introduction
Time difference of arrival (TDOA) and frequency difference of arrival (FDOA) estimation is
aimed at determining the time and Doppler shift between two signals. It is widely used in many
applications such as radar, sonar, and communication [1]. In passive localization for instance,
these two parameters contain the range and relative velocity information of the target, and they
can be used for target localization. So the key element in achieving better localization accuracy
is to accurately determine the TDOA and FDOA measurements.
For joint TDOA and FDOA estimation, the cross ambiguity function (CAF) is one of the classic
methods to realize [2]. In order to get precise estimation result, a considerable number of
sampling points should be taken into account in the CAF calculation, which is shown in Figure
1(a). Due to the TDOA and FDOA estimation quantized by sampling interval and sampling
time, their estimation accuracy is limited fundamentally by the sampling interval and sampling
time. Therefore, some signal interpolation methods [3][4] have been put forward to reduce this
limitation (as shown in Figure 1(b)). Reference [4] used splines to produce a continuous time
representation of a reference signal and then computes an analytical matching function between
this reference and a delayed signal. However, with the increasing of the number of sample
points for signal, the CAF also must be computed at a higher sampling rate and the computation
cost of this method can be significant.
In order to overcome above shortcomings, the post-interpolation strategy has appeared, as
shown in Figure 1(c). These type of interpolation methods changed the discrete ambiguity
surface into continuous surface near the peak, which can produce the best match with original
discrete data. The more accurate estimations of TDOA and FDOA can be then directly

2
determined by searching the location of the peak in approximate surface. So the TDOA and
FDOA estimation accuracies are no longer forced to lie on discrete samples and sampling time
respectively. Several methods have been described to apply the post-interpolation problem,
such as Lagrange interpolation [5], parabolic interpolation [6], cosine interpolation estimator
[7], et al. These plane interpolation methods can only estimate TDOA and FDOA separately.
To achieve joint TDOA and FDOA estimation, surface interpolation methods have been applied
to the pattern CAF including least square surface fitting, moving least square method, et al [8]-
[9]. Although these above interpolation estimators can achieve sub-sample accuracy, it is not
still high enough especially in high signal-to-noise ratio (SNR).
()
1
rn
()
2
rn
()
,CAF m k
()
1
rn
()
,CAF m k
()
1
rt

()
,CAF m k
()
,CAF t f
ˆ
,
d
f
τ

ˆ
,
d
f
τ

ˆ
,
d
f
τ

Compute CAF
Compute CAF
()
2
rn
Interpolation
Interpolation
Compute CAF
()
a
()
b
()
c
()
1
rn
()
2
rn
Figure 1. Common strategies for TDOA and FDOA estimation.
Motivated by these facts, we propose a new interpolation method for TDOA and FDOA
estimation based on second-order cone programming (SOCP) in this paper. The proposed
method combines the advantages of "soft" constraint and "hard" constraint to construct
optimization models, and obtains the approximate surface through the SOCP, which is an
important part of nonlinear programming, and has the merits of fast convergence and
appropriate computational complexity [6]. The TDOA and FDOA accuracy can be improved
significantly compared with conventional methods. The new method is introduced in section 2,
its performance evaluation is shown in section 3 and a brief conclusion is given in section 4.
This paper contains a number of symbols. Following the convention, we represent the matrixes
and vector as bold italic case letters. The operations
()
T
⋅
denotes the matrix transpose.
|| ||⋅
stands for the Euclidean norm.
1i×
0
and
1i×
1
are both transverse vectors
1i×
of zero and one.
8
i
E
also represents the transverse vector
18×
, which is set to 1 in i
th
element and 0 otherwise.
2. Proposed Method
A. Initial Estimation
The models of discrete received signals from two sensors are given by
() () ()
() ( )
()
()
()
0
11
2
20 2
,1,2,,
d
jfn
rn sn nn nN
rn Asn e nn
πτϕ
τ
−+
=+ =
=⋅ − +

(1)
where
()
s
n
is the transmitted signal,
A
denotes a real attenuation constant and
ϕ
is a constant
phase difference.The
0
τ
and
d
f
resecptively stand for the TDOA and FDOA between the two
received signals. The received modeling assuptions for the noises
()
i
nn
are the zero-mean
wide-sense stationary (WSS) Gaussian process [1], and they are independent from each other,
also independent from
()
s
n
.
Once the discrete time signals from receivers is obtained, there are variety of methods to
estimate the positioning measurements. The CAF is the classical and straightforward method
to estimate TDOA and FDOA between two received signals [2], which is defined as

3
() ()( )
12
12
0
,
s
s
kf
N
j
nT
N
n
CAF m k r n r n m e
π
−−
∗
=
=+

(2)
where
1
s
s
Tf=
is the sampling interval,
m
is the discrete time delay index and
s
mT
τ
=
, the
k
is the discrete Doppler shift index and
s
f
kNT k f==Δ,
s
T
and
f
Δ
are respectively the
time and frequency resolution. Searching the peak of CAF which is
() ()
00 ,
,argmax ,
mk
mk CAFmk=
(3)
We can obtain the TDOA estimation
0
ˆ
s
mT
τ
=
and FDOA estimation
0
ˆ
s
f
kNT=
. According
to the [2][8], the estimation accuracy of this method highly relies on the sampling interval
s
T
and sampling time
s
TNT=
, even though the estimator is optimal and the SNR is high. So it is
necessary to use interpolation method to achieve sub-sample estimation accuracy.
B. Optimization Modelling Using GEA Method
GEA method can combine the advantage of interpolation and fitting, and tends to realize high
accuracy approximation instead of an original discrete data [10]. So in this section, we resorts
to GEA method to establish the optimization models about interpolation surface.
τ
f
o
1
x
2
x
3
x
1
y
2
y
3
y
()
ˆ
ˆ,
d
f
τ
13
z
12
z
11
z
21
z
22
z
23
z
33
z
32
z
31
z
Figure 2. Quadratic surface fitting method.
As shown in Figure 2, the peak of CAF result which means the initial estimation result is located
at
2222
(, , )
x
yz
in the discrete ambiguity-plane. In this new coordinates, the
|(,)|CAF m k
can be
regarded as an approximate quadratic surface around the neighbourhood points of its peak [8],
which is defined as
()
22
12 3456
,Uxy ax ay axy ax ay a=+++++
(4)
where
(1,2,,6)
i
ai=
are the unknown surface coefficients. We use this very simple
approximation to match the original CAF surface and the more accurate TDOA and FDOA
estimation can be performed by finding the peak of the new approximate surface, which the
coordinates of the peak satisfy
()
()
13 4
235
,20
,20
Uxy ax a y a
x
Uxy ay ax a
y
∂=++=
∂
∂=++=
∂
(5)
Solving these equations, we obtain the new TDOA and FDOA estimates expressed as
24 35
0max 2
312
15 43
max 2
312
2
ˆ4
2
ˆ
4
d
aa aa
xaaa
aa aa
fy aaa
τ
−
==
−
−
==
−
(6)

4
In order to obtain the optimal approximate surface, we choose nine points including the peak
point and eight pints around it whose values are assumed as
(, 1,2,3)
ij
zij=
. Then, we establish
a sampling point selection strategy shown as Figure 3 and every grid stands for each discrete
point shown in Figure 2.
: on the interpolation surface.
: may not be on the interpolation surface.
1
x
2
x
3
x
1
y
2
y
3
y
Figure 3. Sampling point selection strategy.
In Figure 3, black grids represent that the discrete points are on the interpolation surface and
used to construct the "hard" constraint, which means the surface value equal to original value.
We can obtain
() ( ) ()
() ()
1 1 11 2 2 22 1 3 13
3 1 31 3 3 33
,,,,,,
,,,
Uxy z Ux y z Uxy z
Ux y z Ux y z
===
== (7)
While the white grids represent that each discrete point may or may not be on the interpolation
surface and used to construct the "soft" constraint, which means that minimize the sum of error
norms between surface value and original value. We have
() () () ()
1 2 12 2 1 21 2 3 23 3 2 32
min : , , , ,Uxy z Ux y z Ux y z Ux y z−+ −+ −+ −
(8)
Combining (7) with (8) together yields the optimization model
()
() ()
()
33
11
1 3 13 3 1 31
3
1
min ,
,,,
.. ,0
ij ij
ij
ii ii
i
Uxy z
Uxy z Ux y z
st Uxy z
==
=
−
==

−=




(9)
The above optimization model is convex problem which has only one apex point so that it can
be solved by SOCP. Due to its rapid convergence and reasonable computational complexity,
this solution tool is a key technique and research focus in solving optimization problems
recently [6]. It is worth noting that SOCP can be easily tailored to obtain the interpolation
surface. Therefore, in next section, we convert (9) into standard SOCP model to solve.
C. Accurate TDOA and FDOA Estimation Based on SOCP
From [6], the standard SOCP model is defined as
min
,1,2,,
..
T
T
iii i c
di N
st +≤ + =

=


qy
Ay b c y
Fy g

(10)
where
α
∈
yR
is the optimization variable,
α
∈
qR
,
()
1
i
i
β
α
−×
∈AR
,
1
i
i
β
−
∈bR
,
i
α
∈cR
,
i
d∈
R
,
1
g
λ
×
∈
R
. The number of constraint conditions is
c
N
.
,,
i
α
βλ
respectively represent
the length
of every vector and
R
represents the set of real numbers.
The inequality constraint of (10) can be rewritten by

5
,1,2,,
i
T
i
i
c
i
i
d
y
Qcone i N
β
 
+∈ =
 


c
b
A

(11)
where
i
Qcone
β
is the second-cone of the real field. The equality constraint of (10) can be
reformulated by
{}
λ
−∈gFy 0
(12)
where
{}
λ
0
is the zero-cone model.
Utilizing a set of auxiliary variable
21 12 23 32
,,,
η
ηηη
, the optimization problem (9) can be
equivalently written as
() ()
() ()
() ()
() ()
()
21 12 23 32
1 2 12 12 2 1 21 21
2 3 23 23 3 2 32 32
1 1 11 1 3 13
2 2 22 3 1 31
33 33
min
,,,,
,,,
.. , , ,
,,,
,
Uxy z Ux y z
Ux y z Ux y z
st Uxy z Uxy z
Ux y z Ux y z
Ux y z
η
ηηη
ηη
ηη
+++
−≤ −≤

−≤ −≤

==

==

=


(13)
Then, according to (10)-(12), the (13) can be rewritten in SOCP model by introducing an new
auxiliary variable
[]
14 14 16
T
×××
=q100
and optimization variable
T
TT


=


ya
η
{}
12
22
816 816
12 21
1 8 12 1 8 21
34
22
816 816
23 32
18 23 18 32
1
18
min
00
,
00
,
..
1,
T
TT
TT
T
ii ii
Qcone Qcone
zz
Qcone Qcone
st zz
zi
××
××
××
××
×
 
 
+∈ +∈
 
 
−−
 
 
 
 
+∈ +∈
 
 
−−
 
 

−∈=

qy
EE
yy
nn
EE
yy
nn
ny
00
00
00
00
00
{} {}
11
31 1 8 31 13 1 8 13
2, 3
TT
zz
××









 
−∈−∈
 
ny ny，0000
(14)
where
[]
()
[]
123456
22
21 12 23 32 11 13 31 33
1,1,2,3
T
T
ij i j i j i j
T
aaaa a a
xyxyxy ij
ηηηηηηηη
=

==

=
a
n
η
(15)
After using SeDuMi [11] to solve the optimization variable
y
, we can obtain the approximated
surface
(, )Uxy
and its coefficients. Finally, the more accurate TDOA and FDOA estimation
can be performed by using (6) to obtain the apex of the
(, )Uxy
.
D. Computation Complexity Analysis
If
O
represents the computation complexity;
o
N
is the number of original discrete data of CAF;
n
is the order of interpolation function;
M
is the number of undetermined coefficients of
approximate surface function;
C
is the number of interpolated points; The number of the

6
inequality constraints is
c
N
and
co
NC N

. The computation complexity [12] of proposed
method is
()
()
()
2
2
1
21
c
N
ci cc
i
ON ONNC M
αβ
=
=−+

.
Table 1. Computation complexity of interpolation methods
Interpolation method Computation complexity
Plane interpolation Methods
()
OCn
Method in [8]
()
3
OM
Surface interpolation Methods
()
o
ONM
Proposed Method
()
()
2
21
cc
ONNC M−+
Table 1 indicates the computation complexity comparison among the proposed method and
other conventional interpolation methods. From the Table 1, even though the computation
complexity of proposed method is not the minimum, it does not increase much compared with
plane interpolation methods. Moreover, compared with 3-D interpolation methods, the
proposed method is much lower. This is because our new method just uses nine points to solve
this problem rather than using the original data sequence as other interpolation methods use.
3. Performance Evaluation
This section contains numerical simulations to demonstrate the proposed method and to
compare its performance with that of other different methods as well as the CRLB [2]. TDOA
and FDOA will be added to the received signal, which is assumed to be BPSK modulated with
symbol rate of 2MHz, carrier frequency of 1.5GHz and sampling frequency of 9MHz (T
s
=
111.11ns). The estimation accuracy of TDOA and FDOA are respectively defined as
5000 2
0
1
( ) (1/ 5000)
i
i
RMSE
τττ
=
=−

and
2
5000
1
( ) (1/ 5000)
i
ddd
i
RMSE f f f
=
=−
, where
0
τ
and
d
f
represent the real TDOA and FDOA, which respectively equal to
2μs
and
1KHz
,
i
τ
and
i
d
f
are the TDOA and FDOA of the i
th
simulation result, 5000 is the number of independent
Monte-Carlo runs.
(a) Using CAF only (b) After using proposed method
Figure 4. A schematic diagram of interpolation in CAF.
The first simulation is to demonstrate the proposed method. Fig. 4(a) is the results of using
CAF only and Figure 4(b) is the result of proposed method. It can be observed that after using
interpolation method, the original discrete data of CAF is transformed into continuous surface
near the peak, which produces the best match with discrete function. So the TDOA and FDOA

7
estimation can be obtained by using (6) and their accuracies are no longer forced to lie on
discrete samples and sampling time.
(a) TDOA estimation (b) FDOA estimation
Figure 5. Comparison of RMSEs of estimation obtained by different interpolation methods versus SNR,
which varies from -10dB to 30dB. The number of snapshots is 5000.
The second simulation is to compare the proposed method with other interpolation algorithms.
Fig. 5 shows a comparison of RMSEs of TDOA and FDOA estimation using the proposed
method and other interpolation methods. We plot curves of RMSEs corresponding to SNR using
different interpolation methods, as well as the CRLB [2]. The plane interpolation method
denotes the method in [7], whereas the surface fitting and moving least square method
represents the algorithm in [8] and [9] respectively. It can be observed that the RMSEs of all
the algorithms also decrease as the SNR increases but the estimation precision of the proposed
method is much closer than that of the plane interpolation methods and surface interpolation
methods especially in high SNR. Therefore, the proposed method is much closer to the CRLB
than the other three algorithms, meaning an improvement in estimation performance in both
TDOA and FDOA estimation.
4. Conclusion
A novel joint TDOA and FDOA estimation method based on the SOCP is presented in this
paper. The proposed method combines the advantages of "soft" constraint and "hard" constraint
to build optimization models and changes the discrete ambiguity surface into continuous surface
by using SOCP. The more accurate TDOA and FDOA estimation can be directly obtained by
computing peak coordinates formula of approximate surface. Simulation results indicate that
this estimator achieves sub-sample accuracy, and outperforms other conventional interpolation
methods in terms of estimation precision over a board range of conditions.
Acknowledgement
The authors would like to thank the anonymous reviewers for their careful reading and
constructive suggestions which provide an important guidance for our paper writing and
research work. This work is supported by the National Natural Science Foundation of China
under grant 61703433.
RMSE(ns)
RMSE(Hz)

8
References
[1] Kim, Dong Gyu, et al, "Computationally Efficient TDOA/FDOA Estimation for Unknown
Communication Signals in Electronic Warfare Systems", IEEE Transactions on Aerospace &
Electronic Systems, 2017, vol. 99, pp. 1-1.
[2] Stein, S, "Algorithms for ambiguity function processing", IEEE Transactions on Acoustics
Speech & Signal Processing, 2003, vol. 29, pp. 588-599.
[3] Teixeira, C. A., M. G. Ruano, and W. C. A. Pereira, "An Interpolation-Free and Fitting-Less
Sub-sample Time-Delay Estimation Algorithm", The International Conference on Health
Informatics, 2014, pp. 288-291.
[4] Viola, F, and W. F. Walker, "A spline-based algorithm for continuous time-delay estimation
using sampled data", IEEE Transactions on Ultrasonics Ferroelectrics & Frequency Control,
2005, vol. 52, pp. 80-93.
[5] Tjahjono, Anang, et al, "Modelling non-standard over current relay characteristic curves using
combined lagrange polynomial interpolation and curve fitting", International Seminar on
Intelligent Technology and ITS Applications IEEE, 2017.
[6] Liu, Zhixin, et al, "A Novel Time Delay Estimation Interpolation Algorithm Based on Second-
Order Cone Programming", IEICE Transactions on Communications, 2016, vol. 99, pp. 1311-
1317.
[7] De Jong, P. G. M, et al, "Determination of tissue motion velocity by correlation interpolation
of pulsed ultrasonic echo signals", Ultrasonic Imaging, 1990, vol. 12, pp. 84-98.
[8] Tao, Ran, W-Q. Zhang, and E-Q. Chen, "Two-stage method for joint time delay and Doppler
shift estimation." IET Radar, Sonar & Navigation, 2008, vol. 2, pp. 71-77.
[9] Zhang, Huaiqing, et al, "Measurement data fitting based on moving least squares
method." Mathematical Problems in Engineering, 2015, vol. 2015.
[10] Zhang, Wenjuan, G. Zhang, and J. Yang, "Generalized extended interpolation method", Basic
Sciences Journal of Textile Universities, 2007, vol. 20.
[11] Sturm, Jos F, "Using SeDuMi 1.02, a MATLAB toolbox for optimization over symmetric
cones", Optimization methods and software, 1999, vol. 11, pp. 625-653.
[12] Bai Y, Ma P, Zhang J, "A polynomial-time interior-point method for circular cone programming
based on kernel functions", Journal of Industrial & Management Optimization, 2017, vol 12(2)
, pp. 739-756.