A novel signal processing approach for LEO space debris based on a fence-type space surveillance radar system

Abstract

The increase in space debris can seriously threaten regular activities in the Low Earth Orbit (LEO) environment. Therefore, it is necessary to develop robust, efficient and reliable techniques to understand the potential motions of the LEO debris. In this paper, we propose a novel signal processing approach to detect and estimate the motions of LEO space debris that is based on a fence-type space surveillance radar system. Because of the sparse distribution of the orbiting debris through the fence in our observations, we formulate the signal detection and the motion parameter estimation as a sparse signal reconstruction problem with respect to an over-complete dictionary. Moreover, we propose a new scheme to reduce the size of the original over-complete dictionary without the loss of the important information. This new scheme is based on a careful analysis of the relations between the acceleration and the directions of arrival for the corresponding LEO space debris. Our simulation results show that the proposed approach can achieve extremely good performance in terms of the accuracy for detection and estimation. Furthermore, our simulation results demonstrate the robustness of the approach in scenarios with a low Signal-to-Noise Ratio (SNR) and the super-resolution properties. We hope our signal processing approach can stimulate further work on monitoring LEO space debris.

Full text

A novel signal processing approach for LEO space debris based
on a fence-type space surveillance radar system
q
Jian Huang
a
, Weidong Hu
a
, Mounir Ghogho
b
, Qin Xin
c,⇑
, Xiaoyong Du
a
, Weiwei Guo
a
a
ATR Key Lab, College of Electronic Science and Engineering, National University of Defense Technology, Changsha, Hunan 410073, PR China
b
School of Electronic and Electrical Engineering, The University of Leeds, UK
c
Faculty of Science and Technology, University of the Faroe Islands, Torshavn, Faroe Islands
Received 3 February 2012; received in revised form 17 July 2012; accepted 19 July 2012
Abstract
The increase in space debris can seriously threaten regular activities in the Low Earth Orbit (LEO) environment. Therefore, it is nec-
essary to develop robust, efficient and reliable techniques to understand the potential motions of the LEO debris. In this paper, we pro-
pose a novel signal processing approach to detect and estimate the motions of LEO space debris that is based on a fence-type space
surveillance radar system. Because of the sparse distribution of the orbiting debris through the fence in our observations, we formulate
the signal detection and the motion parameter estimation as a sparse signal reconstruction problem with respect to an over-complete
dictionary. Moreover, we propose a new scheme to reduce the size of the original over-complete dictionary without the loss of the impor-
tant information. This new scheme is based on a careful analysis of the relations between the acceleration and the directions of arrival for
the corresponding LEO space debris. Our simulation results show that the proposed approach can achieve extremely good performance
in terms of the accuracy for detection and estimation. Furthermore, our simulation results demonstrate the robustness of the approach in
scenarios with a low Signal-to-Noise Ratio (SNR) and the super-resolution properties. We hope our signal processing approach can stim-
ulate further work on monitoring LEO space debris.
Ó2012 COSPAR. Published by Elsevier Ltd. All rights reserved.
Keywords: Signal detection; Parameter estimation; LEO debris surveillance; Fence-type radar systems; Sparse reconstruction
1. Introduction
LEO debris, the amount of which has been steadily
increasing, presents serious risks to existing space systems.
Many efforts had been made to use a fence-type radar sys-
tem for LEO debris surveillance (Hu et al., 2006, 2007;
Huang et al., submitted for publication; Khakhinov et al.,
2009; Michal et al., 2005; Montebugnoli et al., 2009;
Schumacher et al., 1998). This type of radar system forms
a“fence-like”electromagnetic area that can have an extre-
mely large coverage area and is a promising alternative for
capturing high-speed LEO objects compared to phased-
array radar and optical telescope surveillance systems. A
well-known fence radar system is the US Naval Space Sur-
veillance System (NAVSPASUR), which transmits single-
frequency, continuous-wave signals and forms a 0.02°
beamwidth in the latitudinal direction and a geocentric
50°width in the longitudinal direction. NAVSPASUR has
been shown to be capable of detecting LEO debris with
inclinations larger than 33°, including approximately 80%
of the space objects in the United States Space Command
(USSPACECOM) catalogue database (Schumacher et al.,
1998). The Graves in France is also a new fence-type radar
system that consists of four phased-array radars with each
0273-1177/$36.00 Ó2012 COSPAR. Published by Elsevier Ltd. All rights reserved.
http://dx.doi.org/10.1016/j.asr.2012.07.020
q
Project supported by the National High Technology Research and
Development Program of China under Grant No. 2010AA0965 and the
National Natural Science Foundation of China under Grant No.
61002021.
⇑
Corresponding author. Tel.: +298 352575.
E-mail addresses: huangjian@nudt.edu.cn (J. Huang), wdhu@nudt
.edu.cn (W. Hu), m.ghogho@ieee.org (M. Ghogho), qinx@setur.fo
(Q. Xin), xydu@nudt.edu.cn (X. Du), weiweiguo@nudt.edu.cn (W. Guo).
www.elsevier.com/locate/asr
Available online at www.sciencedirect.com
Advances in Space Research xxx (2012) xxx–xxx
Please cite this article in press as: Huang, J., et al. A novel signal processing approach for LEO space debris based on a fence-type space sur-
veillance radar system. J. Adv. Space Res. (2012), http://dx.doi.org/10.1016/j.asr.2012.07.020

of them forming an electromagnetic fence with a 10°–30°
beamwidth in elevation and 8°in azimuth. With the individ-
ual beam scanning in a pre-arranged sequence, this system
can cover the entire range of 180°in azimuth (Michal
et al., 2005). Currently, there exist several other fence-type
surveillance radar systems such as the Italian Medicina
space target surveillance system (Montebugnoli et al.,
2009) and the Russian Irkutsk ISR (Khakhinov et al.,
2009). In Hu et al. (2006, 2007), a double-fence radar system
was investigated for LEO debris surveillance that can form
a V-shaped electromagnetic beam in the latitudinal direc-
tion, unlike the typical fence-type radar systems mentioned
previously (Huang et al., Submitted for publication).
The fence-type surveillance radar system is well known
for its high efficiency in detecting LEO space debris. In gen-
eral, the success ratio of space debris surveillance heavily
depends on the performance of the signal detection
and the parameter estimation in the radar system. How-
ever, the accelerated motion of orbiting debris with respect
to the radar leads to a frequency modulation at the Dopp-
ler frequency and consequently shortens the coherent accu-
mulated time interval, especially for high carrier
frequencies.
In addition, because of the limitation of hardware
devices for the antennas, the electromagnetic fence usually
spreads very wide in the longitudinal direction while
remaining quite narrow in the latitudinal direction. This
results in a short time window for space debris to cross
the fence. For instance, the time duration on average for
space debris to cross the electromagnetic fence in NAV-
SPASUR is approximately 0.1 s (Knowles et al., 1982),
which poses a significant challenge to the detection and
estimation procedures in a fence-type radar system.
Recently, a substantial amount of work has been con-
ducted on signal processing for space debris surveillance
systems. In Markkanen et al. (2002, 2005), a generalized
match function (GMF)-based method was proposed for
signal detection in the EISCAT radar. In Maccone
(2007), a fast Karhunen-Loe
`ve transform was proposed
to improve the signal detection performance in the Medici-
na bistatic radar. In Isoda et al. (2006), a signal detection
method based on coherent accumulation with a linear fre-
quency modulation (LFM) of the transmitted signal was
successfully applied to the Kamisaibara radar. The signal
models used in the different surveillance radar systems
are not required to be exactly the same; thus, the mecha-
nisms used to determine the optimal signal detection and
parameter estimation can be significantly different.
Based on the observation of the sparsity, i.e., only a few
pieces of space debris can cross the electromagnetic fence in
a very short time interval, we can formulate the space deb-
ris motion parameter estimation problem as a sparse signal
reconstruction one with respect to an over-complete dictio-
nary that will be constructed later.
The sparse signal reconstruction approach has been suc-
cessfully applied in a variety of applications such as radar
imaging (Cetin, 2001). Moreover, this approach can achieve
high accuracy in motion parameter estimation as well as a
good super-resolution and robustness, especially for the
low SNR scenarios. Sparse signal reconstruction represents
the signal as a linear combination of the minimum number
of generated elements. A crucial step in sparse signal recon-
struction is to construct an appropriate over-complete dic-
tionary, which is problem-dependent. For the estimation
of the space debris Direction of Arrival (DOA) h, radial
velocity vand acceleration a, the dictionary can be con-
structed on a grid with all possible velocities vand acceler-
ations afor each possible DOA; i.e., a three-dimensional
space can be defined on the grid. Nevertheless, we further
exploit the fact that there exists an implicit relationship
between the DOA and the radial acceleration by consider-
ing the physical orbital motion constraints. This enables
us to construct the dictionary using only a two-dimensional
hvspace, which significantly reduces the size of the
dictionary.
The remainder of the paper is structured as follows. Sec-
tion 2introduces the system models for space debris, which
includes the general signal model, the sparse property, the
orbital constraints and the discrete signal model. An inverse
problem formulation based on sparse signal reconstruction
is presented in Section 3, and the simulation results are pro-
vided in Section 4to demonstrate the effectiveness of our
proposed approach. Finally, Section 5presents our
conclusions.
2. Observation model
2.1. Signal model of space debris
A fence-type radar normally transmits single-frequency,
continuous-wave signals (Hu et al., 2006, 2007; Huang
et al., Submitted for publication; Khakhinov et al., 2009;
Michal et al., 2005; Montebugnoli et al., 2009; Schumacher
et al., 1998). Here, it should be noticed that according to
the specific fence-type radar system being studied, the sin-
gle-frequency, continuous-wave (CW) signal is slightly
modulated by a pseudo-noise (PN) code to obtain the
range measurement. However, during the short time for
signal processing, the signal model can be assumed equiva-
lent to the unmodulated CW signal after demodulation,
i.e., rðtÞ¼expðj2pf0tÞ, where f0is the carrier frequency.
The backscattered signal at a specific time tby the debris
is then given by:
sðtÞ¼b0rðtsðtÞÞ ð1Þ
where b0is the signal amplitude, which is assumed to re-
main constant over the short duration required for the deb-
ris to cross the fence, and sis the time delay. When a piece
of debris crosses the fence, its slight jerk can be neglected.
However, the acceleration should be considered because of
its influence and the impact on the performance of the sig-
nal detection and accumulation (Yuan and Hu, 2009),
especially for high carrier frequencies (e.g., a replacement
for NAVSPASUR is being developed that will use S-band
2J. Huang et al. / Advances in Space Research xxx (2012) xxx–xxx
Please cite this article in press as: Huang, J., et al. A novel signal processing approach for LEO space debris based on a fence-type space sur-
veillance radar system. J. Adv. Space Res. (2012), http://dx.doi.org/10.1016/j.asr.2012.07.020

frequencies (Fact Sheet, 2002)). Therefore, a constant
acceleration model can be applied to describe the motion
of the space debris while crossing the fence (Yuan and
Hu, 2009). In the special case where the time interval is
not sufficiently short, one can use a piecewise-constant
acceleration model. When the mono-static radar is taken
into account (Hu et al., 2006, 2007; Huang et al., Submit-
ted for publication), the propagation delay can be bounded
by:
sðtÞ1
cðqþ2vt þat2Þð2Þ
where we use the fact that the radial velocity of the debris is
much slower than the speed of light (c¼3108m/s) and
where ais the debris acceleration with respect to the obser-
vation station, vand qrepresent the velocity and the range
of the debris, respectively, with respect to the observation
station at the instant t¼0.
From Eqs. (1) and (2), we can obtain the baseband
received signal as
sðtÞ¼b0rect tsðtÞ
T

expðj2pf0Dþj2pft þjpct2Þð3Þ
where D¼q=c;f¼2v=kis the Doppler frequency
caused by the debris radial velocity and c¼2a=kis the
chirp rate caused by the debris acceleration, with kdenot-
ing the wavelength of the transmitted signal and Tbeing
the duration of the received signal.
Furthermore, an array antenna is commonly used to
receive the echo signal in a fence-type surveillance radar
for the DOA measurements of the debris (Hu et al.,
2006, 2007; Huang et al., Submitted for publication; Kha-
khinov et al., 2009; Michal et al., 2005; Montebugnoli
et al., 2009; Schumacher et al., 1998). In this work, we
assume a uniform linear array of Komni-directional ele-
ments with an antenna spacing d¼k=2. For the space deb-
ris located in the far field of the antenna array, the
assumption of a planar wave front is applicable. Although
the received signal of the moving space debris is in the form
of a linearly frequency-modulated signal, the bandwidth is
narrow with respect to the carrier frequency. Thus, the
envelope delay between the antenna elements can be
neglected, and only the phase difference caused by the
range delay will be considered. In the case of multiple
pieces of debris, the received signal is a superposition of
the returns from all the debris. At the observation time
sample tn, for each antenna element in the presence of
noise, Eq. (3) can be expressed as
yðk;tn;f;h;cÞ¼X
Q
q¼1
sqðk;tn;fq;hq;cqÞþnðk;tnÞ;
k¼1;2;;K;tn2½T=2;T=2;
n¼1;2;;Nð4Þ
where Qis the number of pieces of debris crossing the fence
simultaneously and nðk;tnÞis the measurement noise at the
kth antenna element at time tn;yðk;tn;f;h;cÞis the received
signal. The noise samples are assumed to be independent
and identically distributed (i.i.d.) in both space and time,
and follow a Gaussian distribution with a zero mean and
a variance r2, (i.e.), Nð0;r2Þ. Here the signal received at
the kth antenna element from the return of the qth piece
of debris is given by
sqðk;tn;fq;hq;cqÞ¼bqexp j2pf0
dsin hq
cðk1Þ

expðj2pfqtnþjpcqt2
nÞð5Þ
The problem investigated here is to estimate the DOAs
hq, the Doppler frequencies fqand the chirp rates cqfrom
the received signal yðk;tÞreturned by the debris, which is
typically an ill-posed inverse problem. Thus, it is necessary
to regularize the ill-posed problem with additional domain-
dependent prior information to obtain stable and interpret-
able solutions.
2.2. Sparsity constraint on the distribution of space debris
Using the ORDEM2000 model (Liou et al., 2002)
released by NASA, a comparison of the different models
in Sdunnus et al. (2004) and the current population of
space debris (Liou and Johnson, 2008), the number of
pieces of space debris that simultaneously cross the fence
can be approximately analyzed. Here, we divide the orbital
altitude from 300 km to 1000 km into 8 orbit layers with
100 km between layers. Moreover, the density of the space
debris is the highest (approximately 40,000 pieces larger
than 1 cm) at an altitude of 800 km. For the fence-type
radar system, the duration for the space debris crossing
the fence in the latitudinal direction is on the order of
T1¼101s, and the orbital period is on the order of
T2¼1:5h ¼5400 s. Therefore, the coverage ratio in the
latitudinal direction is approximately 1/54,000. To meet
the requirement of seamless coverage, the geocentric angle
of the fence in the longitudinal direction has to exceed the
rotation angle of the earth during one orbital period of
each item of debris. The coverage ratio in the longitudinal
direction is then 1.5/24. Assuming that the space debris
density in each layer is the same as the one corresponding
to the 800 km layer, the average number of articles of space
debris crossing the fence simultaneously can be bounded by
Number ¼40;000 1=54;000 1:5=24 8¼0:37037
In addition, for the space debris cloud generated by the
collision between Iridium 33 and Cosmos 2251 in February
2009 (Liou et al., 2010), the density in December 2009 was
approximately 8.4 times higher than the current density of
the observed space debris, according to our statistics, so the
maximum number of pieces of space debris crossing the
fence is approximately
Numbermax ¼8:40:37037 ¼3:11111
J. Huang et al. / Advances in Space Research xxx (2012) xxx–xxx 3
Please cite this article in press as: Huang, J., et al. A novel signal processing approach for LEO space debris based on a fence-type space sur-
veillance radar system. J. Adv. Space Res. (2012), http://dx.doi.org/10.1016/j.asr.2012.07.020

Based on the above analysis, it can be argued that the
items of space debris crossing the fence simultaneously
are sparsely distributed in the observed scenario during
the processing time interval. This sparsity property addi-
tionally motivates us to use the well-known sparse signal
reconstruction approach.
2.3. Orbital constraints on the space debris
Fig. 1 depicts the observation geometry of the scenario
in the Earth-Fixed (EF) coordinate frame. The geocenter
is located at O, and the position of the observation station
is at P. The plane spanned by the electromagnetic fence is
denoted by PS1S2, and the place where the debris crosses
the fence is denoted by S.Ris the position vector of the
transmitter, and q¼qq0is the vector from the observa-
tion station to the debris, where qis the length of qassoci-
ated with the unit-length vector q0. The position of the
debris at ris then described by vector r¼qq0þR.
We consider the second-order motion model of debris
that is given by
€
r¼€
qq0þ€
Rð6Þ
In the case without the perturbation, the movement of
the debris should satisfy the following equation:
€
r¼r
r3lð7Þ
where lis the gravitational constant for the earth and ris
the range from the debris to the geocenter.
Combining Eqs. (6) and (7), we obtain:
a¼€
q¼q0
r
r3lþ€
R
 ð8Þ
where €
R¼x2
Eðx;y;0ÞTis the acceleration of the observa-
tion station resulting from the rotation of the earth,
xE¼7:292115 105rad/s is the mean motion of the
earth, and xand ydenote the coordinates of the observa-
tion station in the X-axis and the Y-axis in the ECF coor-
dinate frame, respectively.
For the fence-type surveillance radar system, the range q
from the debris to the observation station can be obtained
by the range measuring technique for continuous wave sig-
nals, e.g., through a PN code ranging measurement. The
beam width of the fence in the direction of latitude is
assumed to be sufficiently small that it may be neglected.
Furthermore, the position vector rand the unit vector q0
can be determined uniquely from the oblique angle of the
fence aand the direction-of-arrival angle hin the longitudi-
nal direction. Accordingly, Eq. (8) can be written as some
function of aand h, say
a¼gða;hÞð9Þ
Fig. 2 illustrates the relationship between the accelera-
tion aand the parameters ða;hÞat a debris range of
400 km. It can be observed that that the change in the obli-
que angles is negligible compared to the DOA angles,
though the beam width of the fence in the latitudinal direc-
tion approaches 1°(the width of NAVSPASUR is approx-
imately 0.02°). It can be argued that the acceleration ais
virtually independent of the angle h. Thus, with the con-
straint condition c¼cðhÞ¼2gða;hÞ=k, we can reduce
the dimension of the parameter vector to be estimated.
2.4. Discrete signal model
According to the orbital constraint mentioned above,
the signal model with three parameters in Eq. (5) can be
transformed into the following signal model with two
parameters
sqðk;tn;fq;hqÞ¼bqexp j2pf0
dsin hq
cðk1Þ

exp j2pfqtnþjpcðhqÞt2
n

ð10Þ
We further discretize the model in Eq. (10) on a grid of
reduced, that is, two-dimensional hfparameter space,
i.e.,
h¼½h1;h2;...;hl;...;hL;l¼1;2;...;Lð11Þ
f¼½f1;f2;...;fm;...;fM;m¼1;2;...;Mð12Þ
Fig. 1. Scenario of the space debris crossing the fence in EF coordinate
frame.
0 20 40 60 80
−9
−8
−7
−6
−5
−4
−3
−2
Direction of arrival angle θ(°)
Acceleration a(m/s2)
Oblique angle α=4°
Oblique angle α=5°
Oblique angle α=6°
Fig. 2. The mapping from the direction-of-arrival angle hto the
acceleration for different values of the oblique angle a.
4J. Huang et al. / Advances in Space Research xxx (2012) xxx–xxx
Please cite this article in press as: Huang, J., et al. A novel signal processing approach for LEO space debris based on a fence-type space sur-
veillance radar system. J. Adv. Space Res. (2012), http://dx.doi.org/10.1016/j.asr.2012.07.020

Define the matrix
Aðtn;fm;hÞ¼
sð1;tn;fm;h1Þ  sð1;tn;fm;hlÞ  sð1;tn;fm;hLÞ
sð2;tn;fm;h1Þ  sð2;tn;fm;hlÞ  sð2;tn;fm;hLÞ
.
.
...
...
...
..
.
.
sðK;tn;fm;h1Þ  sðK;tn;fm;hlÞ  sðK;tn;fm;hLÞ
2
6
6
6
6
4
3
7
7
7
7
5
and the signal received by the array at time tnas
ytn¼
yð1;tnÞ
yð2;tnÞ
.
.
.
yðK;tnÞ
2
6
6
6
6
4
3
7
7
7
7
5¼Aðtn;f1;hÞAðtn;fm;hÞ  Aðtn;fM;hÞ½
x¼A0ðtnÞxð13Þ
Thus, the received signal vector during the processing inter-
val is expressed in a matrix form as
y¼
yt1
yt2
.
.
.
ytN
2
6
6
6
6
6
4
3
7
7
7
7
7
5¼
A0ðt1Þ
A0ðt2Þ
.
.
.
A0ðtNÞ
2
6
6
6
6
4
3
7
7
7
7
5xþn¼A00xþnð14Þ
where A00 is the measurement matrix. In this signal model,
if there is an article of debris with parameters fm;hl, a peak
will appear in the ðm1ÞLþlelement of the vector x.
3. Signal detection and parameter estimation approach
3.1. Sparse reconstruction for signal detection and parameter
estimation
In this section, we present our sparse signal reconstruc-
tion framework for debris detection and parameter estima-
tion. Sparse signal reconstruction minimizes the number of
elements from an over-complete dictionary while still line-
arly approximating the signal well. This technique has been
successfully applied in a variety of applications including
radar imaging (Cetin, 2001), spectrum estimation (Cabrera
and Parks, 1991) and DOA estimation (Cetin et al., 2002).
Based on the observation of sparsity discussed in Sec-
tion 2, what is sought is to reconstruct the sparse vector
xby representing it using a few non-zero elements. This
is formulated as the following optimization problem:
min
xkxk0;s:tkyA00xk2
26b2ð15Þ
where the parameter breflects the noise level.
Unfortunately, this ‘0-norm optimization problem is
NP-hard (Muthukrishnan, 2005). However, the problem
can be relaxed to a convex ‘1-norm optimization problem
(Chen et al., 2001)
min
xkxk1;s:tkyA00xk2
26b2ð16Þ
which can achieve the same sparse solution as the l0-norm-
based optimization under certain conditions. The authors
in Cande
`s and Tao (2005), Cande
`s et al. (2006a), Cande
`s
et al. (2006b), Donoho and Huo (2001) and Elad and
Bruckstein (2002) have discussed the conditions for the
equivalence between the ‘0-norm optimization and its ‘1-
norm relaxation. In Donoho et al. (2006), it was argued
that a Q-sparsity signal satisfying Q61=4ð1=-þ1Þcan
be exactly recovered from the noisy observations by the
‘1-norm optimization. Here, -represents the maximum
correlation coefficient of two different columns of the mea-
surement matrix A00. It is easy to show that this condition
can be met in our problem.
We solve the problem in Eq. (16) with the second-order
cone (SOC) programming technique (Malioutov, 2003). If
the noise is white and Gaussian with the variance r2, then
1=r2knk2
2v2
NK with v2
NK denoting the chi-square distribu-
tion with NK degrees of freedom. In practice, we choose
bso that the confidence interval ½0;bintegrates to 0.999
probability (Malioutov, 2003). For the estimation of the
noise variance, there are some automatic methods, such
as the cross-validation method (Malioutov, 2003).
Although the range of the Doppler frequency of the
space debris is very wide, the time interval of one item of
space debris crossing the fence is small and the occupied
frequency band is narrow. We only consider parameter
estimation with the pass-band down-sampled signal. The
method of multi-resolution grid refinement is further used
to improve the accuracy of the parameter estimation
(Malioutov, 2003).
3.2. Reconstruction error bounds in the presence of noise
Define the following set of column indices of the mea-
surement matrix A00
KNLM
XG;W¼ðx1;x2;...;xWÞjxw2N;06xw6G;xw<xwþ1

ð17Þ
where G¼LM. Let x12CW1denotes the sparse solution
for the signal model in the absence of noise, and let
y1¼A00x1ð18Þ
In the case of noisy observations, the sparse solution is
denoted by x2¼x1þDx2CW2; so, we have that
y¼y1þn¼A00ðx1þDx2Þð19Þ
Letting W¼W1þW2, and Dx2CW, we have that
(Wohlberg, 2003):
kDxk2
kx1k2
6f1
Wknk2
ky1k2ð20Þ
For xw2XG;W, define
PXG;W¼
dx1
1dx2
2 dxW
G
dx1
1dx2
2 dxW
G
.
.
..
.
...
..
.
.
dx1
1dx2
2 dxW
G
2
6
6
6
6
4
3
7
7
7
7
5ð21Þ
where dxw
g¼dðgxwÞ. Thus, fWcan be expressed as
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veillance radar system. J. Adv. Space Res. (2012), http://dx.doi.org/10.1016/j.asr.2012.07.020

fW¼min
xw2XG;W
gminðA00PT
XG;WÞ
gmaxðA00PT
XG;WÞ
where gminðA00PT
XG;WÞand gmaxðA00PT
XG;WÞare the minimum
and the maximum eigenvalues of matrix A00PT
XG;W, respec-
tively. The parameter fWmeasures the noise sensitivity of
the sparse reconstruction. A smaller value of fWmeans that
there is a larger error in the signal reconstruction and
weaker noise suppression performance. The value of fWde-
pends on the step size of the discrete grid used in the con-
struction of the measurement matrix and hence the
correlation of its columns. Considering W¼2, we compute
fWwith different discrete steps, Dfand Dh, in the frequency
domain and the angle domain, respectively. The result is
shown in Fig. 3.
Fig. 3 indicates that fWgradually tends to 1 as the step
size of the grid increases which means that the vectors in
the columns of the measurement matrix are approximately
orthogonal to each other. The capability for noise suppres-
sion also increases. The method of multi-resolution grid
refinement can be applied to achieve higher accuracy while
avoiding the great computational cost incurred with a den-
ser grid (Malioutov, 2003). When we only consider fWin
the angle domain, it can be observed that the correlation
of the columns of the measurement matrix becomes weaker
with the constraint of orbital knowledge. Thus, the DOA
0 0.2 0.4 0.6 0.8 1
0
0.5
1
Frequency interval Δf(Hz)
ζ2 in
frequency domain
With constraint of orbital knowledge
Without constraint of orbital knowledge
0 1 2 3 4
0
0.5
1
Angle interval Δθ(°)
ζ2 in angle domain
With constraint of orbital knowledge
Without constraint of orbital knowledge
Fig. 3. f2with different discrete grid intervals Df;Dhin the frequency
domain and the angle domain.
Angle (°)
Frequency(Hz)
Sparse reconstruction
15 20 25 30 35
40
45
50
55
60
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Angle(°)
Frequency(Hz)
Space−time processing after acceleration compensation
10 20 30 40
40
45
50
55
60
100
200
300
400
500
Angle (°)
Frequency(Hz)
Sparse reconstruction
20 25 30 35
44
46
48
50
52
54
56
58 0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Angle(°)
Frequency(Hz)
Space−time processing after acceleration compensation
10 20 30 40
40
45
50
55
60
100
200
300
400
500
Fig. 4. Signal detection results using sparse reconstruction and space-time processing after acceleration compensation.
6J. Huang et al. / Advances in Space Research xxx (2012) xxx–xxx
Please cite this article in press as: Huang, J., et al. A novel signal processing approach for LEO space debris based on a fence-type space sur-
veillance radar system. J. Adv. Space Res. (2012), http://dx.doi.org/10.1016/j.asr.2012.07.020

estimation performance improves, but this has no impact
on the correlation of the columns in the frequency domain.
Despite the difference in units, the discrete steps Dhand
5Dfresult in similar correlation values and noise suppres-
sion performance. These theoretical results will be illus-
trated by simulations in the following section.
4. Simulation results
In this section, we conduct several experiments to demon-
strate the effectiveness of our proposed method. The simula-
tion scenario is as follows: (1) the plane of the
electromagnetic fence is inclined 5°to the south; (2) the range
from the debris to the radar station is 400 km; (3) the time
interval for crossing the fence is 0.4 s; (4) the carrier fre-
quency is 3 GHz in the S-band and bq¼1. The input
SNRin is defined as SNRin ¼20log10 ðjbqj=rÞand the SNR
of the output after matched filtering is then SNRMF ¼
20log10ðffiffiffiffiffiffiffiffiffiffiffiffiffiffi
NK
pbq=rÞ¼27:53 dB þSNRin, where N¼
81 and K¼7 in this paper.
The comparison between the proposed sparse recon-
struction method and the spatial-time processing method
(after acceleration compensation) in terms of detection
performance, for the case where SNRin ¼0 dB, is shown
in Fig. 4. It is shown that the proposed method with the
aid of prior knowledge of the orbital motion and sparsity
performs better, in terms of both signal detection and
parameter estimation, and has the capability for super-
resolution of closely spaced debris.
In what follows, we evaluate the performance of the pro-
posed method quantitatively. The following two aspects
are studied: (1) the effect of parameter selection; and (2)
the resolution of multiple targets.
4.1. Influence of parameters selection on signal detection and
parameter estimation
The Cramer-Rao lower bound (CRLB) for our parame-
ter estimation problem in the case of a single target is
derived in Appendix A. We compare the accuracy of the
proposed sparse reconstruction with the CRLB based on
the following simulation conditions: ðf;hÞ¼ð50 Hz;25Þ
and the discrete steps of the measurement matrix are
Df¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
CRLBðfÞ
p=2;Dh¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
CRLBðhÞ
p=2.
Let the two-dimensional angle-frequency plane X¼
ðxijÞLMrepresent the result of the sparse reconstruction.
Because it is difficult to evaluate the performance of the
parameter estimation based on sparse reconstruction with
0.8 0.9 1 1.1 1.2
0
0.2
0.4
0.6
0.8
1
Ratio of estimated noise variance to actual noise variance
Correct sparse reconstruction probability
SNRin=−15dB
SNRin=15dB
0.9 0.95 1 1.05 1.1
0
0.5
1
1.5
2
2.5
3
3.5
Ratio of estimated noise variance to actual noise variance
RMSE of angle estimation(°)
SNRin=−15dB
SNRin=15dB
0.9 0.95 1 1.05 1.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Ratio of estimated noise variance to actual noise variance
RMSE of freuquency estimation(Hz)
SNRin=−15dB
SNRin=15dB
(a)
(c)
(b)
Fig. 5. The influence of the estimated noise variance value on sparse reconstruction.
J. Huang et al. / Advances in Space Research xxx (2012) xxx–xxx 7
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traditional detection metrics, we combine the results of
extensive simulations and the definition in Christian et al.
(2010) to redefine the test criterion for correct sparse recon-
struction as follows:
20log10
maxi;jjxijj
mediani;jjxijj

Peð22Þ
where e¼40dB is selected in this paper.
It can be observed from Fig. 4 that the proposed sparse
reconstruction method has the advantage of being able to
suppress the side-lobes. Because only an output SNR
above 40 dB is considered as a correct sparse reconstruc-
tion, which is more rigorous than that in traditional thresh-
old testing, the correct sparse reconstruction probability
can be used to quantify the detection performance. Fur-
thermore, according to the statistical analysis of the simu-
lation results, when a single target is determined to be a
correct sparse reconstruction, the probability that the sin-
gle peak splits into multiple false peaks is always zero.
In Section 3.1, the noise variance r2must be estimated
before the sparse reconstruction; the difference between
the estimated noise variance c
r2and the actual noise
variance r2will affect the performance of the sparse recon-
struction, which is shown in Fig. 5. This figure shows that
the Root Mean Square Error (RMSE) of the sparse recon-
struction in the low SNR regime is more sensitive to the
estimated value of the noise variance than that in the high
SNR regime. The estimated values of the parameters are
close to the actual values when the noise variance uncer-
tainty is small, but the performance of the parameter esti-
mation degrades when the estimation error of the noise
variance becomes large.
The relationship between the angle hand the chirp rate c
derived from the orbital knowledge constraint reduces the
number of unknown parameters from three to two, which
facilitates signal detection and parameter estimation. If
the orbital knowledge constraint is not taken into account,
the chirp rate would need to be estimated. The effects of the
uncertainty in the chirp rate on the performance of the
sparse reconstruction are shown in Fig. 6. The figure shows
that this uncertainty greatly influences the probability of
correct sparse reconstruction and the accuracy of the
parameter estimation. For a high SNR, the dominating fac-
tor is the mismatch between the assumed and the actual sig-
nal models due to the chirp rate uncertainty, whereas for a
low SNR, the dominating factor is the additive noise.
Furthermore, the accuracy of the angle estimation is more
sensitive to the estimated value of the chirp rate than that
0.9 0.95 1 1.05 1.1
0
0.2
0.4
0.6
0.8
1
Ratio of estimated acceleration to actual acceleration
Correct sparse reconstruction probability
SNRin=−15dB
SNRin=15dB
0.98 0.99 1 1.01 1.02
0
0.2
0.4
0.6
0.8
1
1.2
1.4
Ratio of estimated acceleration to actual acceleration
RMSE of angle estimation(°)
SNRin=−15dB
SNRin=15dB
0.98 0.99 1 1.01 1.02
0
0.05
0.1
0.15
0.2
Ratio of estimated acceleration to actual acceleration
RMSE of frequency estimation(Hz)
SNRin=−15dB
SNRin=15dB
(a)
(c)
(b)
Fig. 6. The influence of chirp rate estimation on sparse reconstruction.
8J. Huang et al. / Advances in Space Research xxx (2012) xxx–xxx
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of the frequency estimation, which coincides with the
remarks in Section 3.2; i.e., the orbital knowledge constraint
mainly enhances the noise suppression capability in the
angle domain, and has little influence in the frequency
domain.
We conducted 500 Monte Carlo runs and estimated the
probability of correct sparse reconstruction. The results are
depicted in Fig. 7, which additionally shows a comparison
between the RMSEs for angle estimation and frequency
estimation and their corresponding CRLBs.
As far as parameter estimation performance is con-
cerned, only the correct sparse reconstruction is used to
evaluate the accuracy of the parameter estimation.
Although the results are very sensitive to the estimated
(a)
(c)
(b)
Fig. 7. Performance of parameter estimation obtained with sparse reconstruction.
0 5 10 15 20
−0.5
0
0.5
Angle interval of double targets |θ1−θ2|(°)
Ratio of estimated angle bias to
angle interval σθ/|θ1−θ2|
Target 1 in SNR
in=−15dB
Target 2 in SNR
in=−15dB
Target 1 in SNR
in=15dB
Target 2 in SNR
in=15dB
3|σθ|=0.5|θ1−θ2|
0 2 4 6 8 10
−0.5
0
0.5
Frequency interval of double targets |f1−f2|(Hz)
Ratio of estimated frequency bias to
frequency interval σf/|f1−f2|
Target 1 in SNR
in=−15dB
Target 2 in SNR
in=−15dB
Target 1 in SNR
in=15dB
Target 2 in SNR
in=15dB
3|σf|=0.5|f1−f2|
(a) (b)
Fig. 8. Parameter estimation bias for two targets when SNRin ¼15 dB, 15 dB.
J. Huang et al. / Advances in Space Research xxx (2012) xxx–xxx 9
Please cite this article in press as: Huang, J., et al. A novel signal processing approach for LEO space debris based on a fence-type space sur-
veillance radar system. J. Adv. Space Res. (2012), http://dx.doi.org/10.1016/j.asr.2012.07.020

noise variance c
r2when compared with the CRLB, they
indicate that the proposed method can effectively diminish
the influence of noise and thus improve the performance
of the parameter estimation. Because the sparse reconstruc-
tion is a biased parameter estimation (Malioutov, 2003), if
an appropriate c
r2is chosen, it may produce estimation
errors smaller than the CRLB in the case of a low SNR.
The reasons behind this outcome may be that the estimation
in the low SNR regime is biased, and thus, the CRLB is no
longer a lower bound. In the case of a high SNR, although
the parameter estimation performance is good, the estima-
tion error does not reach the CRLB because a decrease in
the step size of the search grid increases the correlation of
the columns of the measurement matrix. This unfavorably
affects the performance of the sparse reconstruction.
4.2. Simulation analysis of the resolution for multiple targets
In this section, we investigate the super-resolution capa-
bility of the proposed method. Because it is difficult to
describe the resolution in the two-dimensional frequency-
angle plane directly, we separately consider the resolution
of two targets in the frequency and the angle domains.
The estimation bias for the two targets versus the spacing
sizes is depicted in Fig. 8.
We assume that the parameter estimation errors follow a
normal distribution. If 3 maxðrh1;rh2Þ
60:5jh1h2jor
3 maxðrf1;rf2Þ
60:5jf1f2j, the two targets are resolved.
The minimum values of jh1h2jand jf1f2jthat satisfy
the aforementioned inequalities represent the minimum
resolvable angle and frequency intervals, respectively. In
Fig. 8, the sign of the bias represents the departure direction
of the estimation error with respect to the actual value. With
the increase of SNR or the spacing sizes between the two
targets, the resolution capability improves. In the angle
domain, resolution with a high probability is achieved by
the proposed method when the spacing is 10°and
SNRin ¼15 dB. The conventional methods, such as MU-
SIC and Capon cannot resolve the two coherent sources
using the same uniform linear array setting even when
SNRin ¼20dB (Malioutov, 2003). In the frequency do-
main, the Rayleigh resolution is 1=T¼2:5Hz (Christian
et al., 2010), which is easily achieved by the sparse recon-
struction when SNRin P15 dB. From Fig. 8, we can
claim that the proposed sparse reconstruction method has
the capability of super-resolution in both the angle domain
and the frequency domain. The statistical analysis of the
resolution capability in the angle and the frequency do-
mains for different SNRs is shown in Fig. 9.
5. Conclusion
Target detection and parameter estimation are funda-
mental tasks for space debris surveillance and cataloguing.
Based on the sparse property of the observed scenario, we
proposed a novel signal processing approach based on
sparse signal reconstruction in a fence-type radar system.
We have exploited knowledge of the debris orbits to reduce
the size of the original over-complete dictionary, thus sig-
nificantly lowering the computational cost. Extensive simu-
lations were carried out to demonstrate the effectiveness of
the proposed method and to offer useful technical support
for the feasibility study of new fence-type space surveil-
lance radar systems.
Appendix A. The Cramer Rao low bound of parameter
estimation
Combining Eqs. (5), (10) and (14), we have
sðk;tn;f;h;jbj;UbÞ¼jbjexpðjUbÞexp j2pf0
dsinh
cðk1Þ

exp j2pftnþjpcðhÞt2
n
 ðA:1Þ
where Ubis the phase of b, the probability density function
of the observation samples is
−30 −20 −10 0 10 20 30
5
10
15
SNRin(dB)
Minimum resolvable angle interval(°)
−30 −20 −10 0 10 20 30
1
1.5
2
2.5
3
3.5
SNRin(dB)
Minimum resolvable frequency interval(Hz)
(a) (b)
Fig. 9. The resolution of two targets versus the SNRin in the angle and frequency domains.
10 J. Huang et al. / Advances in Space Research xxx (2012) xxx–xxx
Please cite this article in press as: Huang, J., et al. A novel signal processing approach for LEO space debris based on a fence-type space sur-
veillance radar system. J. Adv. Space Res. (2012), http://dx.doi.org/10.1016/j.asr.2012.07.020

Pðy;f;h;jbj;UbÞ¼ 1
ðffiffiffiffiffiffiffiffi
2pr
pÞNKY
K
k¼1Y
N
n¼1
exp
1
r2yðk;tn;f;h;jbj;UbÞsðk;tn;f;h;jbj;UbÞ
jj
2

ðA:2Þ
Let n¼½f;h;jbj;UbT. The elements of the Fisher informa-
tion matrix are obtained as:
IðnÞ½
ij ¼2
r2Re X
K
k¼1X
N
n¼1
@sðk;tn;f;h;jbj;UbÞ
@ni
"
@sðk;tn;f;h;jbj;UbÞ
@njðA:3Þ
where
@sðk;tn;f;h;jbj;UbÞ
@f¼j2ptnsðk;tn;f;h;jbj;UbÞ
@sðk;tn;f;h;jbj;UbÞ
@h¼j2pf0d
cðk1Þcos hþjp@cðhÞ
@ht2
n
hi
sðk;tn;f;h;jbj;UbÞ
@sðk;tn;f;h;jbj;UbÞ
@jbj¼1
jbjsðk;tn;f;h;jbj;UbÞ
@sðk;tn;f;h;jbj;UbÞ
@Ub¼jsðk;tn;f;h;jbj;UbÞ
8
>
>
>
>
>
>
>
>
>
<
>
>
>
>
>
>
>
>
>
:ðA:4Þ
The Cramer-Rao low bound (CRLB) is
CRLBðfÞ¼½I1ðnÞ11;CRLBðhÞ¼½I1ðnÞ22 ðA:5Þ
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