When Target Motion Matters: Doppler Coverage in
Radar Sensor Networks
Xiaowen Gong, Junshan Zhang, and Douglas Cochran
School of Electrical, Computer, and Energy Engineering, Arizona State University
{xgong9, junshan.zhang, cochran}@asu.edu
Abstract—Radar sensors, which actively transmit radio waves
and collect RF energy scattered by objects in the environment,
offer a number of advantages over purely passive sensors. An
important issue in radar is that the transmitted energy may be
scattered by objects that are not of interest as well as objects
of interest (e.g., targets). The detection performance of radar
systems is affected by such clutter as well as noise. Further, in
many applications, clutter can be substantially stronger than the
signals of interest. To combat the effect of clutter, a popular
method is to take advantage of the Doppler frequency shift (DFS)
extracted from the echo signal due to the relative motion of a
target with respect to the radar. Unfortunately, a sensor coverage
model that only depends on the distance to a target would fail
to capture the DFS.
In this paper, we set forth the concept of Doppler coverage for
a network of spatially distributed radars. Specifically, a target
is said to be Doppler-covered if, regardless of its direction of
motion, there exists some radar in the network whose signalto-noise ratio (SNR) is sufficiently high and the DFS at that
radar is sufficiently large. Based on the Doppler coverage model,
we first propose an efficient method to characterize Dopplercovered regions for arbitrarily deployed radars. Then we design
an algorithm for deriving the minimum radar density required
to achieve Doppler coverage in a region under any polygonal
deployment pattern, and further apply it to investigate the regular
triangle based deployment.
Index Terms—radar sensor network, Doppler effect, deterministic deployment, critical sensor density
I. I NTRODUCTION
Wireless sensor networks have potential in a wide range
of applications, such as border control, homeland security,
and intruder detection. Traditional sensors in such applications
operate by passively collecting natural radiation emitted or
reflected by an object. In contrast to such passive sensors, a
radar (RAdio Detection And Ranging) performs detection by
actively emitting radio waves and receiving echoes reflected
by objects. Due to this salient feature of active sensing, radars
provide a number of benefits over traditional passive sensors.
For example, an ultra-wideband (UWB) radar has a larger
sensing range than either infrared or magnetic sensors. With
the emergence of cheap and compact radar devices, it is becoming feasible to deploy a network of radar sensors working
in concert1 [1]. Indeed, the radar network has been recognized
This research was supported in part by the U.S. National Science Foundation under Grants CNS-0917087 and CNS-1117462, AFOSR project FA955010-1-0464, and DoD MURI project No. FA9550-09-1-0643.
1 For brevity, we may use “radar”, “sensor”, and “radar sensor” interchangeably throughout this paper.
V
V
φ1
f1 > f 0
φ2
f1 < f 0
Fig. 1. The received radar signal frequency f1 is higher than the emitted
frequency f0 when the target is approaching and lower than f0 when it is
receding. Furthermore, for a given speed V , |f1 − f0 | increases as angle ϕ1
(< 90◦ ) decreases or angle ϕ2 (> 90◦ ) increases.
as a promising paradigm for numerous applications, including
road traffic monitoring [2] and earthquake sensing [3].
One important issue for sensor networks is how well an
object of interest (target) is monitored, referred to as the
coverage problem [4]. Previous studies have mostly assumed
that a sensor’s capability of detecting a target is quantified by
its distance to the target. As a result, the coverage region of a
sensor is typically modeled as a disk or a sector. However, this
distance-based coverage model is barely valid for radar sensors
in the presence of clutter. Simply put, clutter refers to echoes
reflected by undesired objects in the monitored environment,
such as rocks, trees, and walls. The clutter can be by many
orders stronger than the echo from a target, making the target
detection very difficult, if not impossible.
Perhaps the most widely used method for detecting a target
in the midst of strong clutter is by taking advantage of the
Doppler effect. In many applications, the target of interest
is a moving object such as an aircraft, ship, ground vehicle,
or person, while most of the objects that cause clutter are
stationary or slow-moving compared to the target. The Doppler
frequency shift (DFS) is the frequency difference between
the emitted and received radar signals due to the relative
velocity between a radar and a moving target. As illustrated
in Fig. 1, compared to the emitted frequency, the received
frequency is higher when the target is approaching, and is
lower when it is receding. For a given speed of a moving
target, the amount of frequency variation is intimately related
to the target’s direction of motion with respect to the radar.
Specifically, for a given target speed, it is maximized when
the target is moving directly toward or away from the radar,
and diminishes with increasing angle between the direction of
motion and the direction to the radar, until reaching zero when
the target is moving at right angles to the radar. By extracting
the DFS, a radar is enabled to detect the target from a Doppler
r
Si
up-Doppler-covered
θ
d~1
θ
P
down-Doppler-covered
d~2
Fig. 2.
The up-Doppler and down-Doppler-covered angular ranges are
bounded between the dashed lines: d⃗1 is not Doppler-covered by Si ; d⃗2
is down-Doppler-covered by Si .
perspective, i.e., the magnitude of the DFS indicates whether
the target is present.
As none of the existing sensor coverage models take into
account the Doppler effect for moving target detection, we
develop a novel Doppler coverage model to capture this
feature of radar sensors. A target is Doppler-covered if and
only if, for any direction of motion, the target is within some
radar’s sensing range, and furthermore, the angle between the
direction of motion and the direction from the target to that
radar is close enough to 0 or π. In other words, if the target is
Doppler-covered, then no matter which direction it is moving,
there always exists some radar that can detect both sufficiently
high SNR and sufficiently large DFS from its received signal.
This Doppler coverage model differs significantly from the
traditional distance-based coverage model, and gives rise to
the following new challenges. First, the Doppler coverage of
a target depends on the angular positions of all the radars
(whose sensing ranges include the target) with respect to the
target. This coupling in both distance and direction complicates the Doppler coverage. Second, a radar can contribute
to the Doppler coverage of a target in two ways: up-Doppler
coverage with positive DFS, and down-Doppler coverage with
negative DFS. In particular, as illustrated in Fig. 2, the moving
directions that are Doppler-covered by a radar form two
separate and opposite angular ranges. This geometric structure induces additional complexity into the Doppler coverage
problem.
With the proposed Doppler coverage model, it is natural to ask a fundamental question: How can the Dopplercovered regions for arbitrarily deployed radar sensors be
characterized? The answer to this question can be used to
evaluate the coverage of sensors under random deployment.
In practice, one efficient sensor deployment strategy is deterministic deployment, in which we can place sensors at
desirable locations. Another related problem is how to deploy
the minimum possible number of radar sensors in a region
such that the entire region is Doppler-covered. The main thrust
of this paper is devoted to answering these two questions.
We summarize the main contributions as follows. First,
we introduce a novel Doppler coverage model for moving
target detection employing the Doppler effect in radar sensor
networks. Next, we propose an efficient method to characterize
the Doppler-covered regions for arbitrarily deployed radar
sensors. Then we design an algorithm CSR for deriving
the minimum sensor density required to achieve Doppler
coverage in a region under any polygonal deployment pattern,
and further apply it to investigate the regular triangle based
deployment. To the best of our knowledge, this work is the
first to investigate the Doppler effect for the coverage of radar
networks.
The rest of this paper is organized as follows. Section II
introduces the model of Doppler coverage. In Section III,
we characterize the Doppler-covered regions for arbitrary
deployed radar sensors. We investigate in Section IV the minimum radar density required to achieve Doppler coverage in
a region under deterministic deployment. Section V provides
some numerical results. The paper is concluded in Section VII.
II. M ODEL AND N OTATION
We consider a set of monostatic2 radar sensors deployed in
a region of interest F for detecting a target. For convenience,
we also use Si to denote the location (point) of a sensor Si .
We denote ∥U V ∥ as the (Euclidean) distance between two
points U and V . We denote the angle between two vectors
(directions) v⃗1 and v⃗2 as α(v⃗1 , v⃗2 ).
For a radar Si , the received SNR from a target located at a
point P is given by [5]
SNR =
K
∥Si P ∥4
(1)
where K denotes radar constant which reflects physical layer
characteristics of the radar (e.g., transmit power, radar cross
section3 , transmitter/receiver antenna power gains). The sensing range r of a radar can be determined based on (1) such that
the SNR received by the radar is higher than some predefined
threshold when the target is within its sensing range. A point
P is said to be covered by a sensor Si (in the SNR sense) if
it is within the sensing range of Si , i.e., ∥Si P ∥ ≤ r.
Let V be the speed of a moving target, and ϕ be the angle
at which the target is moving with respect to a radar. Then the
DFS detected by the radar, denoted by ∆f , is
2V cos ϕ
f0
(2)
c
where f0 is the frequency of the emitted radar signal and c
is the speed of light. We assume that the target’s speed V
is known or is lower bounded by some known value. This
knowledge can be obtained from some general information
about the target (e.g., knowing that the target is a car or a
plane).
For ease of exposition, we assume that all radars have
homogeneous physical characteristics such that they have the
same radar constant. We also assume that radars share a set
∆f =
2 A monostatic radar is comprised of co-located radar transmitter and
receiver, which is the typical radar configuration.
3 Radar cross section measures the amount of radar signal energy reflected
by an object due to its physical characteristics such as shape, size, and
material.
complementary safe region
Si
S2′
S1
2θ
S3
π − 2θ
P
P1′
P1
P2
2θ
P2′
S3′
S2
S1′
Sj
safe region
Fig. 3. The coverage list LP = (S1 , S2′ , S3 , S1′ , S2 , S3′ ) is constructed
from SP = {S1 , S2 , S3 } based on the angular positions of the sensors with
respect to P .
of orthogonal radio resources (e.g., different frequencies or
time slots, orthogonal waveforms) through proper spatial reuse
such that mutual interferences can be avoided. We further
assume that the transmitted and reflected radar signals are
omni-directional. Therefore, a radar’s coverage region is a disk
with radius r. However, our results can be easily extended to
the case of sector-based coverage region.
Definition 1: (Doppler Coverage) A direction d⃗ from a
point P is Doppler-covered (D-covered) by a sensor Si if P is
−→
−→
⃗−
⃗−
covered by Si and α(d,
P Si ) ≤ θ or π − θ ≤ α(d,
P Si ) ≤ π
where θ ∈ [0, π/2) is a predefined parameter called effective
Doppler angle. A point P is Doppler-covered if any direction
from P is Doppler-covered by some sensor.
Given the target’s speed V or a lower bound on V , θ can
be determined based on (2) such that the DFS detected by
a radar is larger than some predefined threshold when the
target’s direction of motion is D-covered by that radar. For
convenience, we say that a direction d⃗ from P can be Dcovered by a sensor Si if it is D-covered by Si given that
P is covered by Si . Similarly, we say that a point P can be
D-covered by a set of sensors S if any direction from P can
be D-covered by some sensor in S. We say that a region is
D-covered if every point in it is D-covered.
III. C HARACTERIZATION OF D OPPLER C OVERAGE
In this section, we propose an efficient method to characterize D-covered regions for arbitrary deployed sensors. Our
results in this section can be used to evaluate the Doppler
coverage of deployed sensors under random deployment.
We denote a set of sensors that cover a point P as SP .
We also use SA to denote a set of sensors that cover a
region A if any sensor in SA covers all the points in A.
For any point P , we can define a coverage list constructed
from SP = {S1 , . . . , Sn }, as follows and illustrated in Fig. 3.
First, we construct an image point Si′ for each Si ∈ SP
−−→
with respect to P such that the direction of P Si′ is opposite
−−→
to P Si . We pick any Si ∈ SP (or its image Si′ ) and
−−→
place it in the coverage list. Then we rotate P Si around
P in the counter-clockwise (or clockwise) direction until it
Fig. 4. The safe region of Si and Sj is the area outside that enclosed by arcs
ú ú
ú
ú
arcs S
P S and S
P S .
Si P1 Sj and Si P1′ Sj ; the complementary safe region is the area enclosed by
i 2 j
′
i 2 j
−−→
−−→
points to the same direction as P Sj (or P Sj′ ) for some
Sj ∈ SP , and place Sj (or Sj′ , respectively) in the list next
to Si . We continue rotating until all the sensors in SP and
their images are placed in the list sequentially, denoted by
LP . One can easily check that LP is a circular list and
can be written as LP = (s1 , . . . , sn , s′1 , . . . , s′n ), where si
represents some sensor Sj ∈ SP or its image Sj′ , and s′i
represents the counterpart of si , i.e., the image Sj′ or the
sensor Sj , respectively. For convenience, we may write LP
in a compact form without ambiguity as LP = (s1 , . . . , sn )
by including only half consecutive points and let sn+1 = s′1 .
For example, (S1 , S2′ , S3 , S1′ , S2 , S3′ ) can be written for short
as (S1 , S2′ , S3 ), (S2′ , S3 , S1′ ), or (S3′ , S1 , S2′ ). Also for convenience, we use S(si ) to denote the sensor represented by
si if si represents a sensor, or the sensor whose image is
represented by si if si represents an image. For example,
S(S1 ) = S(S1′ ) = S1 .
Lemma 1: A point P is Doppler-covered by SP if and only
if ∠si P si+1 is less than or equal to 2θ for any neighbor si
and si+1 in the coverage list LP constructed from SP .
Proof: Suppose the condition holds. Then for any direction
d⃗ from P , there exists a sensor Si or an image Si′ such that
−−→ ⃗
−−→ ⃗
−−→ ⃗
α(P Si , d)
≤ θ or π − α(P Si , d)
= α(P Si′ , d)
≤ θ. Thus P is
D-covered.
Suppose Si and Sj are neighbors in LP and ∠Si P Sj > 2θ.
Then there exists a direction d⃗ from P which is parallel to
−−→ ⃗
−−→ ⃗
the bisector of ∠Si P Sj such that α(P Si , d)
= α(P Sj , d)
>
−−→ ⃗
−−→ ⃗
θ. It follows that α(P Sk , d) > θ and π − α(P Sk , d) =
−−→ ⃗
> θ for any Sk ∈ SP . Thus P is not D-covered.
α(P Sk′ , d)
Lemma 1 gives a sufficient and necessary condition for a
point to be D-covered. We need to verify for which points in
the region F this condition holds. To this end, we introduce
the following concepts. For two sensors Si and Sj , we define
safe region (SR) to be the area in which ∠Si P Sj ≤ 2θ for any
point P , and complementary safe region (SSR) to be the area
in which π − ∠Si P Sj ≤ 2θ for any point P . Accordingly, the
areas complementary to the safe region and complementary
safe region are called the unsafe region and complementary
S4
S4
S1
S1
S4
S1
O
O
P2
P1
S2
S3
S3
S2
S3
S2
(a)
(b)
Fig. 5. P1 and P2 are both covered by SP1 = SP2 = {S1 , S2 , S3 , S4 }
but have different coverage lists: LP1 = (S1 , S4 , S3 , S2 ), LP2 =
(S1 , S3′ , S4 , S2′ ).
Fig. 6. The dashed arcs are parts of the boundaries of the SRs and SSRs
constructed for finding D-covered regions within sub-region (a) ∆S2 OS3 and
(b) ∆S3 OS4 ; the shaded area is D-covered since it is the intersection of the
SRs and SSRs within the sub-region.
unsafe region, respectively.
Lemma 2: For two sensors Si and Sj , the safe region is
the closed region complementary to the open region bounded
′
by two arcs Si Sj and Si Sj such that ∠Si P Sj = 2θ for
′
any P ∈ Si Sj ∪ Si Sj ; the complementary safe region is the
′
closed region bounded by two arcs Si Sj and Si Sj such that
′
∠Si P Sj = π − 2θ for any P ∈ Si Sj ∪ Si Sj .
the points in Ai have the same coverage list.
ø
ø
ø ø
ø
ø
ø ø
ø′
Proof: As illustrated in Fig. 4, the two arcs Sø
i Sj and Si Sj
are segments of the circumscribed circles of triangles △Si P Sj
and △Si P ′ Sj where ∠Si P Sj = ∠Si P ′ Sj = 2θ (or π − 2θ),
and P and P ′ are two points on different sides of the line
segment Si Sj . The proof of the desired property follows from
elementary geometry.
For convenience, if si and sj both represent sensors or both
represent images, we denote by R(si , sj ) the safe region of
two sensors S(si ) and S(sj ); otherwise, if si and sj represent
a sensor and an image, we denote by R(si , sj ) the complementary safe region of S(si ) and S(sj ). For example, R(S1 , S2 )
and R(S1′ , S2′ ) both denote the safe region of S1 and S2 ;
R(S1′ , S2 ) and R(S1 , S2′ ) both denote the complementary safe
region of S1 and S2 . We use Rc (si , sj ) to denote the region
complementary to R(si , sj ), i.e., the unsafe or complementary
unsafe region of S(si ) and S(sj ).
Lemma 3: For a region A covered by SA , A is partitioned
by all the lines passing through a pair of sensors in SA into a
set of sub-regions such that all the points in a sub-region have
the same coverage list constructed from SA .
Proof: For any point P ∈ A, depending on P is located at which side of the line l passing through Si and
Sj in SA , the relative orders of Si , Sj , Si′ and Sj′ in
the coverage list LP constructed from SA either has the
form LP = (. . . , Si , . . . , Sj , . . . , Si′ , . . . , Sj′ , . . .) or LP =
(. . . , Si , . . . , Sj′ , . . . , Si′ , . . . , Sj , . . .). Suppose Ai is a partition
of A divided by all the lines passing through a pair of sensors
in SA . Then any two points P1 and P2 in Ai are located at
the same side of each line, and hence the relative orders of
any pair of sensors in SA and their images are the same in
LP1 and LP2 . One can easily verify that there exists a unique
coverage list which satisfies all these relative orders. Thus all
Algorithm 1: Doppler-covered regions
input : region F , set of deployed sensors S
output: D-covered regions in F
1 Partition F into a set of sub-regions A such that each
sub-region Ai ∈ A is covered by the same set of sensors
SAi ;
2 foreach sub-region Ai ∈ A do
3
Partition Ai into a set of sub-regions Ai such that all
the points in a sub-region Aji have the same coverage
list LAj constructed from SAi ;
i
4
5
6
7
foreach sub-region Aji ∈ Ai do
return a D-covered region
Aji ∩ (∩nk=1 R(sk , sk+1 )) where
LAj = (s1 , · · · , sn );
i
end
end
Theorem 1: The Doppler-covered regions in F can be
determined by Algorithm 1.
Proof: Since ∠si P si+1 = ∠s′i P s′i+1 for any i ∈
{1, . . . , n}, by Lemma 1, it suffices to check if ∠si P si+1
for any i ∈ {1, . . . , n} is less than or equal to 2θ. Then by
Lemma 2, the intersection of the SRs and SSRs constructed in
Algorithm 1 within the sub-region Aji is a D-covered region.
Following a similar argument based on Euler’s formula as
in [6], the number of sub-regions in A is O(|S|4 ). The number
of sub-regions in Ai is O(|SAi |4 ) and the reason is as follows.
The number of lines passing through a pair of points in a set
of n points is at most n(n − 1)/2. By induction, one can
easily show that n lines can partition the plane into at most
n(n+1)/2+1 regions. Therefore, all the lines passing through
a pair of sensors in |Ai | sensors partition SAi into O(|SAi |4 )
sub-regions.
Fig. 6 gives an example to illustrate the idea of line 5
in Algorithm 1. In Fig. 6 (a), we find the D-covered region
in a sub-region ∆S2 OS3 where the coverage list is LP =
(S1 , S3′ , S2 , S4′ ) for any P ∈ ∆S2 OS3 . Hence, we construct
the SR boundary for S1 and S4 , the SSR boundary for S1 and
S3 , S2 and S3 , S2 and S4 , respectively. For brevity, we only
plot the parts that intersect with ∆S2 OS3 . The intersection
area (shaded) of the corresponding SRs and SSRs within
∆S2 OS3 is the D-covered region within ∆S2 OS3 . Similarly,
in Fig. 6 (b), we find the D-covered region within a sub-region
∆S3 OS4 with coverage list LP = (S1 , S2 , S4′ , S3 ) for any
P ∈ ∆S3 OS4 . In this case, we construct the SR boundary for
S1 and S2 , the SSR boundary for S1 and S3 , S2 and S4 , S3
and S4 , respectively.
IV. C RITICAL S ENSOR D ENSITY FOR D OPPLER C OVERAGE
UNDER D ETERMINISTIC D EPLOYMENT
In this section, we consider deterministic deployment where
sensors can be precisely placed at any locations in the region
F . We design an algorithm to derive the minimum sensor
density required for the entire region to be D-covered under a
particular deployment pattern.
A. Problem Formulation
We consider a polygonal deployment pattern comprised of
basic patterns (polygons) of the same shape and size (e.g.,
triangles, rectangles, hexagons) where sensors are deployed
on the vertices of the polygons (as illustrated in Fig. 7).
Under such a deployment pattern, there exists a unit region
U ⊂ F such that U is D-covered if and only if the entire
region F is D-covered. For example, as illustrated in Fig. 7,
a unit region for the regular triangle pattern can be a regular
(equilateral) triangle, or can be just a partition of a regular
triangle divided by the perpendicular bisectors of its three
sides. This is because for any point P in F covered by any
set of sensors SP , there exists a point P ′ in U covered by
some set of sensors SP ′ such that the relative positions of
sensors in SP ′ with respect to P ′ are the same as those in SP
with respect to P , and hence the D-coverage on P and P ′ are
equivalent. As a result, it suffices to focus on the D-coverage
in a unit region.
We assume the region F is sufficiently large and thus
the boundary effect can be ignored. Given the shape of the
polygons, a Doppler effective angle θ, and a sensing range r,
our goal is to find the minimum sensor density required to Dcover F , which is equivalently to find the maximum possible
size l∗ (θ, r) of the polygons under which F is D-covered. It
can be easily shown that, given any θ, l∗ (θ, r) is a one-toone increasing function of the sensing range r. Therefore, it
suffices to find the minimum sensing range r∗ (θ, l) required to
D-cover F as a function of θ and l, from which we can obtain
−1
l∗ (θ, r) as the inverse function (r∗ ) (θ, r) with θ being a
parameter. In the next subsection, we design an algorithm
CSR for finding the Critical Sensing Range r∗ (θ, l). We will
use r∗ instead of r∗ (θ, l) for brevity.
B. CSR Design
As described in Algorithm 2, the design of CSR essentially
follows a divide and conquer approach. Before proceeding to
unit region
P
unit region
P
P′
P′
(a)
(b)
Fig. 7. The relative positions of P and P ′ are the same in their respective
unit regions, which are partitions divided by the dashed lines within (a) a
regular triangle and (b) a square, such that the Doppler coverage on P and
P ′ are equivalent.
Algorithm 2: CSR (Critical Sensing Range)
input : effective Doppler angle θ
output: critical sensing range r∗
1 S ← ∅;
2 repeat
3
S ← arg minSi ∈S c dmax
Si (U );
4
S ← S ∪ S;
5 until U can be D-covered by S;
+
max
6 r ← dS (U );
′
c min
+
7 S ← {Si ∈ S |dS (U ) ≤ r };
i
−
min
8 r ← minSi ∈S ′ ∪S dS (U );
i
+
′
9 S ←S ∪S ;
−
max
−
10 S ← {Si ∈ S|dS (U ) ≤ r };
i
∗
−
11 r ← r ;
12 Partition U into a set of sub-regions U such that all the
points in a sub-region Ui ∈ U have the same coverage
list LUi constructed from S + ;
13 foreach Ui ∈ U do
14
Split LUi into partial coverage lists L1 , · · · , Lk such
that Lj = (sj , · · · , sj+1 ) for any j ∈ {1, · · · , k}
where S(sj ) ∈ S − for any j ∈ {1, · · · , k} and
k = |S − |;
15
foreach j ∈ {1, · · · , k} do
16
r∗ ← max{r∗ , PCSR (Ui , Lj )};
17
end
18 end
∗
19 return r ;
“divide”, CSR first constructs a set of sensors S by iteratively
including the sensor requiring the critical sensing range to
cover U , until U can be D-covered by S. In other words, if
the sensing range r is large enough such that U is covered by
S, then U is also D-covered by S. To verify this condition, we
can use a procedure similar to Algorithm 1: First we partition
U into a set of sub-regions U such that all the points in a subregion Ui ∈ U have the same coverage list LUi constructed
from S; then U can be D-covered by S if and only if Ui ∩
Rc (sj , sj+1 ) = ∅ for any neighbor sj and sj+1 in each LUi .
max
We denote by dmin
Si (A) , minP ∈A ∥Si P ∥ and dSi (A) ,
maxP ∈A ∥Si P ∥ the minimum sensing range required for a
sensor Si to cover any point in a region A and all the points
in A, respectively. We also denote by S c the set of sensors
not included in S.
Based on S, CSR can find a lower bound r− and an upper
bound r+ such that r− ≤ r∗ ≤ r+ , and further find S − as a
set of sensors that must cover all points in U when r = r∗ ,
and S + as a set of sensors such that those not included in S +
must not cover any point in U when r = r∗ . Thus S + \ S −
includes all sensors that possibly cover some points in U when
r = r∗ . S − and S + serve as the basis for the “divide” steps
described below.
The “divide” steps are two-fold. First, using Lemma 3,
U is partitioned into a set of sub-regions U such that all
the points in a sub-region Ui ∈ U have the same coverage list LUi constructed from S + . Second, each coverage
list LUi is split into k partial coverage lists (not circular)
L1 , · · · , Lk , where k is the number of sensors in S − , such
that for any j ∈ {1, · · · , k}, Lj starts from some sj with
S(sj ) ∈ S − and ends at the next sj+1 with S(sj+1 ) ∈ S − .
For example, if S + = {S1 , · · · , S10 }, S − = {S2 , S5 , S6 , S9 },
and LUi = (S3 , S4′ , S2 , S8 , S6′ , S1 , S7′ , S5 , S10 , S9′ ), then LUi
can be split into L1 = (S2 , S8 , S6′ ), L2 = (S6′ , S1 , S7′ , S5 ),
L3 = (S5 , S10 , S9′ ), L4 = (S9′ , S3′ , S4 , S2′ ).
Taking a region V and a partial coverage list L =
(s1 , · · · , sn ) as inputs, a separate algorithm PCSR is used
to find the minimum sensing range required to D-cover all
the directions, if there is any, from any point P ∈ V and
−−→
−−−−→
between P sj and P sj+1 that can not be D-covered by S(s1 )
or S(sn ). Then CSR returns the maximum among r− and the
outputs of PCSR applied over all partial coverage lists for all
sub-regions in U.
As described in Algorithm 3, PCSR is also based on
a divide and conquer approach. PCSR first checks if all
−−→
−−→
the directions from any P ∈ V between P s1 and P sn
can be D-covered by S(s1 ) or S(sn ). If yes, it returns 0;
otherwise, PCSR partitions V , if necessary, to a set of subregions V such that for any sub-region Vi ∈ V, there exists
some sensor S(sk ) closest to all the points in Vi among
the sensors in {S(s2 ), · · · , S(sn−1 )}. This partition can be
done by constructing the perpendicular bisectors of all the
line segments, if necessary, connecting a pair of sensors
in {S(s2 ), · · · , S(sn−1 )}. In this case, PCSR returns the
maximum among the outputs of PCSR recursively applied
over all sub-regions in V.
If the input region V does not need to be partitioned, then
c
PCSR returns the maximum of dmax
S(sk ) (V ∩R (s1 , sn )) and the
outputs of PCSR applied recursively for region V ∩Rc (s1 , sn )
with partial coverage list (s1 , · · · , sk ), (sk , · · · , sn ), respectively. This is to guarantee that S(sk ) can cover all the points
in V ∩ Rc (s1 , sn ), from which there exist directions between
−−→
−−→
P s1 and P sn that can not be D-covered by S(s1 ) or S(sn ).
Next we prove the correctness of CSR.
Theorem 2: The critical sensing range r∗ can be found by
CSR.
Proof: First we show that CSR can find r− , r+ , S − , and
Algorithm 3: PCSR (CSR for a Partial coverage list)
input : region V , partial coverage list L = (s1 , · · · , sn )
output: critical sensing range r for V and L
1 r ← 0;
c
2 if V ∩ R (s1 , sn ) ̸= ∅ then
3
S ← {s2 , · · · , sn−1 };
4
if @sk ∈ S such that ∥P S(sk )∥ ≤ ∥P S(si )∥ for any
P ∈ V and any si ∈ S then
5
Partition V into a set of sub-regions V such that
for any Vi ∈ V, ∃sk ∈ S such that
∥P S(sk )∥ ≤ ∥P S(si )∥ for any P ∈ Vi and any
si ∈ S;
6
foreach Vi ∈ V do
7
r ← max{r, PCSR(Vi , L)};
8
end
9
end
10
V ← V ∩ Rc (s1 , sn );
11
Split L into partial coverage lists L1 = (s1 , · · · , sk ),
L2 = (sk , · · · , sn );
12
r ← max{dmax
S(sk ) (V ), PCSR(V, L1 ),PCSR(V, L2 )};
13 end
14 return r;
S + correctly. Based on the construction of S (line 2-5), when
r = dmax
S (U ) where S is the last sensor included in S, U is D+
covered by S. This implies that r∗ ≤ dmax
S (U ) = r . From the
′
+
value assignments of S and S (line 7 and 9), one can see that
+
+
any Si with dmin
Si (U ) ≤ r must be in S . Therefore, any Si ∈
+ c
min
+
(S ) must satisfy dSi (U ) > r ≥ r∗ , and hence does not
cover any point in U when r = r∗ . Let S ′′ = {Si |dmin
Si (U ) <
minSj ∈S ′ ∪S dmin
(U
)}
be
the
set
of
all
sensors
that
cover
some
Sj
min
′
points in U when r = minSj ∈S ∪S dSj (U ). One can observe
that S ′′ ⊆ S \ {S}. Then U can not be D-covered by S ′′
since it can not be D-covered by S \ {S}. This implies that
−
r∗ ≥ minSj ∈S ′ ∪S dmin
Sj (U ) = r . Then it follows from the
−
value assignment of S (line 10) that S − is a set of sensors
that must cover all the points in U when r = r∗ ≥ r− .
Let r′ be the actual output of CSR. To show that r′ is
equal to the critical sensing range r∗ required to D-cover U ,
it suffices to show that 1) U is D-covered when r = r′ and
2) U is not D-covered when r = r0 for any r0 < r′ . Part
1) can be shown as follows. First we observe that CSR can
always terminate. The only possible case where CSR can not
terminate is when PCSR is input with a region V and a partial
coverage list (s1 , s2 ) of length 2 but V ∩ Rc (s1 , s2 ) ̸= ∅.
However, this can not occur since s1 and s2 must be neighbors
in the coverage list of points in V constructed from S + , and
V ⊆ U can be D-covered by S + . Next, we observe that the
output of any execution of PCSR is set to ganrantee that
all the points in V ∩ Rc (s1 , sn ) are covered by S(sk ) when
r = r′ . Thus, for any execution of PCSR, all the points in V
are covered by S(s1 ) and S(sn ) when r = r′ . One can observe
from the designs of CSR and PCSR that any direction d⃗ from
I
J
E
A
A
D
Q
Q
E
A
N
O
Q
O
Q
M
B
G
C
O
R
O
C
B
B
K
S
L
T
d
N
L
G
C
d
N
L
W
X
′
C
F
(a)
(b)
H
F
H
(c)
Fig. 8. CSR applied for the regular triangle pattern: (a) π/3 ≤ θ < π/2 (b) π/4 ≤ θ < π/3 (c) π/6 ≤ θ < π/4. Each dashed arc is part of the SR or
′
SSR boundary of the two sensors connected by it. The following notes are for sub-figure (c). N R is part of AB. N L is part of B C. N L is part of AE. The
′
perpendicular bisector of DG intersects N L, OC at K, S, respectively. JF intersects N L, N L , OC at W , X, T , respectively. The input regions in (4) and
(5) are indicated by their boundary (segments) as follows. U1 = △OQT ; U2 = △T QC; U11 : N Q, QT , T R, RN ; U12 : N Q, QX, X N ; U13 = △OQS;
U14 = △SQT ; U15 : N Q, QK, K N ; U16 : KQ, QW , W K.
ö
öö
ö
÷
ø
öö
ö
öö
÷
ö
PCSR[U, (A, F ′ , E, B ′ )] = max{∥EM ∥, PCSR[U, (A, F ′ , E)], PCSR[U, (E, B ′ )]} = max{∥EM ∥, 0, 0} = ∥EM ∥
(3)
PCSR[U1 , (A, F ′ , J, E, B ′ )] = max{∥ER∥, PCSR[U11 , (A, F ′ , J, E)], PCSR[U11 , (E, B ′ )]}
= max{∥ER∥, max{∥F Q∥, PCSR[U12 , (A, F ′ )], PCSR[U12 , (F ′ , J, E)]}, 0}
= max{∥ER∥, max{∥F Q∥, 0, 0}, 0} = ∥F Q∥
(4)
PCSR[U1 , (B ′ , G, D′ , C)] = max{PCSR[U13 , (B ′ , G, D′ , C)], PCSR[U14 , (B ′ , G, D′ , C)]}
= max{max{∥DK∥, PCSR[U15 , (B ′ , G, D′ )], PCSR[U15 , (D′ , C)]},
max{∥GK∥, PCSR[U16 , (B ′ , G)], PCSR[U16 , (G, D′ , C)]}}
= max{max{∥DK∥, 0, 0}, max{∥GK∥, 0, 0}} = ∥GK∥
(5)
any point P ∈ U will be checked in some execution of PCSR
where d⃗ can be D-covered by S(s1 ) or S(sn ). Therefore, d⃗
must be D-covered when r = r′ .
C. CSR Case Study: Regular Triangle
Next we show part 2). Suppose r = r0 where r0 < r′ . If
r = r− , the desired result directly follows from r0 < r′ =
r− . Otherwise, r′ must be assigned the value of dmax
S(sk ) (V ∩
Rc (s1 , sn )) during some execution of PCSR[V, (s1 , · · · , sn )].
We observe that there must exist a point P ∈ (V ∩ Rc (s1 , sn ))
c
′
with ∥P S(sk )∥ = dmax
S(sk ) (V ∩ R (s1 , sn )) = r > r0 , and a
−
−
→
−
−
→
direction d⃗ from P and between P s1 and P sn such that d⃗
can not be D-covered by S(s1 ) or S(sn ), and hence also can
not be D-covered by any Si ∈ S + \ S. Due to the partition
step in PCSR, we must have r0 < ∥P S(sk )∥ ≤ ∥P Si ∥ for
any Si ∈ S. Hence d⃗ is not D-covered by any Si ∈ S since
P is not covered by Si . Thus, d⃗ is not D-covered by any
Si ∈ S + . Since P ∈ V ⊆ U can be D-covered by S + , and P
is not covered by any Si ∈ (S + )c when r = r∗ , there must
exist some Sj ∈ S + such that P is covered by Sj and d⃗ is
D-covered by Sj when r = r∗ . Thus Sj must be from S.
Then it again follows from the partition step in PCSR that
r∗ ≥ ∥P Sj ∥ ≥ ∥P S(sk )∥ = r′ > r0 . This implies that P is
not covered by any Si ∈ (S + )c and hence d⃗ is not D-covered
by any Si ∈ (S + )c . Therefore, d⃗ is not D-covered by any
In this subsection, we apply CSR to investigate the regular
triangle based deployment. This case study also serves as an
example to illustrate how CSR works. We focus on the Dcoverage of a unit region U = △COQ as shown in Fig. 8.
For any input θ ∈ [0, π/2), we observe a lower bound that
l ≤ r∗ . The reason is that when r < l, there exists a point
P sufficiently close to any given sensor such that P is only
covered by that sensor, and thus must not be D-covered. In
what follows, we present the results for θ ∈ [π/6, π/2).
Consider the case θ ∈ [π/3, π/2) as illustrated in Fig. 8(a).
In this case, r∗ can be found without using CSR as follows.
When r = l, U is covered by {A, B, C} with coverage list
(A, B ′ , C). Since the SSRs of A and B, B and C, A and C,
all include U , U is D-covered. This implies that r∗ ≤ l. Then,
using the lower bound l ≤ r∗ , we conclude r∗ = l.
Consider the case θ ∈ [π/4, π/3) as illustrated in Fig. 8(b).
Then, using CSR, we have r− = l, S − = {A, B, C},
S + = {A, B, C, E, F, G, H}, LU = (A, F ′ , E, B ′ , G, C, H).
It follows from U ⊆ R(C, A′ ) and U ⊆ R(B ′ , C) that
PCSR[U, (C, H, A′ )] = 0 and PCSR[U, (B ′ , G, C)] = 0.
Since ∥EP ∥ ≤ ∥F P ∥ for all P ∈ U and U ⊆ R(A, E),
′
sensor, from which we conclude that U is not D-covered. 0.4
0.2
0
200
400
600
800
Number of deployed sensors
1000
0.6
0.4
θ = π/6
θ = π/4
θ = π/3
0.2
0
200
400
600
800
Number of deployed sensors
1000
1
0.8
0.6
0.4
0.2
0
1500
θ = π/6
θ = π/4
θ = π/3
2000
2500
3000
Number of deployed sensors
Probability of doppler coverage
0.6
0.8
Probability of doppler coverage
0.8
1
θ = π/6
θ = π/4
θ = π/3
Doppler covered percentage
Doppler covered percentage
1
1
0.8
0.6
0.4
θ = π/6
θ = π/4
θ = π/3
0.2
3500
0
200
400
600
800
1000
Number of deployed sensors
1200
Fig. 9. Doppler-covered percentage Fig. 10. Doppler-covered percentage Fig. 11. Probability of Doppler cov- Fig. 12. Probability of Doppler coverage vs sensor number: r = 5.
erage vs sensor number: r = 10.
vs sensor number: r = 5.
vs sensor number: r = 10.
÷
U ⊆ R(E, B ′ ), we have (3), where AB intersects OC at M
c
′
and ∥EM ∥ = dmax
E (U ∩ R (A, B )). Therefore, the output of
∗
−
CSR is r = max{r , 0, 0, ∥EM ∥} = ∥EM ∥. By geometric
calculation, the closed form expression for r∗ can be found as
r∗ = l
̂ √
3
1
1
−
−
2
sin 2θ tan 2θ
Œ2
to overcome the boundary effect. We use three sets of effective
Doppler angles: θ = π/6 (30◦ ), θ = π/4 (45◦ ), and θ = π/3
(60◦ ). We assume that sensors are deployed uniformly in the
region F under random deployment.
A. Doppler-covered Percentage
+ 1.
Consider the case θ ∈ [π/6, π/4) as illustrated in Fig. 8(c).
Then, using CSR, we have r− = l, S − = {A, B, C},
S + = {A, B, C, D, E, F, G, H, I, J}, LU1 = (A, F ′ , J, E,
B ′ , G, D′ , C, H, I ′ ), LU2 = (A, J, F ′ , E, B ′ , G, D′ , C, H, I ′ ),
where U is partitioned by JF into U1 and U2 . Regarding
U1 , we have PCSR[U1 , (C, H, I ′ , A′ )] = 0, and also (4),
(5), where all the input regions are indicated in Fig. 8(c).
Therefore, the maximum value output by PCSR applied for
U1 is max{∥F Q∥, ∥GK∥, 0} = ∥GK∥. On the other hand,
following similar steps (omitted), the maximum value output
by PCSR applied for U2 is ∥GW ∥. Therefore, CSR returns
r∗ = max{r− , ∥GK∥, ∥GW ∥} = ∥GK∥. The closed form
expression for r∗ is lengthy and thus omitted here.
D. Discussions
We have a few remarks on the complexity of CSR. Following a similar argument to the one in Section III, the complexity
of verifying S once in line 5 is O(|S|5 ), and hence the
complexity of the repeat loop is |S| × O(|S|5 ) = O(|S|6 ).
Since |U| is O(|S + |4 ), the total number of executions of
PCSR in line 16 is |U| × |S − | = O(|S + |4 |S − |). We note
that the complexity of PCSR can be upper bounded by
some function of the length of the input partial coverage list,
which determines the maximum possible number of recursive
executions of PCSR caused by it. As the input θ decreases,
both |S − | and |S + | will increase, and the length of a partial
|S + |
coverage list obtained in line 14 is around |S
− | + 1.
V. N UMERICAL R ESULTS
In this section, we provide some numerical results on the
Doppler coverage. We illustrate the relationship between the
number of deployed sensors and two performance metrics of
the Doppler coverage under random deployment.
We consider a region of interest F with an area of 100×100.
We use two sets of sensing ranges: r = 5 and r = 10. We
deploy sensors in an enlarged area of (100 + 2r) × (100 + 2r)
In this part, we vary the number of deployed sensors from
100 to 1000 for all configurations to illustrate the D-covered
percentage, which is the percentage of D-covered areas in the
region F . For a given random deployment of sensors, we use
Algorithm 1 to find the D-covered regions and then calculate
the D-covered percentage. We run the experiment for 100
times and take the averaged results. Figure 9 and 10 present
the results of D-covered percentage with different numbers of
deployed sensors. As expected, the D-covered percentage is
always higher for a larger θ. The D-covered percentage for
r = 10 is much higher than that for r = 5. The D-covered
percentage increases as the number of deployed sensors grows,
and in particular, the increasing speed is reduced rapidly when
the D-covered percentage is close to 1.
B. Probability of Doppler Coverage
In this part, we vary the number of deployed sensors from
1500 to 3500 for r = 5 and from 200 to 1200 for r = 10,
to illustrate the probability of Doppler coverage. Following
the methodology in [7], we use a grid of discrete points
to approximate the region F . We set the side length of the
grid to 1. The probability of Doppler coverage is defined as
the probability that all the grid points are D-covered. We
calculate it as the ratio between the number of times that
this event occurs, and the total number of experiments that
is set to 100. Figure 11 and 12 present the results of the
probability of Doppler coverage with different numbers of
deployed sensors. Similar observations can be made as in the
D-covered percentage part. By comparing these two figures
with Figures 9 and 10, we observe that many more sensors
are required to achieve a high probability of D-coverage than
to achieve a high D-covered percentage.
VI. R ELATED W ORK
Radar technologies have continued to advance ever since
its invention in the early 20th century [5]. However, radar
networks have begun to draw research interests only in the
past few years, largely driven by the emergence of cheap,
efficient, and compact radar sensors in place of conventionally
expensive and bulky radar systems. For example, in [1], a
platform has been successfully designed and built to integrate ultrawideband radars with mote-class sensor devices.
The existing literature have studied different problems arising
from radar sensor networks, including waveform design and
diversity [8], [9], radar scheduling [10], data management [11],
for a variety of objectives, such as target detection [12],
localization [13], and tracking [14], [15]. In particular, Doppler
effect has also been considered in several works [14], [15].
However, very little attention has been paid to coverage issues
specific to radar sensor networks, especially by exploring the
Doppler effect. Our work fills this void by taking into account
the Doppler effect for moving target detection.
There have been vast literature on sensor network coverage.
A recent and comprehensive survey can be found in [4].
Many existing works focus on area coverage problems, where
the subject to be covered is the entire area of a region.
It has been proved in [16] that the regular triangle lattice
is the optimal deterministic deployment pattern in terms of
requiring the minimum number of sensors to cover a region.
The optimal deployment for achieving both network coverage
and connectivity has been studied in a line of works [17].
Our work in this paper also falls into the category of area
coverage. As the Doppler coverage model is very different
from traditional distance-based models, the existing results can
not be applied to our problem.
A full-view coverage model has been recently introduced
in [18] for camera sensor networks. This model shares a
similar flavor with the Doppler coverage model in our study:
An object is full-view covered if its facing direction is sufficiently close to some camera’s viewing direction. However,
our work is quite different from [18] due to several reasons.
First, our model is motivated by the Doppler effect, which is
an utterly different physical phenomenon. Second, a direction
is D-covered if it is sufficiently close to either the direction
towards or the direction opposite to a radar, rather than
only sufficiently close to a camera’s viewing direction as in
the full-view coverage model. Due to this complication, the
Doppler coverage problem studied here is more complex and
hence challenging. Third, while the critical sensor density
for full-view coverage is derived in [18] specifically for the
regular triangle based deployment, the algorithm designed in
our work can apply for any polygon deployment pattern.
Therefore, our algorithm can be modified to apply for the
full-view coverage model. Based on the full-view coverage
model, barrier coverage for camera sensors has been studied
in [6] and [19]. [20] has investigated the full-view coverage
for heterogeneous camera sensors under random deployment.
VII. C ONCLUSION
The radar network has emerged as a promising paradigm for
many applications. Based on a key observation that the existing
sensor coverage models fail to capture the Doppler effect,
which can otherwise be employed by a radar to distinguish
a moving target from stationary or slow-moving clutter, we
introduce a novel Doppler coverage model. A point P is said
to be D-covered if for any direction d⃗ from P , there exists
some sensor Si such that P is within Si ’s sensing range, and
−−→
the angle between d⃗ and P Si is no greater than θ or no less
than π − θ, where θ is a predefined value that depends on the
target speed and the Doppler resolution of the radar. Based on
this Doppler coverage model, we propose an efficient method
to characterize D-covered regions given arbitrary deployed
radar sensors. We also design an algorithm CSR to derive the
minimum sensor density required to Doppler cover a region
under any polygonal deployment pattern, and further apply it
to investigate the regular triangle based deployment.
Our results in this paper can be used to evaluate the coverage
of any deployed radar sensor network where a worst-case
performance guarantee is needed. We are currently studying
the sensor density required for a region to be Doppler-covered
under random deployment.
R EFERENCES
[1] P. Dutta, A. Aroraand, and S. Bibyk, “Towards radar-enabled sensor
networks,” in ACM/IEEE IPSN 2006.
[2] B. Donovan, D. J. McLaughlin, and J. Kurose, “Principles and design
considerations for short-range energy balanced radar networks,” in
IGARSS 2005.
[3] “Earthscope: Exploring the structure and evolution of the north
american continent.” [Online]. Available: http://www.earthscope.org
[4] B. Wang, “Coverage problems in sensor networks: A survey,” ACM
Computing Surveys, vol. 43, Oct. 2011.
[5] M. Skolnik, Introduction to radar systems. McGraw-Hill, 2002.
[6] Y. Wang and G. Cao, “Barrier coverage in camera sensor networks,” in
ACM MOBIHOC 2011.
[7] S. Kumar, T.-H. Lai, and J. Balogh, “On k-coverage in a mostly sleeping
sensor network,” in ACM MOBICOM 2004.
[8] Q. Liang, “Waveform design and diversity in radar sensor networks:
Theoretical analysis and application to automatic target recognition,” in
IEEE SECON 2006.
[9] J. Liang and Q. Liang, “Orthogonal waveform design and performance
analysis in radar sensor networks,” in IEEE MILCOM 2006.
[10] T. Hanselmann, M. Morelande, B. Moran, and P. Sarunic, “Constrained
multi-object markov decision scheduling with application to radar resource management,” in Infomation Fusion 2010.
[11] M. Li, T. Yan, D. Ganesan, E. Lyons, P. Shenoy, A. Venkataramani, and
M. Zink, “Multi-user data sharing in radar sensor networks,” in ACM
SenSys 2007.
[12] S. Bartoletti, S. Conti, and A. Giorgetti, “Analysis of UWB radar sensor
networks,” in IEEE ICC 2010.
[13] E. Paolini, A. Giorgetti, M. Chiani, R. Minutolo, and M. Montanari,
“Localization capability of cooperative anti-intruder radar systems,”
EURASIP Journal on Advances in Signal Processing, 2008.
[14] B. Kusy, A. Ledeczi, and X. Koutsoukos, “Tracking mobile nodes using
RF Doppler shifts,” in ACM SenSys 2007.
[15] J.-H. Lim, I.-J. Wang, and A. Terzis, “Tracking a non-cooperative mobile
target using low-power pulsed Doppler radars,” in IEEE Local Computer
Networks (LCN) 2010.
[16] R. Kershner, “The number of circles covering a set,” American Journal
of Mathematics, vol. 61, p. 665671, 1939.
[17] X. Bai, Z. Yun, D. Xuan, T.-H. Lai, and W. Jia, “Optimal patterns for
four-connectivity and full coverage in wireless sensor networks,” IEEE
Trans. Mobile Computing, vol. 9, p. 435448, 2010.
[18] Y. Wang and G. Cao, “On full-view coverage in camera sensor networks,” in IEEE INFOCOM 2011.
[19] H. Ma, Y. Meng, D. Li, Y. Hong, and W. Chen, “Minimum camera barrier coverage in wireless camera sensor networks,” in IEEE INFOCOM
2012.
[20] Y. Wu and X. Wang, “Achieving full view coverage with randomlydeployed heterogeneous camera sensors,” in IEEE ICDCS 2012.