506 IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 46, NO. 2, FEBRUARY 2008
Fig. 2. Geometry of the L-shape antenna array imaging system.
Let ϕ denote the phase of s
∗
1
(t)s
2
(t)
ϕ 2πf ×
xd
Rc
=
2πxd
Rλ
(8)
and after rearranging, we have
x
ϕλR
2πd
. (9)
The value of x can be estimated from the phase of s
∗
1
(t)s
2
(t).
However, since the phase is of modulo 2π, we must make s ure
that |2πxd/λR| <πholds in order to ensure that there exists
a unique relation between x and ϕ. Therefore, the cross-range
unambiguous window is
x ∈ X =
−
λR
2d
,
λR
2d
. (10)
For example, in the case where f =10GHz, d =1m, and
R =10km, the unambiguous distance will be λR/d = 300 m.
B. Principle of Cross-Range Measurement Using Multiple
Antenna Elements
The L-shape antenna array configuration is shown in Fig. 2. It
can also be of a cross shape where the antenna elements extends
to the negative x- and z-axis. Let the transmitting antenna be at
the origin and the 2K − 1 receiving antennas be located at (kd,
0, 0) and (0, 0, kd), where k =0:K − 1. The distance between
point P to the reference antenna at the origin (zeroth antenna)
and point P to the kth antenna in the x-axis direction is
∆R
k
= R
0
− R
k
=
x
2
+ y
2
+ z
2
−
(x − kd)
2
+ y
2
+ z
2
R
0
+ R
k
=
2xkd
R
0
+ R
k
−
k
2
d
2
R
0
+ R
k
. (11)
It can be seen that, apart form the near-constant term
(k
2
d
2
/(R
0
+ R
k
)) (which is nearly independent of x), ∆R
k
is an approximately linear function of k. Since d is known and
R
0
and R
k
can be estimated by a wideband signal, therefore,
(k
2
d
2
/(R
0
+ R
k
)) is approximately known and can be com-
pensated. The received array signal can then be expressed as
s = s
0
(t) ×
1,e
j2π
xdk
λR
,...,e
j2π
xd(K−1)
λR
T
(12)
where s
0
(t) is the received signal of antenna zero. The spatial
frequency is
f
s
= xd/(λR). (13)
Having obtained the spatial frequency, it is easy to get the
cross-range distance x
x =
λR
d
f
s
. (14)
Similarly, we can get the z-coordinate of a scatterer using
the antenna array on the z-axis. After we have obtained the
x- and z-coordinates of a scatterer and together with the range
information, the 3-D coordinates can be obtained.
The above method looks like the linear array DOA esti-
mation method. In linear array DOA estimation, the angle is
limited to [0,π]. In order to ensure that there is no ambiguity
in DOA estimation, d ≤ λ/2 must be fulfilled. For the array
configuration of this paper, the cosine value of the unambiguous
angle is limited to (X/R)=[−(λ/2d), (λ/2d)], and the phase
compensation of (πk
2
d
2
/λR) must be carried out when the
values of k, d are large and/or λ, R are small. However, for the
case of conventional DOA estimation, since d = λ/2, therefore,
ϕ =(πk
2
d
2
/λR)=(πk
2
λ/4R) is a small value in the far-field
and can be omitted.
As the unambiguous distance is λR/d, the cross-range reso-
lution is λR/(K × d) if conventional Fourier analysis is used.
Generally, many antennas are needed to attain fine resolution.
However, if superresolution spectrum analysis such as the
RELAX and maximum-likelihood methods are used [23], [24],
the required number of antennas can be reduced.
III. T
HREE-DIMENSIONAL IMAGING BASED
ON
ARRAY DOA ESTIMATION
In Section II, we have described the cross-range coordinate
measurement of a single scatterer by using an antenna array
transmitting a narrowband signal. In this section, we discuss the
coordinates measurement method for multiple scatterers using
a wideband signal.
A. Wideband-Signal Model
Let the transmitted wideband linear-frequency modulated
signal be
˜s(t)=˜χ(t)exp
j2π
ft+
µ
2
t
2
, |t| <
T
2
(15)
where f is the carrier frequency, µ is the chirp rate, and T is
the chirp pulse duration. ˜χ(t) is the envelope of the transmitted
signal which satisfies ˜χ(t)=1,for|t| < (T/2), and ˜χ(t)=0,
for |t| > (T/2). The received signal is
s(t)=ρ× ˜χ(t−τ )exp
j2π
f(t−τ )+
µ
2
(t−τ)
2
. (16)