IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 17, NO. 11, NOVEMBER 1999 1829
Angle and Time of Arrival Statistics for
Circular and Elliptical Scattering Models
Richard B. Ertel and Jeffrey H. Reed, Senior Member, IEEE
Abstract— With the introduction of antenna array systems
into wireless communication networks comes the need to better
understand the spatial characteristics of the channel. Scattering
models provide both angle of arrival (AOA) and time of arrival
(TOA) statistics of the channel. A number of different scattering
models have been proposed in the literature including elliptical
and circular models. These models assume that scatterers lie
within an elliptical and circular region in space, respectively.
In this paper, the joint TOA/AOA, the marginal TOA, and the
marginal AOA probability density functions (pdf’s) are derived
for the elliptical and circular scattering models. These pdf’s
provide insight into the properties of the spatial wireless channel.
Index Terms—Antenna arrays, propagation.
I. INTRODUCTION
F
OR beamforming and emitter localization applications in
which an array of sensors is used to receive communi-
cation signals, it is necessary to have channel models that
provide angle of arrival (AOA) and time of arrival (TOA)
information about the multipath components. To meet this
challenge, geometrically based single bounce channel models
have been proposed. These models assume that scatterers are
randomly located in two-dimensional space according to a
joint probability density function (pdf). For convenience, this
joint pdf will be referred to as the scatterer density function
or just scatterer density. From the assumed scatterer density,
it is possible to derive the joint TOA/AOA, marginal AOA,
and marginal TOA pdf’s of the multipath components.
Various scattering models have been proposed in the litera-
ture [1]–[6]. Here we consider only the circular and elliptical
models described in [1] and [2], respectively. The first assumes
a uniform pdf for the position of scatterers inside an ellipse,
in which the base station and the mobile are located at the
foci [2]. The model was proposed for microcell environments
where multipath components may originate near both the
mobile and the base station. The second model assumes a
uniform pdf for the position of scatterers within a radius about
the mobile [1]. Such a scatterer density is more appropriate in
macrocell environments in which antenna heights are typically
higher than surrounding scatterers, making it less likely for
multipath reflections to occur near the base station.
In this paper, the joint TOA/AOA pdf, marginal AOA pdf,
and marginal TOA pdf at both the base station and mobile are
Manuscript received April 10, 1998; revised November 11, 1998.
The authors are with the Mobile and Portable Radio Research Group-432
NEB, Bradley Department of Electrical and Computer Engineering, Virginia
Polytechnic and State University, Blacksburg, VA 24061 USA.
Publisher Item Identifier S 0733-8716(99)08959-3.
Fig. 1. Scatterer geometry.
derived for the elliptical and circular scattering models. The
pdf’s for the elliptical model were previously derived in [7].
Here a more general approach is used that allows the pdf’s
for both models to be derived using a common approach. The
joint TOA/AOA pdf and the marginal TOA pdf for the circular
model are original contributions of this paper.
II. P
ROBLEM GEOMETRY AND ASSUMPTIONS
Fig. 1 shows the geometry and notation used to derive the
TOA and AOA pdf’s of the scattering models. The mobile is
separated from the base station by the distance
. Although
only one scatterer is shown in the figure, it is assumed that
there are many scatterers, the locations of which are described
by the statistical scatterer density function
. The
results derived here apply to the ensemble of randomly located
scatterers and make no assumptions regarding the number of
scatterers present.
It is assumed that the effective antenna patterns are om-
nidirectional for both transmit and receive. In practice, the
derived pdf’s should be used in conjunction with knowledge of
the actual antenna radiation patterns. The following additional
assumptions are common to both the elliptical and circular
models [1].
1) The signals received at the base station are plane waves
which propagate in the horizontal plane.
2) Scatterers are treated as omnidirectional reradiating el-
ements.
3) The signals that are received at the base station have
interacted with only a single scatterer in the channel.
III. J
OINT TOA/AOA PDF’S
We first derive the most general joint pdf’s and then show
how these pdf’s simplify for particular scatterer densities. Let
0733–8716/99$10.00 1999 IEEE
1830 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 17, NO. 11, NOVEMBER 1999
the coordinate system be defined such that the base station
is at the origin and the mobile lies on the
axis, as shown
in Fig. 1. It will be useful to express the scatterer density
function with respect to the polar coordinates
as an
intermediate step in deriving the joint TOA/AOA pdf. The
polar coordinates are related to the rectangular coordinates via
the following set of equations
(1)
(2)
(3)
(4)
where
denotes the location of the scatterer.
The joint pdf
is found using [8]
(5)
where
is the Jacobian transformation given by
(6)
Substituting (6) into (5) gives
(7)
If we restrict
to be positive, then (7) becomes
(8)
The next step is to find a relationship between
and the
delay
of the multipath component. Applying the law of
cosines to the triangle shown in Fig. 1 gives
(9)
The total path propagation delay is given by
(10)
Squaring both sides of (10) and solving for
results in
(11)
The joint TOA/AOA pdf is given by
(12)
where
is the Jacobian transformation
(13)
Substituting (13) into (12) gives
(14)
Next, we may express the joint TOA/AOA density in terms of
the original scatterer density function using (8) and (11). The
resulting joint pdf becomes
(15)
where
is given by (11). Equation (15) expresses the joint
TOA/AOA observed at the base station in terms of an arbitrary
scatterer density.
When the scatterers are uniformly distributed within an
arbitrarily shaped region
with an area , then the scatterer
density function is given by
and
else.
(16)
In this special case, the joint TOA/AOA pdf reduces to
(17)
where the appropriate range of
and are assumed.
Equation (17) gives the joint TOA/AOA pdf observed at the
base station. Due to the symmetry of the geometry shown in
Fig. 1, the joint TOA/AOA pdf at the mobile will have the
same form as the TOA/AOA pdf observed at the base station.
The only difference between the two is the range of
and
where the pdf is nonzero.
The relationship between
and is identical in form to
the relationship between
and , namely
(18)
Repeating the derivation with respect to the polar coordinates
at the mobile gives
(19)
Note, however, that the range of
and for which (17) is
valid will be different from the range over which
and is
valid in (19).
ERTEL AND REED: ANGLE AND TOA STATISTICS FOR CIRCULAR AND ELLIPTICAL SCATTERING MODELS 1831
Fig. 2. Calculating AOA pdf.
IV. MARGINAL PDF’S
A. AOA Probability Density Function
Although the AOA pdf could possibly be found by integrat-
ing the joint TOA/AOA pdf over
, a more straightforward
approach is to integrate the polar coordinate system represen-
tation of the scatterer density function given in (8) with respect
to
over the range to
(20)
Fig. 2 shows the definitions of
and for an
arbitrary scatterer region. For a uniform scatterer density with
(21)
The functions
and may be obtained from
the polar coordinate representations of the boundary of the
scatterer region. When the base station is located inside of the
scatterer region,
and the AOA pdf reduces to
(22)
The same equations apply at the mobile when the scatterer
density is referred to the polar coordinate system
defined at the mobile (see Fig. 1). Hence, the AOA pdf at
the mobile is given by
(23)
where the functions
and define the boundary
of the scatterer region. When the mobile is located inside of
the scatterer region,
, and the AOA pdf at the
mobile becomes
(24)
These results will be used to find the AOA pdf’s for the
elliptical and circular scattering models in the corresponding
sections that follow.
Fig. 3. Calculating TOA cumulative distribution function.
B. TOA pdf
Deriving a general TOA pdf for an arbitrary scatterer density
function is more difficult. Integrating the joint TOA/AOA pdf
over AOA, even for the case of a uniform scatterer density
function, is nearly intractable and does not yield manageable
results. A second, more promising approach is to first derive
the TOA cumulative distribution function (cdf) and then take
the derivative with respect to
to obtain the desired TOA pdf.
The TOA cdf may be calculated as the probability of a scatterer
being placed inside the ellipse corresponding to a delay equal
to
. The area of overlap of the ellipse with the scatterer region
is illustrated in Fig. 3. In general, calculating the probability
of a scatterer being placed inside the ellipse would involve
a double integral of the scatterer density function over the
overlap region.
For the case of a uniform scatterer density function, the
TOA cdf is simply
(25)
Taking the derivative with respect to
gives the desired TOA
pdf
(26)
The function
and the approach used to find this area
are dependent upon the particular scatterer density function.
The TOA pdf observed at the base station is exactly the
same as the TOA pdf seen at the mobile, since the distance
traveled by a multipath component between the mobile and
the base station by way of a scatterer is independent of the
perspective.
V. E
LLIPTICAL SCATTERING MODEL
The assumed geometry for the elliptical scattering model
(Liberti’s model) is shown in Fig. 4 [2], [7]. Scatterers are
uniformly distributed inside the ellipse with foci at the base
station and mobile. The major axis of the ellipse is given
by
, where is the maximum delay associated with
scatterers within the ellipse, and
is the speed of light.
The model is appropriate for microcell environments where
antenna heights are relatively low. With low antennas, the base
station may receive multipath reflections from locations near
the base station as well as around the mobile. The elliptical
model has the physical interpretation that only multipath
components that arrive with a maximum delay of
are
1832 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 17, NO. 11, NOVEMBER 1999
Fig. 4. Elliptical scatterer density geometry.
considered. Ignoring paths with longer delays may be justified,
since such signal components will experience greater path loss
and hence will have relatively low power compared to those
with shorter delays. Therefore, provided that
is chosen
sufficiently large, nearly all of the power of the multipath
signals of a physical channel will be accounted for by the
model.
The parameters
and shown in Fig. 4 are the semi-
major axis and semiminor axis values which are given by
(27)
(28)
The ellipse shown in Fig. 4 may be described by either the
Cartesian equation
(29)
or by the polar coordinate equation [9]
(30)
For each of the pdf’s shown below,
m and the
maximum delay was
s. Each of the histograms
shown to validate the closed form expressions of the pdf
are generated by placing 50000 scatterers uniformly into the
ellipse, measuring the desired property (AOA or TOA), and
then creating a histogram containing 75 bins. The number
of points in each bin is divided by the total number of
points to give the normalized histogram. In all cases, there
is good agreement between the closed form expression and
the normalized histogram.
A. Joint TOA/AOA pdf (Elliptical Model)
1) Joint TOA/AOA pdf at the Base Station: The joint TOA/
AOA pdf was derived in Section III for the case of a uniform
scatterer density. The result given in (17) is repeated here for
convenience
(31)
The area of an ellipse is given by
. Substituting
this expression into (31) and explicitly stating the range of
validity gives (32), shown at the bottom of the page. The case
when
is considered separately, due to the zero/zero
condition which occurs when substituting
and
directly into the first expression of (32). However, when
, several terms may be cancelled, leaving only
(33)
which may be evaluated at
.
For verification purposes, the analytically derived pdf’s are
compared against scatter plots. Fig. 5(a) and (b) shows the
joint TOA/AOA pdf present at the base station for
m and s. Fig. 5(a) is a plot of the closed form
expression for the joint pdf given in (32). Fig. 5(b) is a
scatter plot created by randomly placing 10000 scatterers in
the ellipse using a uniform distribution and then plotting the
corresponding TOA/AOA pairs. The scatter plot shows a high
concentration of points where the joint pdf is high. Also, both
plots show a high concentration of scatterers with relatively
small delays near the line-of-sight.
2) Joint TOA/AOA pdf at the Mobile: Due to the symme-
try of the ellipse with respect to the base station and the
mobile, by inspection, the same TOA/AOA pdf will be valid
when related to the mobile perspective. The resulting joint pdf
as a function of
and is shown in (34) at the bottom of
the next page.
B. AOA pdf (Elliptical Model)
1) AOA pdf at the Base Station: The polar equation defin-
ing the boundary of the scatterer region is
(35)
Since the base station is located inside the scatterer region,
using (22), the AOA pdf is
(36)
else
(32)
ERTEL AND REED: ANGLE AND TOA STATISTICS FOR CIRCULAR AND ELLIPTICAL SCATTERING MODELS 1833
(a)
(b)
Fig. 5. Elliptical model: joint TOA/AOA pdf.
The elliptical AOA pdf and the normalized histogram are
plotted in Fig. 6(a) for
m and s. As
expected from the joint TOA/AOA pdf, Fig. 6(a) shows that
the scatterers are concentrated near line-of-sight.
2) AOA pdf at the Mobile: Again due to the symmetry of
the ellipse, the same pdf applies at the mobile, namely
(37)
where
in the direction toward the mobile and increases
in a clockwise direction.
C. TOA pdf (Elliptical Model)
As derived earlier, the TOA pdf may be calculated using
(38)
where
is the area of intersection of the ellipse corre-
sponding to a delay of
with the uniform scatterer region. For
else.
(34)
