IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 46, NO.

3, MARCH 1998 309

Doppler Characterization for LEO Satellites

Irfan Ali, Member, IEEE, Naofal Al-Dhahir, Member, IEEE, and John E. Hershey, Senior Member, IEEE

Abstract— Mobile ground-based terminals observe significant

Doppler on the forward channel when communicating through
low earth orbit (LEO) satellites. This paper deals with the
analytic derivation of the Doppler shift measured by a user
on the surface of earth on a signal transmitted by a circular
orbit LEO satellite. Two simplifications are performed to obtain
the analytical expression of the Doppler shift as a function
of time. First, during the visibility duration of the satellite at
a terminal, the trajectory of the satellite with respect to the
earth is approximated by a great circle arc. Second, the angular
velocity of the satellite with respect to the user is assumed to be
constant. Numerical results validate the approximations. Another
result of our analysis is an expression for the visibility window
duration of a satellite at a terminal as a function of the maximum
elevation angle. An algorithm for estimating the parameters of the
Doppler curve based on a couple of Doppler and Doppler-rate
measurements is also presented.
Index Terms— Doppler characterization, Doppler estimation,
LEO satellite, orbital approximation, satellite visibility.

Fig. 1. Actual and approximate Doppler-time S-curve for maximum eleva-

I. INTRODUCTION tion angles 11.4 , 30.3 , 50.9 , and 90 .
OR SATELLITE communications through low earth orbit
F (LEO) satellites, mobile units (terminals) or earth stations
observe significant Doppler, which has to be estimated and
authors attempt to characterize Doppler-time curves. In [1] the
authors consider the simple case of circular LEO satellites in
compensated for, to enable reliable communication. In this let-
the equatorial plane and Doppler observed by points on the
ter we mathematically characterize the Doppler shift observed
equator. However, they did not consider the general case of
at points on earth for circular orbit satellites.
inclined circular orbits and points not on the ground trace.
The Doppler estimation algorithm presented here can easily
In [2] the authors derived expressions for the time-evolution
be implemented in the terminal’s processor. Thus, mobile
of elevation angle and Doppler for elliptical orbit satellites.
terminals, at the onset of satellite visibility, can predict the
However, they did not parameterize Doppler curves observed
shape of the Doppler-time variation over the remainder of the
by points on earth.
visibility duration. This information could be used to improve
the performance of the terminal’s phase-lock loop. Moreover,
the terminal can also estimate the duration of the visibility II. DOPPLER CHARACTERIZATION
window and the instant of maximum elevation. This could be For LEO satellites, the Doppler frequency at terminals
used as a basis for multiple-access by scheduling transmission exhibits well-behaved variation with time that can be param-
of packets from the terminal at higher elevation angles to eterized by the maximum elevation angle from the terminal
the satellite. A more elaborate multiple-access scheme based to the satellite during the visibility window. This S-shaped
on Doppler characterization has been proposed in [4]. The variation is depicted in Fig. 1 for maximum elevation angles
Doppler characterization can be used to predict the visibility- ranging from 11.4 to 90 for a terminal located at latitude
time function of a satellite at a terminal [5]. Effective power 39 N and longitude 77 W. The satellite follows a circular orbit
conservation at the terminal, by switching the power supply (eccentricity 0) of altitude 1000 km and inclination 53 The
off during the nonvisibility periods and then switching it on minimum elevation angle for visibility is assumed to be 10 .
based on the visibility-time information, can be implemented. Doppler shift is captured in terms of normalized Doppler shift
Previous research has primarily focused on methodology which is equal to , where is the relative velocity of the
to compensate for Doppler shifts. Only in [1] and [2] did the satellite with respect to the terminal and is the speed of light.
Time is expressed relative to the zero Doppler instant. The
Paper approved by M. Luise, the Editor for Synchronization of the IEEE
Communications Society. Manuscript received May 27, 1997; revised October zero Doppler instant is the time during the visibility window
24, 1997. at which the elevation angle from the terminal to the satellite is
The authors are with the General Electric Corporate Research and Devel- at its maximum value and the satellite is at its closest approach
opment Center, Niskayuna, NY 12309 USA (e-mail: aliirfan@crd.ge.com;
aldhahir@crd.ge.com; hershey@crd.ge.com). to the terminal. The Doppler frequency shift is shown only
Publisher Item Identifier S 0090-6778(98)02118-7. for the visibility duration of the satellite at the terminal; the
0090–6778/98$10.00  1998 IEEE
310 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 46, NO. 3, MARCH 1998

Fig. 2. Satellite geometry during visibility window at location on earth. (a) Basic satellite geometry. (b) Plane triangle !" . (c) Spherical triangle #$ %

visibility duration increases as the maximum elevation angle visibility window duration is less than 14 min, whereas the
to the satellite increases. orbit period is 1.75 h. Hence, for the visibility window duration
the deviation of the satellite’s orbit from a great-circle arc is
A. Analysis Strategy small.
The slant range is determined by the law of cosines
The first step in our analysis is to derive, from geometry, the
applied to the plane triangle , redrawn in Fig. 2(b)
equation for the observed Doppler shift for a given terminal
location and a maximum elevation angle. The analysis is (1)
performed as seen from the terminal’s location, i.e., in the earth
centered fixed (ECF) coordinate frame, using trigonometric Let denote the instant when the terminal observes max-
formulas for spherical triangles, i.e., triangles formed by arcs imum elevation angle, and is the angular distance
of great-circles. To use the spherical triangle laws, we make the between and measured on the surface of earth along
assumption that in the ECF frame the satellite’s orbit during the ground trace. From the cosine law of sides applied to the
the visibility window can be approximated by a great-circle spherical right triangle [Fig. 2(c)]
arc. We then show that the variation in the angular velocity
of the satellite in the ECF frame is very small ( 3%) for (2)
most LEO circular orbits and, hence, can be approximated Differentiating the above expression and substituting it in
by a constant. We next derive the equation for the visibility the expression for the derivative of the slant range, we have
window duration of a satellite for a given maximum elevation
angle.
(3)

B. Doppler Equation
Also, from Fig. 2(b), the central angle at epoch of maximum
Consider the geometry of Fig. 2(a). The coordinate system elevation angle satisfies
is an ECF coordinate system. denotes the location of the
terminal which observes a maximum elevation angle A (4)
segment of the ground trace and the corresponding segment of
the satellite’s orbit is shown in the figure. is the subsatellite Now is the angular velocity of the satellite in the ECF
point at the instant the terminal observes maximum elevation frame; hence, , where is the angular
angle. velocity of the satellite in the ECF frame. Substituting in
In the ECF frame the satellite’s orbit is not a great-circle (3) and noting that normalized Doppler is given by
due to the rotation of the earth (see [3, Figs. 2–15, pp. 72]). , we obtain (5), shown at the bottom of the next page.
However, the visibility window at a point on earth for a From (5), we observe that the normalized Doppler is a
LEO satellite is small compared to the orbit period. For function of the maximum elevation angle and the angular
example, for a circular orbit altitude of 1000 km, the maximum velocity of the satellite in the ECF frame.
IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 46, NO. 3, MARCH 1998 311

Fig. 3. Satellite’s tangential velocity in the ECI frame.

C. Satellite Velocity km, the variation is 0.3073% and it is 2.466% for orbit altitude
In the earth centered inertial (ECI) frame the angular veloc- km. Hence, for low to medium orbit altitudes, the
ity is constant; however, the angular velocity of the satellite magnitude of the tangential velocity of the satellite in the ECF
in the ECF frame varies with latitude due to earth’s rotation. shows small variation and can be approximated by a constant.
Consider the geometry of Fig. 3. Let denote the inclination We approximate by its value at the highest latitude,
of the orbit. The angular velocity of the satellite in the i.e.,
ECI coordinate system is denoted by The corresponding
tangential velocity of the satellite in the ECI frame is denoted (8)
by . denotes the velocity of the satellite in the ECF (9)
frame when the subsatellite point is at latitude degrees.
denotes the velocity at latitude degrees due to earth’s We should note here that ; hence, we
rotation. is the angular velocity of the Earth’s rotation. have approximated by its minimum value.
Clearly

(6) D. Satellite Visibility Window Duration

Using the triangle law of cosines and the relation Let denote the time when the satellite just becomes visible
to the terminal. The angle of elevation to the satellite at
is the minimum elevation angle for visibility.
From the cosine law of sides applied to the right triangle
(7) [Fig. 2(c)]

Numerically, for LEO satellites, the absolute variation of the

satellite’s velocity in the ECF frame for a given orbit is very
small. For an orbit of inclination and the altitude (10)
km, km/s and
km/s. The percentage variation of the with respect to Using the angular velocity approximation and noting that the
is only 0.168%. For orbit altitude total visibility window duration of the satellite at the terminal

(5)
312 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 46, NO. 3, MARCH 1998

Fig. 4. Actual and approximate satellite visibility window duration.

Fig. 5. Accuracy of the Doppler-time approximation as a function of the
maximum elevation angle for different orbit altitudes.
is , we have
III. DOPPLER CURVE ESTIMATION
In this section we show how to process Doppler and Doppler
rate measurements to compute estimates of the zero Doppler-
time and the associated maximum elevation angle
Denoting the right-hand side of (5) by , differentiating
and manipulating the resulting expression, it can be shown that
(11)
(13)
E. Numerical Results
The configuration for the numerical results consists of a where Since the right-hand side of (13)
satellite in a circular orbit (eccentricity 0) of altitude 1000 is independent of time
km and orbit inclination of 53 The terminal is assumed to
be located at 39 N and 77 W (Washington, DC). A computer
orbit generation program was used to generate exact Doppler-
time curves.
Fig. 1 consists of plots of exact normalized Doppler-time (14)
and the analytic Doppler-time (5) approximation for a range
of maximum elevation angles. In Fig. 4, we provide numerical where the subscripts 1 and 2 denote measurements made at
results for the visibility window duration versus maximum sampling instants and , respectively. Since and
elevation angle. In Fig. 5, we plot results of the approxi-
are related by , we
mation error, in terms of coefficient of determination [6, p.
can rewrite (13) as follows:
449], between the Doppler-time approximation and the exact
Doppler-time curves. The coefficient of determination is
defined as (15)
where we have defined
(12)
(16)
where and with
denoting data points for actual Doppler and denoting Using the trigonometric identities for and
those for estimated Doppler. it can be readily shown that
We consider three orbit altitudes of 1000, 5000, and 10 000
km. The orbit inclination is 53 in all three cases. The analytic (17)
approximations are excellent fits to the exact Doppler-time
curves. We observe that the approximation error increases as The zero Doppler-time is computed from the relation
the altitude of the orbit increases due to larger variations in (18)
the satellite’s ECF velocity, as discussed in Section II-C.
IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 46, NO. 3, MARCH 1998 313

From (4), we have rate measurements is provided. Applications of the Doppler

characterization to multiple-access and for power conservation
at terminals are also discussed. Numerical results validate our
(19)
approximations.
The enhancement of our estimation algorithm in the pres-
where is computed from (4) and ence of noise is a good direction for further investigations.
(since one sign is
rejected based on physical considerations).
REFERENCES
IV. CONCLUSIONS [1] M. Katayama, A. Ogawa, and N. Morinaga, “Carrier synchronization
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relative motion of circular orbit LEO satellite. We showed shift for nongeosynchronous communication satellites,” Int. J. Satellite
that Doppler-time curves can be classified based solely on the Commun., vol. 9, pp. 123–136, 1991.
[3] W. L. Pritchard, H. G. Suyderhoud, and R. A. Nelson, Satellite Com-
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and satellite. We also derived an expression for the visibility systems,” Int. J. Satellite Commun., to be published.
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LEO satellites,” submitted for publication.
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