An empirical model was generated to estimate diversity gain on Earth-space propagation paths as a function of Earth terminal separation distance, link frequency, elevation angle, and angle between the baseline and the path azimuth. The resulting model reproduces the entire experimental data set with an RMS error of 0.73 dB

Hodge, D. B.

English

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11.8. Hodge
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The Ohio State University
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ElectroSrience Laboratory
Department of Electrical Engineering
Columbus, Ohio 	 43212
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ABSTRACT
An empirical model has been generated to estimate diversity
gain on earth-space propagation paths as a function of earth terminal
separation distance, link frequency, elevation angle, and angle between
the baseline and the path azimuth. This analysis utilized 34 diversity
experiments which have been conducted in Canada, England, Japan, and
the United States during the past decade. The resulting model repro-
duces the entire experimental data ;set with an RMS error of 0.73 dB.
The separation distance dominates the dependence of the diversity
gain. The dependence on link frequency is small but significant. No
s	 identifiable dependence on baseline orientation was 'found.
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TABLE OF CONTENTS
Pagex$
I.	 INTRODUCTION 1
}
^! t II.	 DIVERSITY GAIN 2
t,
III.	 EXPERIMENTAL DATA BASE 4
ar
IV.	 DATA ANALYSIS D
V.	 CONCLUSIONS ^S
VI,	 RECOMMENDATIONS 16
REFERENCES 17
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LIST OF FIGURES
PAGE
pothetical example showing the definitions of
versity gain and diversity improvement.	 I
eometry of path diversity system. 	 7
aefficient a versus average single site fade
epth, A.	 9
oefficient h verses average single site fade
epth, A.	 to
ormalized diversity gain, Gf, versus link
requency,	 f.	 17
n
rmalized diversity gain, G R , versus elevation
Ole ) 	f3.	 13
ormalized diversity gain, G A , versus orientation A,
 the baseline relative to the propagation path.	 14
* F
	
X yf
	
I.	 INTRODUCTION
a	
^
	
1
Path diversity has been found to b.e a useful
	 in some cases,
necessary technique to provide acceptable reliability on earth-space
communication links operating above 10 GNz. This technique takes
	
3i	
advantage of the tendency of rainfall to occur in limited spatial re-
gions in order to overcome the deleterious effects of rainfall attenu-
a
	
}C
	 ation. This is accomplished by placing two earth terminals such that
the probability of rainfall attenuation occurring simultaneous on both
paths is significantly less than the probability of rainfall attenuation
occurring on either individual path. Although the redundant hardware
normally associated with a single earth terminj1 may be removed to the
remote diversity site in order to hold costs down, the costs of the
additional site, the connecting link and the switching capability must
be weighed against the resulting increase in link reliability.
3
Little progress has been trade in the development of theoretical
models applicable to this method. Therefore, a need still remains to
establish an empirical model which will provide the link designer with
sufficient information for design purposes. Such a model was developed
earlier [1]; this 'model was based on the experimental results available
in the early 1970 1 s. This model established the dependence of diversity
gain on the separation distance between the earth terminals. However,
the question of the dependence of diversity gain on link frequency
remained. And, furthermorey this early model predicted a fixed diversity
gain for extremely deep fades which has been found to be incorrect in
subsequent experiments.
The publication of results of a number of diversity experiments
have now made it possible to reexamine and improve this empi 0cal model.
The results of thirty-four experiments were used in this studyto
establish the dependence of diversity gain on separation distank,,e,
fink frequency, elevation angle, and the angle between the baseline
and the-path azimuth. The dependence on separation distance is strongest
1
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with the other dependences being weaker but significant. The model was
also generalized to remove the constraint that the diversity gain in-
crease as rapidly as the attenuation for deep attenuations.
II.	 DIVERSITY GAIN
Diversity gain, GD , is a measure of the reduction in attenuation
exceeded for a fixed percentage of time as a result of selecting the
larger of the two received signals at two separated earth terminals on
an instantaneous basis. In order to determine the diversity gain it is
first necessary to find the average single site attenuation associated
with each fixed percentage of time as shown in Figure 1. Diversity gain
is then the difference between the average single site attenuation and
the diversity (joint) attenuation for each fixed percentage of time.
Since the average single site attenuation distribution establishes a
unique relationship between attenuation and percentige time, the diversity
gain may be viewed as either a function of average single site attenu-
ation or of percentage time, The former is generally found to ba. morn
useful and will be utilized in the following.
Diversity improvement, I D , is another parameter which may be use-
ful for describing diversity performance. It can M defined as the
ratio of the average single site percentage time to the diversity
percentage time for fixed attenuation as shown in Figure 1. However,
this approach suffers from two disadvantages. First, diversity improve-
ment cannot be established for very deep fades, i.e,, small percentages
of times, due to experimental limitations. This is a consequence of
the fact ghat, for very deep fade levels, the diversity attenuation
distribution usually falls well below the temporal resolving capability
of the experiment. Secondly, this parameter is based on measurements
which occur over markedly different intervals of time. Thus, for
experiments of limited duration, the uncertainties it these levels are
quite different. This property leads to randomness which is not as
evident in the use of diversity gain. Theoretically, of course, these
two approaches are equivalent given ideal data sets.
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FIXED
ATTENUATION
I
t
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i- IDQ 	 SINGLE SITE ATTENUATION
z	 `	 DISTRIBUTIONW 
0	 — FIXED PERCENTAGE. TIME
aW.	 I G p AVERAGE SINGLE SITE
I	 i	 ATTENUATION DISTRIBUTION
JOINT ATTENUATION
DISTRIBUTION
0
0 ATTENUATION EXCEEDED
Figure I. of diversity gainpandsdiversitye i improvement.
I}
a.t
III. EXPERIMENTAL DATA BASE
The results of thirty-four diversity experiments were utilized
for the analysis presented below. These experiments are tabulated in
Table 1 along with the references from which the data were obtained.
The experiments utilized were restricted to direct satellite beacon
measurements and radiometric emission measurements where no frequency
scaling was utilized. The frequencies associated with these experiments
ranged from 11.6 to 30 GHz, and the earth terminal separation distances
ranged from 17 to 46.5 kin. The conventional azimuth, a, and elevation,
p, angles of the propagation paths are also listed in Table l along with
the orientation, Y, of the baseline separating the earth terminals.
The orientation of the baseline is measured from true north as is con-
ventional for the azimuth angle of the path. The orientation of the
baseline relative to the propagation path, A, will be defined as the
difference between a and y taken such that
0 < o < 900.
These definitions are shown in Figure 2.
The diversity gains, G D , associated with these experiments were
extracted for average single site attenuation values of 2, 3, A,
dB up to the 'highest value available for each experiment.. This procedure
yielded 312 values of diversity gain. These values along with the
parameters shown in Table 1 were then stored in a disk file to permit
convenient processing.
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Figure 2. Geometry of path diversity system,
7
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IV.	 DATA ANALYSIS
As in the earlier analysis, C1j, a nonlinear regression was
performed to find the coefficients, a and b, which provided the least
mean square error fit between the experimental data set and the ana-
lytic form,
{
d	
/!^d	 a^^^e#Yd^	 ^/)
t
f 1(
where d is expressed fir. km .	 This analysis was performed as a function
of average single site attenuation, As so that two families of chef-
ficients, a(A) and b(A), were obtained for integer values of A from
2 to 10 dB.	 The resulting coefficients are shown in Figures 3 and 4.
The coefficients shown in these figures are actually the final result
of the iterative application of the procedure described in this section.
The process was, however, very stable with only minor changes occurring k
after the first iteration.
These coefficients were then modeled by the following analytical ..^}
expressions:
a r 0.64A .. 1.6	
(l_d"0.11A)
	 (3)
b	 0.585
	
(1-e^0.098A )	 (4)
where A is expressed in dB and the numerical constants were determined
by a regression analysis. These expressions are of the same forms as
those used in the earl ,jer model except that the leading coefficient
in the expression for a was not forced to be unity. This eliminates
the problem in the earlier model associated with deep fades.
Having found the dependence, G d ., of the diversity gain on single
site fade depth, the experimental data set was normalized to remove
ORIGINAL PAGE IS
OF POOR QUALIV
8
4
2
0
10
2	 4	 6	 8	 10 12 14
A [dB I
SINGLE SITE FADE DEPTH
Figure 3 * Coefficient a versus average single siti fade depth, A.
9
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0.5 1
0.4—
b 0.3
0.2
0.1
0
0	 2	 4	 6	 8 10	 12 14
A	 [dB]
SINGLE SITE FADE DEPTH
V
Figure 4. Coefficient b versus average single site fade depth, A.
10
ORIGI14AL PAG IE IS
OF pOOR QUALITY
41
r.
y.
its depoAance on separation distance, d, and average single site -fade
depth, A * The rzosulting values,
G
Gf	
D
d
shown in Figure 5, were then used to establish the coefficients in the
following analytic expression by a regression analysis:
Gf	1.64 
e"O.Usf	 (6)
where f is expressed In Gliz. This expression shows a moderate frequency
dependence declining with increasing frequency.. Equation (6) is plotted
in Figure 6 along with the normalized data.
Next, the frequency dependence was also normalized out of the
original data set and the dependence on elevation angle, 0 1 was sou ghto,
G
(7)
uduf
The values of G along with the model resulting from a linear regression
a 'i'val ys I $
G	 0.00492 0 + 0.834	 (8)
are shown in Figure G. p is expressed In degrees. This regression
indicates a weak increasing dependence on elevation ►igle.
Finally, this process was repeated seeking the dopendence on
the orientation of the baseline relative to the propagation path'
12	
G 0
OA d
A very weak increasing linear dependence was found as shown in
Figure 7:
ll
43
Gf
2
1
ORIGINAL PAGE 19
OF POOR QUALITY
0
u 10	 20	 30	 40
f [GHz
L! NK FREQUENCY
Figure S. Normalized diversity gain, G f , versus
link frequency, f.
50
12
uG	
+
I
V
0 10 20 30 40 50 60 70 60 90
ELEVATION ANGLE , P [DEG.]
Figure 6. Normalized diversity gain, G O , versus
elevation angle, 0.
13
F
1
1
OR14INAL PAGE IS
OF POOR QUALITY
32
G A
rN
oRIGINAL PVUE 0'
or. POOR QUALITY
T-	 I
0 10 20 30 40 50 60 70 80 go
ORIENTATION ANGLE , A L DEG, j
Figure 7. Normalized diversity gain, G , ver^,,,.4s- orientation A,
of the baseline relative to the propagation path.
4C
a
5
e
E
^^
1
k
s
t
y k7
4
GA n 0.00177 A * 0.887
	
(10)
where A is expressed in degrees.
The dependence of the diversity gain on the orientation of the
baseline itself was also examined. However, no discernible dependence
was found,
The entire preceding analysis was performed iteratively. The
changes found during the second iteration were quite small, and those
found during the third iteration were negligible.
Finally, the resulting model,
Gp = GdGfG0GA
was compared with the original data set. The resulting RMS error was
0.73 dB. This value indicates that the model produces a satisfactory
characterization of diversity gain over a wide range of separation dis-
tances, frequencies, elevation angles, and relative baseline orien-
tations.
V.	 CONCLUSIONS
Thirty-four diversity experiments were studied to establish an
empirical model for the prediction of diversity gain. The resulting
model characterizes diversity gain as a function of average single
site fade depth, separation distance, link frequency, path elevation
angle, and orientation of the baseline relative to the propagation path.
The model reproduces the original data set with an RMS error of 0,73 dB
x
15
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Vt. RECOMMENDATIONS
it is recommended that the improved empirical model presented
here be used for the calculation of diversity gain on earth-space
propagation paths.
()
\\ ^
REFERENCES
1. D. B. Hodge, "An Empirical Relationship for Path Divers ity Gain,"
IEEE Transactions on Antennas and Propagation, Vol, AP-24 1 1976,
P. 250.
2. n.n. Hodge., "A 15.1 GHz Satellite-to-Ground Path 1iversity
Fxperimpnt= Utilizing the ATS.5 Satellite," Radio Ici pncr,
Vol. 9, 1174 1 p. 1.
3,	 D. B. Bodge, D. M. Theobol d and Rw C. Taylor, "ATS .. G Millimeter
Wavelength Propagation Experiment," Report 3863»6, 1976, The Ohio
State University Electro5cie?nce Laboratory.
4. W. J. Vogel, A. W. Straiton, B. M.Fannin, and N. K. Wagner,
"Attenuation Diversity Measureme^its at GC mnd 3^ .%3U^," °: diw
Science, Vol. 11, 1976, p 167.
5. C. W. Bostian, ".'he SIRIO Diversity Experiment," Virginia poly-
:::	
technic Institute and State University, EE Dept. Technical
Report, June, 1961.
6	 Y Hosoya, K. Satoh, T. Nagatsu, "Propagation Characteristics on
Slant Paths in 20 and 30 Gliz Rands," Review of the Electrical
Comiunications Laboratories, Vol. 28 8 Nos. 7-8, July-August, 1980,
P * 5SD,,
7.	 R. W. Wilson, "A Three Radiometer Path Diversity Experiment,"
[dell System Tp4hnical Journal, Vol. 49, 1970 0 p. 1239.
t3.	 R. W. Wilson and W. L. Manmel , "Results from a Throe Radi once ter
r	 ;;
	
Path Diversity Experiment," Proceedings of Conference on Propa-
gation of Radio Waves at f=requencies above 10 G'Nz, London., 1973,
p 23.
17
9. D. A. Grays "Earth-Space Path Diversity: Dependence on Baseline
Orientation," 1973 G-AP/URS1 Meetingo Boulder, Colorado, 1973.
10. K. Funakawa and Y. Otsu, "Characteristics of Slant Path Rain
Attenuation at 35 Gliz Obtained by Solar Radiation and Atmospheric
Emission Observations," Journal de Recherches Atmospherique,
Vol, 8, 1974 ► p. 339.
11. J. E. Hall and J. C. Allnutt, "Results of Site Diversity Tests
Applicable to 12 GHz Satellite Communications," TEE Conference
Publication No. 126, "Satellite Comiunication Systems Technology,
London, 1976,
12. J. E. Allnutt, "Slant Path Attenuation and Space Diversity
Results Using 11.6 Gliz Radiometers," Proceedings TEE, Vol, 123,
1976, p. 1197.
13. J. I. Strickland, "Radiometric Measurements of Site Diversity
Improvement at Two Canadian Locations," Proceedings of URSI
Commission F Open Symposium, LaBaule, France, 1977.
14. H. J. Bergmann, "Satellite Site Diversity: Results of a Radio-
meter Experiment at 13 and 18 GHz,"' IEEE Transactions on Antennas
and Propagation, Vol. AP-25, 1977 0 p. 483.
15. W, J. Vogel, Private Communications.
16. D. V. kDgers and G. Hyde, "Diversity Measurements of 11.6 GHz
Rain Attenuation at Etam and Lenox, West Virginia," Comsat 	 ^^
Technical Review, Vol. 9, No. 1 1 Spring, 1979, p. 243.
18
AA


An improved empirical model for diversity gain on Earth-space propagation paths

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