A Statistical Model for a Land Mobile Satellite Link
Abstract-A statistical model for a land mobile satellite link is described. The model assumes that the amplitude of the line-of-sight
component under foliage attenuation (shadowing) is lognormally distrib- uted and the received multipatb interference has a Rayleigh distribution.
Expressions for the level crossing rate and average fade duration are given. Comparison between results calculated from the model and from measurements show reasonably good agreement. The model should be useful for designing communications systems and for simulating propaga- tion effects in the laboratory.
A
I. INTRODUCTION STATISTICAL model for received signal envelope
fading encountered on a land mobile satellite channel is useful for predicting communications systems performance with various modulation schemes. The model will also facilitate the design of propagation simulator. There have been many papers published on the modeling of mobile radio channels in urban environments, a few of which are given in [1]-[9]. The model which is of interest here is for a land mobile satellite channel in a rural environment, where, for most of the time, a line-of-sight signal component is available at the receiver.
This paper describes a statistical model in terms of its probability density function, level crossing rate (LCR) and
average fade duration (AFD). These characteristics were
obtained under the hypothesis that foliage not only attenuates but also scatters radio waves at UHF and at L-band frequen- cies. The model assumes that the line-of-sight component under foliage attenuation (shadowing) is lognormally distrib- uted and that the multipath effect is Rayleigh distributed.
Additionally, these two random processes are assumed to be correlated. Although no attempt is made at physical justifica- tion of the model, results derived from the model are
compared with those derived from measurements and are found to be good engineering approximations.
The probability density function of the model facilitates the calculation of fade margins in the designs of communication systems. The results of LCR and AFD provide information relevant to modem designs and error correction techniques.
Section I1 of the paper describes the statistical model and numerical results are given in Section 111. Conclusions are presented in Section IV.
Manuscript received February 20, 1984, revised January 7, 1985.
The author is with the Department of Communications, Communications Research Centre, P.O. Box 11490. Station H, Ottawa, Ontario, Canada K2H 8S2. Telephone (613) 998-2574.
LI. STATISTICAL MODEL
The mathematical derivations required to describe the model are given in this section.
A . Probability Density Function
Consider the sum of a lognormally distributed random phasor and a Rayleigh phasor [ l o ] ,
r exp
(jO)=z
exp W O ) + w exp (j4),z ,
w > O ( 1 ) where the phases 4o and 4 are uniformly distributed between 0 and 2a,z
is lognormally distributed, and w has a Rayleigh distribution.If
z
is temporarily kept constant, then the conditionalprobability density function of r is simply that of a Rician vector [ 101 :
where bo represents the average scattered power due to
multipath, and Zo ( a ) is the modified Bessel function of zeroth order. Applying the theorem of total probability, one obtains
From this, p ( r ) is given by
p ( r ) = r/bo
56
exp [ - ( r 2 +z2)/2bo]Zo(rz/b~)p(z) dz. (4) It has been assumed that p ( z ) is lognormal, given bywhere &and p are the standard deviation and mean, respectively.
Thus, substituting ( 5 ) into (4), one obtains
p(r)=r/(bo=)
jm
l/z exp [-(lnz
0
It has been shown [lo] that r is lognormal for large values OO18-9545/85/0800-0122$01 . O O
0
1985 IEEELOO: LAND MOBILE SATELLITE LINK 123
and Rayleigh distributed for small values as given by (7). where B = bob2 - b l , z is the amplitude of the sine wave and For mathematical simplicity, assume that the noise (multi- exp [ - r - pc>2/2d01 for %
‘6
(7) path) spectrum is symmetrical about the line of sight compo-2
the b are the moments [ 131.
r/bo exp [ - r2/2bo] for r
a &.
nent frequency. Thenbl = O , B = bob2, For intermediate values of r, the complicated integral given
by ( 6 ) must be evaluated numerically. The following relation and from 1 1 3 1 9 the joint Probability becomes can be used to allow integration over [0, R ] instead of [ R ,
0 3 1 .
p ( r > R ) =
lm
p ( r ) drR
r(2a) - 3’2
b 0 G
p ( r , f ) = exp [ - ( r 2
+
z2)/2bo] exp [ - i 2 / 2 b 2 ]= 1 -
1
p ( r ) d r .R
0 (8)
lr
exp [rz cos B/bo] dB. (11)- - *
The integral in (1 1) can be recognized as the modified Bessel B. Level Crossing Rate and Average Fade Duration function of zeroth order and is given by 2a10 (rz/bo). Thus the
In this subsection, expressions for the LCR and AFD of the expression given in (1 1) can be separated into two parts:
received signal due to the combined effect of shadowing and multiDath on a land mobile satellite link are derived.
RiEe [I21 has shown that for any wide sense stationary and
random process, NR (which is the LCR), defined as the expected rate at which the envelope crosses a specified signal level R with positive slope is given by the following integral:
p ( i ) = - 1 exp [ - i 2 / 2 b 2 ] . (1 3)
45&
i ) d i ,
From ( 1 2 ) and ( 1 3 ) , it is clear that the rate of change of the (9) signal envelope i is uncorrelated with the envelope, r; that is, p(r, i ) = p ( r ) p ( i ) . Further, (13) shows that the probability where the overdot indicates time derivative and p ( R , i ) is the
joint probability density function of i and r at r = R . If the lognormal variation of the direct sign,al component is ignored, the remaining signal envelope would have a Rician distribution, and it is easy to show under simplifying assump- tions, based on previous work [ 131, that the resultant received signal envelope and its rate of change are statistically independent. Similarly, if the multipath component is ignored, the same statistical independence can be shown [ 9 ] . This gives justification for the reduction of (9) into a more tractable form.
In addition, based on the referenced analyses, it can be shown that the rate of change of the resultant envelope is a Gaussian random process under both fading conditions. This can be used for the determination of LCR’s and the AFD of the combined received signal.
First, to show the desired properties of the received signal under Rician fading conditions, let the lognormal component (z) remain temporarily constant as in Section 11-A. Then the received signal can be modelled as a sine wave plus narrow- band Gaussian noise, and from Rice [ 1 3 ] ,
density function of i is Gaussian.
If one now considers the lognormal component in isolation, it can be shown [9] that the rate of change of r, i , is distributed normally. The derivation is rewritten here for convenience.
Following [9], the joint probability density p ( r , i ) is given by
p ( r , i ) = 1
2-r exp [ - (In r - ~ ) ~ / 2 d ~ - i 2 / 2 d 2 ] (14) where p and do are the mean. and variance of the envelope r, and d2 is the variance of i. One can write (14) as
where
1
p ( r ) = -
s r exp [ - (In r - ~ ) ~ / 2 d ~ ] (15) and
p ( i ) = - 1 exp [ - i 2 / 2 d 2 ] .
This shows that r and i are also uncorrelated for lognormal
iT
exp - [ r 2 / 2 b o - rz cos B/bo+z2/2b0 fading. Equations (15) and (16) are lognormal and normal- - * distribution, respectively.
+
boi2/2B+
2bl sin B/B+
b:z2 sin2 B/2Bbo] dB,It has been shown for both Rician fading and lognormal variation of the line-of-sight signal, that envelope r is (10) uncorrelated with the rate of change of the envelope i. The
expression in (9) for LCR can thus be rewritten as
NR=p(r) i p ( i ) d i , ( 1 7) where p ( r ) is the probability density function of the received envelope due to the combined effects of multipath and shadowing, given by ( 6 ) . In (17), one needs to determine the probability density function of the rate of change of the envelope, p ( i ) . Then to obtain values of the LCR, one must evaluate the integral.
It has been shown in previous paragraphs that the probabil- ity density function of the rate of change of envelope, i, for both multipath and shadowing is Gaussian. Therefore, it seems reasonable to assume that the probability density function of the rate of change of received signal envelope, p ( i ) , comes from two correlated Gaussian random processes. This allows one to obtain results with varying degrees of correlation, for comparison with results derived from measurements.
Let
i combined rate of change of the envelope;
x rate of change of the envelope due to multipath with j rate of change of envelope due to shadowing with
standard deviation &and mean = 0;
standard deviation &%-and mean = 0;
then i = x
+
y .given by [14] as
The correlated Gaussian random process of two variables is
exp
[
-- 1 (X2/b2 - 2pkj/JbZdz + j 2 / d 2 ( 1 8 )2(1 - P 2 )
1
where p is the correlation coefficient.
manipulations, the following integral can be evaluated.
After substituting 2 = i -
3
into ( 1 8) and some algebraicC r n
1 P ( i ) =
[ 2 ~ ( l - p2)(b2
+
2 p G+
d2)]where a 2 in (20) is given by
Using (20), the integral given in (17) can be evaluated, as
m 1 1
i p ( i ) d i =
6
Jb2+2pJbzdzfdzCY‘
Then substituting ( 2 2 ) into (17), the expression of LCR is given by
With the assumption that the fading spectrum due to shadow- ing and multipath is symmetrical and Gaussian [ 131, that is
b2 = bo(2Tfm)’
d2 = do(2Tfm)’ ( 2 4 )
where f m is the maximum Doppler-shift frequency, ( 2 3 ) can be normalized with respect to f m to make results independent of vehicle velocity, giving
( 2 5 ) The average fade duration,
TV
is related to the LCR [ 8 ] by the equationThis completes the mathematical derivation required to obtain expressions for the probability density function, LCR, and
AFD due to the combined effect of multipath and shadowing.
III. NUMERICAL RESULTS
This section presents numerical values for the probability density function, LCR, and AFD which define the model given in this paper. In addition, a comparison with similar functions computed from measured data is discussed. Equa- tions ( 6 ) , (7), and (8) were used to calculate the probability density function, p ( r ) . Many calculations with different values for bo, do, and p were carried out with the objective of fitting results from application of the model to those derived from measurements made. by colleagues on a simulated satellite path. During these measurements a helicopter was flown at a fixed elevation angle of 1 5 ” with respect to the receiver [ l l ] . The measurement site was a rural area with about 35 percent tree cover.
Numerical evaluation of the probability density function in (6) was time consuming, but allowed computation of the LCR and the AFD using ( 2 5 ) and (26).
Fig. 1 shows a comparison of the cumulative distribution function for the received envelope calculated using (8) and that computed from measurements for infrequent light shadowing (sparse tree cover), frequent heavy shadowing (dense tree cover) and overall results (the combined results of light and heavy shadowing). For the case of infrequent light shadowing, the model shows the best fit around the median region and some deviation near the tails of the distribution. Generally, results of the model showed a slightly higher shadowing effect than those from measured data. For the case of frequent heavy shadowing, the results of the model show reasonably good
LOO: LAND MOBILE SATELLITE LINK 125
-35
'
I l l I 1 I I I I I 1 1 1 1 I I I I 1 I II
0.01 0.1 0.5 1 2 5 10 20 30 40 50 60 70 80 90 95 98 99 99.9 99.99 PERCENT OF TIME RECEIVED SIGNAL IS GREATER THAN ORDINATE
Fig. 1. A comparison of measured and calculated values of the probability distribution of signal,
TABLE I MODEL PAR4METERS
C o n d i t i o n s
10 loglo ( b o ) 10 loglo(p)
10 l o g i o ( / b o )
M u l t i p a t h Power M e a n
S t a n d a r d D e v i a t i o n
I n f r e q u e n t l i g h t
r e s u l t s
-3.0 -6 1.0
O v e r a l I s h a d o w i n g
-12.
-17.0 3.5
F r e q u e n t h e a v y s h a d o w i n g
-a.
0.5 0.5
agreement with measured values throughout the fading range.
For the combined results, the fit was poor about the median but reasonably good in the weak signal range which is most important for fade margin calculations. The model parameters were obtained by trial and error to fit measured values. These parameters are given in Table I.
Fig. 2 shows values for the LCR calculated using (6) and ( 2 5 ) for the case of infrequent light shadowing. The results show that the maximum LCR occurs when there is no correlation between the rate of change of the envelope due to multipath and that due to shadowing. Measured values [l 11 for this case are also shown. These results indicate that the model gives a good indication of the LCR, when high correlation,
p = 0.5 to 0.9, between multipath and shadowing is assumed.
The figure also indicates that there is a bound on the LCR when the signal is large. Fig. 3 shows a plot of the AFD calculated from the model and from measurements [ l l ] . Application of the model results in a good approximation to the AFD anticipated on a mobile satellite link. For low signal
10.0
1 .o
w t
w B
Y d 0.1
4
0 z
W 0
3
5
P
0.01
0.001.
CALCULATED
-
MEASUREDI I I I I I i
-20 -15 -10 5 0 5 10
LEVEL RELATIVE TO LINE-OF-SIGHT LEVEL (dB)
Fig. 2 . Level crossing rate-infrequent light shadowing.
100.
10-
z 0
4 3
CALCULATED
-
MEASURED-
-
W
2
s
1.0-9
2
e
W
4
CI
4
0.1
-
I I ' I I I I 1
-25 -20 -1 5 -10 -5 0 5 10
LEVEL RELATIVE TO LINE-OF-SIGHT LEVEL (dB)
Fig. 3. Average fade duration-infrequent light shadowing.
'o""l
10.01 .O
W 4
g
v)
8
g
0.1J 0
E
4 5
z
0.01
0.001' LEVEL RELATIVE TO LINE-OF-SIGHT LEVEL (dB)
Fig. 4. Level crossing rate-frequent heavy shadowing.
lo
-30 -25 -20 -15 -10 -5LEVEL RELATIVE TO LINE-OF-SIGHT LEVEL (dB)
Fig. 5 . Average fade duration-frequent heavy shadowing.
I I I I I I 1
-20 -15 -10 -5 0 5 10
LEVEL RELATIVE TO LINE-OF-SIGHT LEVEL (dB)
Fig. 6. Level crossing rate-overdl results.
LOO: LAND MOBILE S.4TELLITE LIKK
I
~
0 Y Y a u 1 w a
0 1 -
0 0 1
1
I I I I I I 1-25 -20 -15 -10 5 0 5 10
LEVEL RELATIVE TO LINE-OF-SIGHT LEVEL (dB)
Fig. 7. Average fade duration-overall results.
levels, the two curves compare reasonably well. These results also show that the best fit is obtained when p is large.
Fig. 4 shows a plot of normalized LCR versus relative signal level for the case of frequent heavy shadowing.
Measured values are also shown. The values calculated from the model for p between 0.5 and 0.8 bound the values calculated from measured data throughout the relative signal level range. Fig. 5 shows AFD for the case of frequent heavy shadowing; the calculated values, with p = 0.7 give good agreement when compared with measurements. This indicates that the correlation between the rate of change of envelope due to multipath and foliage attenuation is high for heavy shadowing as well as for light shadowing.
Figs. 6 and 7 show the LCR and AFD for the case of the combined light and heavy shadowing with correlation coeffi- cient, p as a parameter. There were no measured values
available for this case. Using measured values for the previous two cases as a guide, however, one can estimate that the measured curve would be closest to the case of p = 0.8, but this remains to be verified.
A comparison between the probability distribution of signal level calculated from (8) and that calculated from measured data for the two extreme cases, infrequent light shadowing and frequent heavy shadowing, has been described. It has been
found that the model can provide a good indication of LCR and AFD throughout the range of relative signal levels indicated in the figures. The model should thus be useful for predicting the performance of communication systems.
127 IV. CONCLUSION
A statistical model for a land mobile satellite link has been described in the paper. Expressions for the LCR and AFD were obtained. Comparisons between values calculated from the model and those calculated from measured data show
reasonably good agreement. It is also shown that the assump- tion of high correlation between the effects of multipath and those caused by shadowing yields results that compare well with values derived from measurements. This is true for both light and heavy shadowing. The model should be useful in estimating the performance of communications systems with various modulation schemes and it should also facilitate the development of a propagation simulator.
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Chun Loo (”62) received the B. Eng. degree in electrical engineering from McGill University, Montreal, PQ, Canada, in 1962, and the M.Sc.
degree in electrical engineering from the University of Ottawa, Ontario, Canada, in 1966.
From 1962 to 1965, he was employed by Com- puting Devices of Canada, Ottawa, where he participated in the development of avionic equip- ment. He was with Northern Electric Research and Development Labs (now Bell Northern Research) from 1966 to 1970 where he was engaged in the theoretical studies of FM distortions, FM threshold characteristics and in computer simulation of communications systems. Since 1970, he has been with the Communications Research Centre, Department of Communications, Ottawa. He did systems analysis under the CTS (Hermes) project a forerunner of direct broadcast satellite, and the ANIK B Pilot project. He is the author of several papers relating to FM and digital transmission through a nonlirlear satellite channel. He is currently doing system studies on various modulation schemes and on channel characterization for mobile satellite communications applicable to the MSAT program.
