' A Statistical Rain Attenuation Prediction ' :' Model With Application to ...

NASA

Tec_ca!

Memorandurn._

1.00243'

'

A Statistical

' :'

Model With Application to the Advanced Communication Technology Satellite Project HI

Rain Attenuation

A Stochastic

Prediction

Rain Fade Control Algorit_for

Link Power via Nordine_Marko_¥_Fi!tering

RobertM, Manning Lewis Research Center Cleveland,

Satellite

Theory

-

Ohio

_

....May 1991 _'(NASA-TM-IOO243) ATTFNUATIBN APPLICATION ,TECHNOLOGY -!STOCHASTIC

NASA

A

STATISTICAL

pReDICTION MODEL TO THE ADVANCED SATELLITE PROjeCT. RAIN FADE CONTROL

RAIN WITH COMMUNICATION 3: A ALGORITHM

N?I-22_9_

Unclas G3/32 FOR

00134B2

_L

_w

A STATISTICAL MODEL WITH COMMUNICATION III-

A Stochastic Link Power

RAIN ATTENUATION PREDICTION APPLICATION TO THE ADVANCED TECHNOLOGY SATELLITE PROJECT

Rain Fade Control Via Noniiiie_ir-Markov

Algorithm Filtering

for Satellite Theory

Robert M. Manning Aeronautics and Space Administration Lewis Research Center Cleveland, Ohio 44135

National

SUMMARY

The dynamic earth-space

and composite

communications

i.e., rain attenuation, counter

Advanced

Technology

schemes

derived

Prediction

Model

and nonlinear

Prediction

Model

discerns

portion

probabilities, and, as shown schemes

which

are employed.

etc.,

Markov

U.S.

for the "spot beam"

of the model,

to optimally

which

etc., combined

in this paper, can be made

filtering

identification

estimate

and predict

theory.

variations

performance

are being

Project

However, parameters

to the design

to the particular

and prediction the levels

of each

met in NASA's

by the implementation

of

Rain Attenuation

The ACTS

of the model

to

bands)

Rain Attenuation

on the order of 0.5 ° in latitude

to the state-variable

specific

with the need

statistical

on

Ka band,

of such frequency

the use of the ACTS

links of ACTS.

is amenable

(typical

(ACTS)

The static portion

yields

is isomorphic

Satellite

that are incurred

the 30/20 GHz

Such requirements

through

climatological

in the continental

predictions

in order components.

Communications

longitude

air scintillation,

the use of dynamic

signal

attenuation

processing

clear

impairments

in and above

after the small link margins

necessitate

of the fading

of the deleterious

optimal

and/or

such degradations

processing

of propagation

links at frequencies

cloud

have been exceeded,

nature

gives precise

the smacture

and

availability

of the dynamic

such as fade duration

approach

of stochastic

of such statistical climatological

control

theory

fade processing

location

at which

they

I.

INTRODUCTION

The development

of the ACTS

in a variety

of such models

performance

requirements

a satellite Project

system

estimate

that now exists, of satellite

is NASA's

after which

Rain Attenuation

the model

is named.

case, attenuation

systematic

measurement

attenuation

levels

deployment

Two

major

) and 2) predict

(defined

with all its attendant

the use of which

out the component

different

manner

problem,

by time diversity It is the purpose Rain Prediction

are robust whereby

beyond

of clear-air

Model

to such scenarios

discrete

points

in time.

required

to obtain

optimal

a situcom-

etc., as

fade mitiga-

fade threshold.

that is not only impaired here, one receives

communications

link, must

since each must be dealt with in a

and scintillation,

if it proves

to be a

transmission).

of this paper

one has available

effects

control

some

scintillation;

satellite

such

the need for the

due to modulation,

at a frequency cloud

what

the received

some pre-established

and/or

rain and scintillation

(e.g., rain fade by power

to employ

in the

as well as

). For example,

due to rain so as to drive

signal and, if one is to have a reliable

separate

terminal

fluctuations

one may be operating

by rain but also by the phenomena

ACTS

satellite

power

is needed

so as to forewarn

here as prediction

of link attenuation

Or, as in the case of ACTS,

the total fading

link using,

here as identification

a short time into the future

Such

are the ability to 1)

errors

(defined

and

(ACTS)

with random

of measurement

tion technique,

Satellite

that may be corrupted

ation may exist for the user of a small remote

the source

requirements

design

(the Ka band);

measurements

will prevail

channel,

30/20 GHz

Technology

[ 1,2,3], another

system

signal fade due to rain on a satellite

of a fade countermeasure

munications

Model

to several

in and above

Communication

the level of a communication

general

was in response

systems

Advanced

Prediction

how results

of the dynamic

can be used in the implementation indicated

attenuation

From

to indicate

above.

measurements

these discrete,

estimates

In particular,

"noisy"

with associated

of not only the satellite

2

of processing

the simplest

observations

portion

of the

schemes

case is considered

measurement

errors

of link attenuation,

link attenuation

that

that corre-

it is

at

spondedto the measurement, butalsoanoptimalpredictionof whattheattenuationvalue will beat thenext (future)samplingtime. This problemis formedwithin thecontextof non-linearMarkovfiltering. Theoptimalitycriterionusedwill betheminimizationof the leastsquareerrorthatexistsbetweenthepredictedvalueof attenuationandthatestimated from the noisymeasurement process.Theonlyfading mechanismthatwill beconsidered hereis thatdueto rain; scintillationandothersimultaneoussignalpowerimpairmentscan alsobeconsideredbut attheexpenseof a muchmorecomplicatedandinvolvedexposition. Suchcaseswill bedifferedto futurepublications.

II.

A BRIEF ACTS

From lar, Section through

OVERVIEW

RAIN

OF

THE

ATTENUATION

the development

of the ACTS

5 of [3]), one models

the parameter

Xa(t),

DYNAMICS

PORTION

PREDICTION

MODEL.

Rain Attenuation

the temporal

evolution

OF

Prediction

THE

Model

of the link attenuation

(in particuA(t)

i.e.,

InA(t

)-

lnA m

x A (t) :

(1) tYlnA

A m is the median

where

logarithm quency t"

given

of attenuation; of operation,

of the link attenuation these two parameters

and geometry

by, in the most general

determined pedient

by a system

to reduce

poration

are specific

of the satellite

link.

case, a multi-component

of first order

to a one component

of the temporal

and o_n A is the standard

smoothing

stochastic model process

as was done in [3], one can consider

cess given

by the single

first-order

differential

equations. consideration

by the extended

(approximately) equation

3

random

of the

location,

It is the XA(t) parameter

differential

induced

particular,

to the particular

Markov

with careful

deviation

fre-

that is

process

which

It is, however, given

is ex-

to the incor-

propagation

the one component

path. pro-

In

dXA dt

where

the random

"white

noise"):

and where

function

_'(t) is governed

7S is a "smoothed"

scendental

7sXA + _/ 27S _ (t )

temporal

_71

geometrical

between

+exp

concerning

isotropy

the coefficients

L is the total propagation

elevation mental

angle,

by the solution

(i.e.,

of the tran-

=

(3)

the propagation

71 and 72 that appear

characteristic

rain cell

one has the following

relation

in Eq.(3):

c ]

path length

and model

path,

2LcosO.]_l/2

constraints

(4)

within the potential

and R C = 4 Km is the characteristic

considerations

the coefficient

-

of rain cell movement,

72=71 where

that is given

with a zero mean

7s]

considerations

size, and long term spatial that exists

parameter

statistics

equation

exp From

by Gaussian

(2)

[3].

rain region,

rain cell size as required By the same considerations,

0 is the link by fundaone has for

71

2V

-

-1

- 0.1336m

in

(5)

_R c

where

v = 14 m/s is the characteristic

For a typical

communications

speed

of the rain cell that is assumed

link with an elevation

sec" 1.

4

in the model.

angle - 40 °, 7S - 0.06 min "1 = 0.001

In what follows, the onecomponentMarkovprocessXA(t) contact

made

to the link attenuation

fade detection

III.

and prediction

algorithm

DEVELOPMENT

OF

PREDICTION

A.

Observation

Here,

the problems

following

AN

and

OPTIMAL

Sampling

ation.

"frequency surement

process scaling"

at another

characterized

an optimal

fade data.

IDENTIFICATION

AND

Model

it is useful

attenuation

measurement

The measurement

measurement

(real time)

with

and prediction to consider

as defined

the measurement

in the introducprocess

via the

model"

is the actual

the associated

FADE

of fade identification

At the outset,

observation

A(t)

will be used to derive

for use with measured

Aobs(t) where

Eq.(1),

by Eq.(2),

ALGORITHM

The

tion will be solved.

A(t) through

given

that exists

uncertainty noise, which

or from whereby

a signal

(6)

on the communications

or "noise", can result

an assumed

frequency,

by a white

= A(t ) + n (t)

and Aobs(t)

function

fade at one frequency

is a random

noise process,

is the observed

from inaccuracies

measurement

function

link at time t, n(t) link attenu-

in the hardware

of the

such as that used in

is derived

from a fade mea-

of time and can also be statistically

viz,

(n (t))=

0

(7)

2 (n

where

an is the standard

has from

deviation

(tl)n

is

(t2))=Cr

n S(t 1 -

of the measurement

t2)

noise.

Remembering

Eq.(1),

one

Eq.(6),

Aobs(t)=A(XA,t

) + n(t)

, A(XA,t

5

)=Amexp((TlnAX

A (t))

(8)

for the observation ferential

model

equation

fade Aobs(t)

where

Eq.(2).

and given

The following Eqs.(8),

quantity

XA(t)

quantity

XA(t + x) that should

Eq.(1),

the random

problem

This is formally tion (or filtering)

x A is governed

prevail

obtain

a problem

and prediction

by the stochastic

can now be defined:

(7), (2), and (1), it is desired

at time t (i.e., identification)

one can then easily

process

to obtain

as well as the extrapolated

at a future

the measured

an estimate estimate

time t + x (i.e., prediction).

the corresponding in the optimal

of a non-linear

given

values

of the

of the Using

A(t ) and A(t + "c).

(with respect Markov

dif-

to a given

continuous

criterion)

random

estima-

process

sam-

pied in time. Since information

the random concerning

Aobs, one must sity P(XA(t)

process

the instantaneous

invariably

I Aobs(t)

vation

of the quantity

easily

obtain

x;(t

is the quantity

) governing Aobs(t).

the random Once

known process

this probability

of the random

) ( referred

x A and it is desired

value of this quantity

deal with a parameter

descriptions

as the average

in question

process

to as the optimal

from the measurement

as the a posteriori

has been

minimized

den-

on the obser-

secured,

x A in the form of statistical estimate

of

probability

at a time t conditioned density

to obtain

one can moments

with respect

such

to the

8c

mean-square

error),

Although latter is actually

the associated

both x(t) sampled

standard

and Aobs(t)

at discrete

deviation

_rx_(t ) of the optimal

are, in general,

continuous

times ti with a corresponding

the samples,

i.e., T = ti - ¥1 • Thus, due to this sampling

uous process

Aobs(t),

sult from

the sampling,

one must also now consider

circumstance

time interval overlaid

random

the

T separating

on the contin-

processes

which

re-

viz,

A obs(tO = A obs(t )_ (t-ti)

One must thus amend

etc.

in the time variable,

process

additional

estimate,

the a posteriori

and consider

p (XA (t),

x A (t_

and

probability

density

xa (t) I A obs (t)),

6

= x A

(t)S

P(XA(t)

(9)

(t -t i ).

t Aobs(t)

viz, the probability

) to reflect density

this

gov-

erningthecontinuousrandomprocessXA(t) on the value

B

of the discrete

The

•

measurement

Statistical

Identification

the random

process

and the sampled

of Aobs(t)

process

x A (t)

conditioned

•

of The

Prevailing

Link

Attenuation

Level

Now sition

(and conditional)

with the evolution counterpart

XA(t),

probability

density

of the sampled

value

by Eq.(2),

p (Xa (t)[

associated

probability

with the first order

differential

_P(XA (t )l'XA (ti))-_t

the differential

operator

_a (t3)

giving

the results

density

is given

equation

Eq.(2),

Dxa[P(XA

Dx, _

has associated

with it a tran-

the statistics

connected

(ti) at a time ti to the value of its continuous

X'A

x a (t) a later time t > ti . Following

and (54) of [3]), this transition

where

described

in [2,3] (in particular, by the Kolmogorov

equation

i.e.,

(t )l'_A (ti))] ,

t > ti

the Kolmogorov

(XA) ] for

Eqs.(53)

(10)

equation

is given

by

2

f Oxa_e

(XA)]---

)tS _-_

Af

(3(,A)} + _ZS

probability

density

of measured question

fashion,

governing

observations

pling

model.

model

density

the random

model

Equation

In particular,

of Eq(9),

_X A

p

processes

A obs (t), each of which

via the observation

case, the Kolmogorov servation

the probability

of Eq.(6)

of Eq.(10) amending

(11)

2

_XA

In an analogous

(XA)

(X A

(t),

x A (t) I A obs (t))

Xa (t) and x A (t) conditioned is connected

and the sampling must be augmented

the observation

one has that the quantity

7

is a transition

model

with the random model

of Eq.(9).

on the set process In this

with this (non-linear) of Eq.(6)

ob-

with the sam-

in

n (t) = A obs(t ) - A (_A

,t

)

(12) 2

is a zero mean Appendix,

Gaussian

random

variable

with variance

using this fact in an extrapolated

lution of x a (t ), one obtains a modified

version

a posteriori

an integrodifferential

of the Stratonovich

¢rn (by Eq.(7)). probabilistic

equation

Equation

As shown analysis

in the

for the evo-

for p (Xa (t ), x a (t ) I A oo s (t ))

[4], well known

in Markov

filtering

the-

ory, viz,

Op

(x -

t [(

A(t),XA(t)lAobs(ti)

_DxaP

bt

-- _

_

O(XA"

(t),ti)P(X

)1[

XA(t),XA(t)]Aobs(ti)

A" (t),X

+

(_(XA(t),ti)

--

A" (t)Iaobs(ti))dxa'dxa']X × p(x

a (t),x

a (t)Iaobs(ti)

(13)

)

where

1 2 (Aobs(t i )dP(XA(t)'t')-_

It is important

to note that Eq.(13)

Obtaining bility density

(14)

holds for times t within

from Eq.(13)

commences

the interval

with writing

ti < t < ti+ 1.

the a posteriori

proba-

in the form

P(XA(t

Substituting

a solution

A (_ ,t i ))2

2_ n

Eq.il5)

),'XA(t

)lAobs(ti))=P(XA(t

into Eq.(13),

)IXA(t))P(XA(t

integrating

the result

with respect

)lAobs(ti)

)

(15)

to x, and using the

facts that

and f_

P(XA(I)lXA(t

))dXA=

f_

1

8

Dxa[P(XA(t)lXA(t

))]dx

A

0

(suchintegralsindicateintegrationoverall thepossiblevaluesrealizedby x A t and where p (XA

the latter relation

(t) I xa

is obtained

and its first derivative

(t))

_gPlXa(t)lA°bs(ti))Ot

-- f_

Eq.(15)

_P(XA" (t),ti)P(Xa"

the probability

at x = + **) yields

conditions

that

the relation

) --

(t)laobs(ti)}dxa]P(Xa

Eq.(16)

density

and the boundary

back

p (x a (t)

(t)laobs(ti

into Eq.(13),

one then re-derives

Ix a (t)) thus making

(16)

))

Eq.(10)

the proposed

solution

of

self-consistent. An approximate

second

term within

x A. Since

form which

analytic

expression

the brackets

is simply

it is only a function

the characteristic

can be obtained the average

(O{t

Of t, and if the time interval

time if variation

for this average,

for Eq.(16). I)

One notes

over all possible

that the

values

t - ti is small as compared

one can neglect

it and obtain

of to

a simpler

has the solution

P(XA(t)

where

vanish

[(btxa(t),ti

for ti < t < ti+ 1 . By substituting prescribing

using Eq.(11)

at a f'L_ed time

If

lAobs(ti))=Nexp

the normalization

]

_(_A(t'),ti)dt'

constant

(17)

p('XA(ti)lAobs(ti))

N is given by

-1

N

__

(18)

P (_A (ti ) IAobs(ti))

Equations density

(17) and (18) define function

the "partition"

representation

[5].

9

of the a posteriori

probability

One now admits the probability

density

the well established

Gaussian

p{ x a (t i )1 aob s (ti))

/

and K(t i ) is the standard

exp

estimate, ployed

which

of the random

of this estimate

from the optimal

here since it is the only aposteriori

random

process

tics that govern ferred

is obtained

x A (t)

to combine

the possible

to as the standard

rors of the extrapolated into Eqs.(17)

qb_A(ti

),ti} that appears

series about the optimal

deviation

estimate

into the expansion

Using

Eq.(9)

and using Eq.(19),

(

exp

a(ti)(XA--X")

is em-

sampling

re-

with the er-

by the substitution

of

the function

in Eqs.(17)and th

the

is sometimes

associated

one expands

at the i

composite

19 ([XA

defined

xa

that of Eq.(19).

the extrapolated

time interval,

K(t)

the covariance)

functions

to match

value;

at time ti

Aob s (t i ) to yield the statis-

For this reason,

amenable,

obtained

x A (t i ) available

the actual

quantity

the exponential (t)

process

2 )

that one has at time ti about

To make the problem

up to the second result

for "XA(ti).

(19) _A(ti)--XA(ti)) 2 K(t i )

of the previous

information

and (18) analytically within

estimate

( or, in general,

estimate.

Eq.(19)

about

with the observed

values

that

-(

2zr K(t i )

estimate

deviation

demands

1

¥

x A (t i ) is the extrapolated

which

take the form

P ('XA (ti) I A obs(ti )) = ft/

where

approximation,

(18) into a Taylor

time ti , retaining

terms

in Eq.(14)

and substituting

this

one obtains

from Eqs.(17)

and (18)

2)

('XA? x_-2crx ,(t i )

(t) l A obs (ti)) = (20) 2

* lrff x ,(ti ) exp

10

a2(ti_(ti).)

where

the quantity

defined

shortly)

(YxA(ti) is the error covariance

given

of the optimal

estimate

(the latter will be

by

.

(YxA(ti) -

(

K -1 (t i ) - b (t i )

)1

(21)

and

a(t_=

are coefficients _)s( XA

(ti)'

-3_s(_,

that contain ti )=

_)( XA

(t),

, b(t_

=

_Ps(_'t'_-_

t_]

1 32

the observed

sampled

t )(_lt

Using

--ti).

(22) ]

values

A obs (ti);

here,

Eq.(14),

one then

obtains

a(ti)=((Yl2A_lA(_,tiXAobs(ti)-A(_A,ti)

)

(23)

o-/ b (ti)=

Using pression

the a posteriori

for the optimal

tional data Aob s (ti). estimate, square

--

probability

estimate

density

optimality

here is the simplest,

yields

Eq.(20)

relationship

for the optimal

of Eq.(20),

criteria

one can now obtain

of _A (ti) connected

with the observa-

that can be used to define

viz, the criterion

an ex-

that minimizes

a particular the mean

the relation

x A(ti)=f[_ Substituting

2A(XA,ti))

x *(t_ of the value

Of the many

the one selected error which

A(;,ti_Aobs(ti)-

X A(t'_p(

into this and performing

xA(ti)lAobs(ti))dx the integration

estimate:

11

yields the following

recurrsion

X A (l'_

As mentioned timal estimate ity density given

earlier,

the extrapolated

of the previous

defined

(t3 + tYx(ti)

= _A

sampling

by Eq.(10).

estimate

a (t o

X'A

time ti-1 through

The solution

(24)

(t3 is that obtained

from the op-

the use of the transition

of this equation

is well known

probabil-

[2,3] and is

by =

X A (ti) -- _i "XA(ti-1) exp

(25)

1

where

@i-

exp (-)'s

(ti-

ti-1 )). The extrapolated

N

posteriori

substituting

_A

(t.) at time ti given

the a

*

Eq.(25)

_XAP(XA(ti)

I_A(ti-1))dXA

into this relationship

and integrating,

X A (ti)

which,

estimate

@it

value XA (ti- 1) = XA (ti- 1) is simply

f

Upon

2(1-

when used in Eq.(24), It now remains

trapolated

estimate

K(t).

quantity.

In particular,

=

demonstrates

to obtain

a similar

Equation

one obtains

(26)

(I) i X A (ti-1)

the recurrsive expression

(19) essentially

one has

12

nature

of the estimate.

for the standard provides

a working

deviation definition

of the exfor this

2p (XA(ti)

=

In order

to relate

a value of this parameter

1, one can use Eq.(15)

I Aobs(t i ))d'x

at ti to the a posteriori

values

obtained

to write the relationship

K(ti)=f_f____.(_A(ti)-'XA(ti))2p(_A'(ti),'XA(ti-1)lAobs(ti-1))d_

Using currsive

the previously

derived

expressions

(27)

d'_"

of Eqs.(19)

and (25) in Eq.(27)

yields

the re-

relationship

K(t i ) = 1 + _i that relies on former Equations

values

collectively

outset.

The approximation

of the errors

incurred

the dynamic

samples.

However,

are easily

prediction

from this fade identification

estimate

within

(28)

to the auxiliary

place upper

this limit.

relations

part of the problem

one to write Eq.(17)

the typical

of link fade levels

)

from Eq.(21).

identification

used that allows

in the optimal

in practice

for the dynamic

¢rxA(ti-1 ) - 1

of a x A(ti ) obtained

compose

consecutive

encountered

2(.

(24), (21), (26), and (28), in addition

Eq.(23)

between

at time ti-

of

defined

as well as considerations

limits on the time interval

clock

intervals

It now remains

into the future

based

at the

- 1 second to obtain

[ti-1, ti)

that are

a prescription

on the data afforded

process.

o

C.

The

Prediction

From the transition ing the average problem

some

of Link

results

probability value

Attenuation

of the foregoing, density

defined

as follows:

Short

in particular,

of the attenuation

of the link attenuation

can be succinctly

for

process,

Times

the solution

into

given

one can derive

at a time Tpred into the future. Given

13

the fact that an optimal

the

Future

by Eq.(25) a relation

yield-

In particular, estimate

for

the

XA (ti)

is obtainedfor the attenuationprocessat thetimeti, tion will increase, change

it is of interest

in the attenuation

value of this combined

to obtain

that will occur extrapolated

and on the hypothesis

the extrapolated

estimate

at a later time t i +Tpred.

optimal

estimate

Z_

that the attenuaa (t i +Tpred)

of the

One then has for the total

of the attenuation

process

A

X A (ti +Tpred)

By this formulation

= X A (ti)

of the problem,

+ z_

A (l i

(29)

+Tpred)

one has

A

AX A (t i +Tpred)

=

=

+Tpred)

a (ti +Tpred)

P

fO'Z_a(ti

> Xa (t_ + AX a (t i +Tpred ) I X a (t'_

(X

(ZkXa )

*_

(30) where

the conditional

P

(x

A (fi +Zpred)

probability

> XA

(ti)

=--

density

+ ZIXA

upon

a change

by

.)

(ti +Tpred)

(t'_

=

p (X A

"(t

IX A

fl a

To this end, substituting

is given

)IX A (t i )d x A"

i)+Axa(ti+Tm.a)

Eq.(25)

into Eq.(31)

and this result

of variables,

.------AX A (ti +Tmea)

1 2

(31)

erfc

Y 2D-(T

14

yay-

into Eq.(30)

yields,

- x_(1-

*pred

(,,/213

}f_

erfc,

(32)

(-rpree)Y)dy]

2

where

_precl = exp (I-_s

which

one can assume

contributions conjunction future

and D (Tp,ed)

Tpred)

*(

with Eq.(29),

(l)pred . In the approximation

)

that x A 1--_pred

of the integrands,

-1-

= 0 relative

the integrals

in Eq.(32)

gives for the extrapolated

time t i +Tpred based on the estimate

to the range

of integration

can be analytically

optimal

estimate

in and the

evaluated

of attenuation

and, in for a

at time t i

,

D (Tpred)

XA (t i +Tpred) = tFpred X A (ti)

+

(33)

4

where ptrlJ-red -- 1 - 4O(Zpred) 2zc

A limitation to evaluate

is placed

the integrals

upon

the maximum

of Eq.(32);

in particular,

(1

(34)

IfI)pred )

value of Tpred by the approximation

used

one has

tTln A Tpred