Oct 13, 1989 - COSATI CODES. 18. ... of discrete conditonal probability density functions (pdf's) for a phase state variable shall be ... order to produce a phase reference or estimate, S(n). Here. the ..... used for sifting or masking some of the.
t; 71
,
" ",
1
I,4CT...1,8-7
Fir aI Technical Report 00,66w 10W
NETWORK SIMULATOR DEFINITION cV) (0 0q
1 frmacise University
Dr. Pramod K. Varshney
APPROVED FOR PUBLIC RELEASE; DISTRIBUT ION UNUMITED
.. EA..T1 E E
OCT 13 1989
ROME AiR DEVELOPMEN1 CENTER
Air Foc c. Systems Cummand Grifflss Air Force Base, NY 13441-5700
89 10 18 048
This report has been reviewed by the RADC Public Affairs Division (PA, and is releasable to the National Technical Information Service (NTS). At NTIS it will be releasable to the &eneral public, including foreign natiorvs. RADC-TR-89-167 has been reviewed and is approved for publication.
APPROVED:
/U1
~
JOHN B. EVANOWSKY, Ph.D. Project Engineer
APPROVED: JOHN A. GRANIERO Technical Director Directorate of Coumunications //
FOR THE COMMANDER:
M1 JAMES W. HYDE III Directorate of Plans & Programs
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36 October 1989 IS SUPPLEAENTARY NOTATION Figures are out of sequence because they are part of a more comprehensive report which will entail a stressed tactical communication network being designed for PC's Netwok Dsin Laboratory FROMOCt 86
Final
17.
COSATI CODES FIELD
GROUP
SU-ROUP..-
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18. SUBJECT TERMS (Com ome on revere ,f nhctry and edeef by,bfck number) Networks))
-- C99.JIqCjao1
-Ciest and Evaluation
~~SimulationIaIKz
02 19. ABSTRACT (Contbiue an rtvers if ihEmC6wy and, oden fyby 11-c, numbr)
A white paper was delivered to RADC which provided a detailed analysis of a technique utilizing state transition probability matrices that provided a reali~tir-model-of a digital coounlcations channel. As such, the technique offers a computationally viable method for implementing a large number of realistic channel models. This work
will feed or transition to efforts eventually implementing this technique to perform network survivability analyses. The results will be incorporated in more comprehensive reports resulting from this work and in the actual implementation of these techniques in RADC s Network Design Laboratory. 7- h
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UNCLASSIFIED
UNCLASSIFIED
UNCLASSIFIED
A DECISION DIRECTED PHASE ESTIMATOR MODEL
INTRODUCTION The objective of this effort was to develop techniques for modeling digital errors which occur in a communication system.
The model was to be
computationally efficient for applications which require a large number of replications of the model such as a communication network analysis. In this effort, an important component of a digital communication system, namely a digital phase-locked loop was analyzed.
The analysis produced a stochastic
model of the digital error producing process associated with the system.
A
state transition probability matrix formulation was used which allowed the calculation of a variety of digital error performance measures. In this leport, an analysis will be performed of a digital phaselocked loop.
Using standard analytical procedures and assumptions, a set
of discrete conditonal probability density functions (pdf's) for a phase state variable shall be developed.
This will then form the basis for a
state transition matrix formulation which will be used to calculate vari>Lprobability of error statistics. The standard cross-correlation demodulator described in the literature implies that a carrier with proper phase is present at the receiver. In practice, the correct phase must be determined at the receiver.
One
popular technique for performing this function is the decision directed phase estimate.
It finds its place in this report because of the "memory"
it imparts to the error producing process.
'
-1-
.....i''" a ro e
i ,, " -
t
I
tI
'
Ci n I
are
There be
might
ind
assumed
an
to
be
the
before,
considering received
SYSTEM
A
at
gnsumed spectral used
to
of
presentation,
demodulator
the
shall
of
which
decision will
requires
a
phase
That
iS,
is
system
transformation
phase
by
analysis
receiver.
only
the
the
This
the
assumed.
be
demodulators
phase
of
as
reduced
being
technique
for
vector.
implementation shown
introduced with
input to
clarity
For
dimensionality
signal
is
along
the
a demodulator
MODEL
estimator
s(t),
at
be
of
multiple
to
will
performed
demodulator
generated
typical
notation
be
additive
of
noise
phase
Drevions.Lv.
any
for
useful
of
preser':e
transformation
described
as
reference
the
keyed
appropriate
boundaries
by
will loop.
shift
the
only
that
noise
phase
of
causes
source
applicable
is
analysis
prove
the
analysis
an
phase
binary
using
as
noise
a phase-locked
employing
The
Here,
hypothesized.
white Gaussian
a
underlying
many
be
the to
generate
of an
Figure
a decision 6.
earlier, additive the
white
density
in
of
noise
noise
A reference
in-phase
-2-
and
with
transmitted
waveform,
The
with
phase
Consistent
waveform,
demodulator.
Gaussian N 0.
the
directed
a
z(t),
additive two
sinusoidal
quadrature
is
the
shown
noise
sided
power
waveform
signal,
is
each
is of
0
ldU4 E-4~
411w
410 0.j
*
OE-4e "ID
U-3
C
S
O
I
E
T
D
q
S
SIUA
J
v,
which
signal
vectors.
time
is
n
The
given
in
dissertation,
should
equal
In
the to
equal
is
received converge
phases.
is
to
order
to
the
first-order
the
decision
process.
by
a voltage
received
previous
at
vector
signal
received
this
for
voltage
the
of
vector,
signal
generated
control
received number
phase more
vector.
the
vector
Thus,
as
-reference,
An
one
an
shown
-4-
of
of
or
the
used
should
phase,
or
i(n),
technique
Figure
,
the
This,
of used
signal
loop
phases,
modeled 7.
is
received
estimate,
be
for
measurement
digital
received
shall in
the
analog
reference
filter
phase
averaging
value
sequence
phase
loop
network
signal
2aE.
than
Typically
a
being
viewpoint
the
to
due
expected
process
produce
Here.
on
signal
vector used
the
based
to
is
required
zero.
general,
demodulator
the real
the
ideally
these
]/Re[v(n) ])
that
before,
from
ideally
of
by
Recall, this
the
the
of
phase
tan 1'(Im[v(n)
(n)-
in
phases
the
using
generated
is
VCO
received
The
(VCO).
os,!illator
controlled
the
reference
phased
properly
This
used
be
ultimately
will
of
phase
correct
the
assure
to
phase
is
estimate
phase
accurate
An
waveforms.
reference
each
of
argument
the
in
used
is
$,
estimate,
A
integrator.
cross-correlation
a
in
used
is
which
filter
4(n),
in
S(n).
as of
a digital course,
FII~ST ORDER
LOOP
-5-
FITER,
computes
values
for
the c
a(nT)-
The
transfer
been
derived
first-order
in
for
textbooks
the
system
such
equation:
0(nT)
(nT-T)+
function
difference
as
Gold
shown
in
Figure
and
[1969]
is
7 has given
by jwT H(e
-1/2 )=(i
c-2c
-I
coswT)
exp[jwT-jtan
(sinwT/(coswT-c))!, (4-1)
This
function
in
terms
T.
Note
that
one-half
the
the
of
loop
the
wT(n),
is
random
expressed
phase
model
of
the
filter
discussion
)f
coefficient
c0
c
c
(n)
loop
The
expected (n).
An
the
(n).
filter
which
correct
phase
value
of
the
implementation
random
sequence.
the a
filter,
generic
(4-3)
h(n-1) l
controls
pass
is
recursive
using
the
variables,
reference. the
of
as
C¢
lw
random
phases
0
coefficient
of
to
input loop
4(n)-
The
implementation
first-order
equal
sequence
the
digital
the as
first
normalization
-10-
roll-off indicated order factor
frequency
for
the
in
the
filter.
The
earlier
digital which
assures
that
each phase
estimate
;s
a~sc
defined
relationship
satisfies in
the
some
very
coefficients
are
it
is
that
for
the
known low
pass
case,
respectively,
that
if
as
zero, the
the
Co.1,
and
digital
approaches
1,
filter
approaches
zero.
c l1
and
0(n-l)= (n)-
phase
initial
for
phase
previous
From
c1
i.e.,
then
demodulation
limiting
the
c 0=0,
properties filtering roll-off
the
theory,
frequency
Therefore,
Equations
of
4-3
in
this
and
4-2,
yield
the
the
correct
following
the
(4-4)
considered.
(n)-
is,
],
' (n)
1
interesting
O(n)-
That
(-rv
Since
-T.
-
0
limiting
region
i}
3
composed
of
five
probability
of
falling
given
by
Equation
segments in
}
as
any
is
defined is of
in
noted Equation
a result the
of
random
S(n-l).
(In
given
a
the
is
fact
chat
variable,
of
an
4-10.
the
4-12
Example
probability Equation
that
$(n-l). this
error
is the
previous
P(error(n)I$(n-l)=$
>
is than
4-15
equal
to
c 0 N/4
previously provide
the
states the
states
the
computed
probabitity
of
error
probability.
This
step state
-21-
is
a
function
particular
value
of
calculated
using
of
a
error
as
follows:
number
estimate
-
>
c
a
$3)
symbol
the
(4-15)
N/4)
of
transition that
was
0
probability
Recall
single
symbol
probability
of
Example
a
(n)
probability away.
a
.
that
a
of
value
1.
Equation
these
particular
phase
P(
of
probability
For
4.1,
Express
particular
conditional this
of
in
4-10.
probability
a
shown
one
End
It
(4-14)
3
the
transition
a symbol to
a state
components matrix,
transitions.
error more
of
F1
However.
the
(7 i the
computation single more
in
step than
Squation transition
c 0 N/4
representation the
set
S
of
j for
l,..,N;
the
that
modulo
these
integers
uses
matrix
states,
of
(j,
(i)-
4-15
produce N.
elements, each
such
i
that
elements
For
define
such
a
of
this
transition a
of
convenient
the
set
Sj.i)
as
that
>
ji-jl
c
j
N/4.
modulo
N)
0 ( 4- 16)
Now
Equation
components
of
4-15 the
can
single
be step
expressed
in
tran-ition
matrix
terms
}=
P(error(n)j$(n-l)= i
f
the
(4-17) S
(i)
j
J
The
average
arbitrary
Pu e
time
n
probability is
given
of
a
symbol
vector,
P(inf]-
previous 4-17,
the
into
phase
components (Pi(inf)},
estimate
Equation
4-18
Pe, e'
at
an
by
N Z P(error(n)I$(n-l)-O > P(O(n-l) =il i
Substituting
error,
of for
0(n-1), yields
-22-
the
steady
the and
(4-18) i
state
probability
distribution the
results
in
of
the
Equation
N P
p
=
e
This to
equation
the
sum
state
relates
of
than of
relationship
S
that
the
C 0 N/4
being in
in
a
(i)
transition
units
state
matrix
j
probability
of
(4-19)
1
(inf)[ i
probabilities
more
probability
i1
i.
of from
away, In
form,
error
is
state
weighted
order define
equal
i to by
to
express
the
matrix,
a the
this S,
as
fol lows
S-
} where
(s ij
Thus, the
ij
for
those
values
corresponding
Example
I for all li-jl < ( 0 otherwise
s
4.1
The
i and
element
of
the
j
for
c N/4 0
which
matrix
is
(4-20)
an
error
equal
matrix
S
for
1 1 1 1
this
1 1
I 1 0 0 0 1 1 1 o 0 11110001 1 0
1 0
example
is
given
order
product
to one.
by
1 1 1 1 1
1 1 0 0 1 1 1 0
; N
8,
c
=.5, 0
c
N/4=
to
use
operation,
the
I
0
0 0
(4-21) End
In
occurs,
Continued
00111110 0 0 0 1 10001111 S=
of
>
matrix,
referred
S, to
-23-
a as
relatively
of
Example
unorthodox
congruent
matrix
multiplication rules used
of for
matrix. matrices,
is
used.
standard
matrix
sifting
or
It
term
is
A
a
and
C=
B,
A*B
This
defined
c
;
J
The
can
be
the
NxL
average
now
of
is
the of
not
follow
simply
a device
components two
the
of
identical
a
order
; i,j-
l,..,N
(4-22)
ij
vector of
with error
in matrix
Ptinf ] (-l*S)
-
It
product
b ij
probability
P
does
by:
a
column
be expressed
some
term
ij
Let
algebra.
masking by
operation
all
elements
derived
in
equal
to
Equation
1.
4-19
form as
J
(4-23)
e
As
alluded
dissertation certainly chapter.
error
time.
the An
resulting bit
is
from given
given
earlier, error for
an
measure
an
the
is
error
probability
major with
phase-locked
this memory that
the
models
the
excellent
This
n
on case
Equaticn 4-18, time
to
of
the
the
be
probability of an
error
at
the
previous
-24-
of
memory. loop
This
model
error
this
in
is this
dependence
average probability of
occurred
will
focus
in
the
calculated error at time
n-1
previous
next. an is
a
bit
As
in
arbitrary given
by
P
! error(n-l)>
P(error(n) ele
N
Z
> P($(n-l)= $
P(error(n)J$(n-l)=
i~1
i
error(n-l)> i
(4-24)
The
last
expression
is
the
result
version of
the Chapman-Kolmogorov
process.
That
dependent only probability second
estimate
ends,
occurs.
This
which
to
rather
process present
at
problem.
these
developed.
In
used
for
corresponding
in
is
Before
is
more
inverse of
the
last
given
past
known.
is
problem
this
development,
each of
the phase
times:
-25-
when
the
is
the
phase
an
error
problem scived the
transition
current
to
of
problem, the
given
that
is the
inference
be evaluated,
transitions
The
in each state of
calculated
This
first
the probability
the
system was
Equation 4-24 can phase
and
compute In
n is
It
that $.
to
The
development.
at
used
time
Equation 4-17.
begins,
the
the
by
probability
was
Markov-l
estimate.
given
phase state.
that
concerning
be
the
was
next
state
phase
the
state
matrix
a time
at
than
just
probability
probability of error
requires
phase
the
a
4-24
is
equation
the discrete
for
previous
this
is
a
probability
the
the
probability that
jumping
on
the
in Equation
observed
in
is,
of applying
the
following
estimates 4 at
some
its
theory
past must indices the
be
shall
following
.
, 4(n)
(n-1)=O
,
4(n-2)=4
......
k
(4-25)
.....
j
i
Let
Then, p (n-2) iT (n-2,n-1) ki k (n-l,n-2) -.-------------------p (n-1) ik
(4-26)
i
As
forward and that 3-19
the
is
substituted
chat
assume
Equation
steady
state
so
limiting
state
probabilities
in the the
and
applies
single-step Also,
probabilities.
backward transition
process
the
for
dropped
is
argument
the
usual,
are
yielding
in Equation 4-26,
p (inf) k ik
Now the
--
-
p
(int)
return
i
(4-27) ki
4-24.
to Equation
second conditional
discrete
Using Bayes
theorem on
probability distribution:
error(n-1))-
P(Vn-)*$ i
(4-28) P(error(n-L))
Considering
each of
the
terms
on
-26-
the
right
hand side:
P>=(n-l)
p (inf)
i That of
is,
this
the phase
is the
P{error(n-l))}
Finally,
the
last
N E k 1
)
an
error results
conditioned
value "from"
This
probability
probability first in
Equation
the steady
in the
p (inf) k
steady
S
state:
z (k)
(4-30) ki
i
term:
i
is,
state probability distribution
Also,
P = e
P(error(n-l)J$(n-l)=$
That
.i
steady
estimates.
(4-29)
uses
the
discussed 4-26,
=
2 S (i) k
if
the
p
(n-l,n-2)
phase
a phase estimate inverse earlier.
then Equation
state:
-27-
(4-31)
ik
jumped
cON/4 units
single
step
Therefore, 4-27
"to"
into
the away.
cransitio.n substituting
Equation
4-31.
p (n-2) k
-k p (n-1) i
P(error(n-l)j$ n-1.-4 S (i) k
(n-2,n-l)
I
ki
p (inf) k
(4-32)
---
S (i) k
Now,
p (inf) i
4-32
and
substituting Equations 4-17
ki
into
Equation 4-24
IT
p (inf)
yields:
p
(inf) k
N
-
---
S (i)
i-1
j e
N Z k-I
e
p
i
z S (k)
(inf) k
ki
(inf)
p
S (i) k
ij
ki
i N Z
[
(inf ) -r
p
T
j
ki
k
S (i)
ij
S (i)
i--
k --------------------------------------P
(4-33)
e
Thus, the
the
previous
probability dependent
bit matrix
errors
can
has
been
given
an error
calculated
using
a transition
formulation. be
computed
-28-
in
an error
probability of
average
Other
in a similar
measures fashion.
for
MISSION of Rome Air Development Center RADC plans and executes research, development, test and S selected acquisition programs in support of Command, Control, Communications and Inteligence (C1) actities. Technical and S engineering support within areas of competence is provided to ESD Program Offices (POs) and other ESD elements to perform effective acquisition of C! systems. The areas of technical competence include communications, command and control, battle management information processing, surveillance sensors, intelligence data collection and handling, solid state sciences, electromagnetics, and propagation, and electronic reLiabdity/maihtainabilityand compatibility.
a?
