非线性卫星通信链路建模与性能评估：Volterra级数方法
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"Modeling and Performance Evaluation of Nonlinear Satellite Links - A Volterra Series Approach"
这篇论文主要探讨了非线性卫星链路的建模与性能评估，采用了一种基于Volterra级数的方法。Volterra级数在分析具有记忆效应的非线性系统时，能将已知的带通信号复包络和带通线性系统的低通等效概念扩展应用。作者S. Benedetto、E. Biglieri和R. Daffara是意大利理工学院电子与电信研究所的研究员。
首先，论文提出了一种分析方法，利用Volterra级数来表示整个信道，从而能够处理具有记忆特性的非线性。这种方法的核心在于，它能够将带通信号的复杂包络理论和带通线性系统的低通等效原理应用于非线性环境。
接着，论文将这些通用结果应用于数字卫星通信系统，研究在非线性信道中运行的这类系统。特别关注的是卫星中的在线放大器，以及地球站中的发射和接收滤波器的影响。同时，论文还考虑了上行链路和下行链路的噪声因素。
该方法的主要优点在于其分析的普遍性和实用性。通过Volterra级数，可以对各种复杂的非线性行为进行建模，同时考虑到通信系统的实际组件（如卫星放大器和地面站滤波器）对性能的影响。这使得对系统性能的评估更为准确，对于理解和优化卫星通信系统的性能至关重要。
此外，这种方法还能帮助设计者预测和控制非线性失真，从而提高数字通信系统的传输效率和可靠性。在卫星通信领域，由于信号经过大气层和卫星转发器时会遇到各种非线性效应，因此，这种建模和评估技术对于确保高质量的远距离数据传输具有极大的价值。
"Modeling and Performance Evaluation of Nonlinear Satellite Links - A Volterra Series Approach" 这篇论文为非线性卫星通信系统的性能分析提供了一个强大的工具，对于未来卫星通信系统的设计和优化具有重要的理论指导意义。
ship
for
nonlinear
channels
with
memory
satisfying
certain
regularity
conditions.
Such
a
relationship
takes
the
form
y(t)
=
X,1
10.
dT,
...
Jf0,dT,
h,(T,,
...,
Ti)
.nfl,X(t
Tr)
(3.1)
where
x(t)
and
y(t)
are
the
input
and
the
output
of
the
channel,
respectively.
The
functions
h,(
...
),
1
l
<
00,
that
generalize
the
impulse
response
of
linear
systems,
completely
describe
the
channel
behavior.
Thus
the
problem
of
characterizing
a
nonlinear
system
with
memory
is
equivalent
to
the
problem
of
com-
puting
these
functions,
usually
referred
to
as
Volterra
kernels.
In
the
following
we
shall
assume
that
the
Volterra
kernels
we
are
dealing
with
are
symmetric
functions
of
their
arguments,
i.e.,
any
permutation
of
the
arguments
does
not
change
the
function
values.
This
symmetry
assumption
does
not
entail
any
loss
of
generality
[17].
We
shall
now
derive
a
form
of
Volterra
series
which
is
suitable
to
represent
bandpass
channels
like
the
one
modeled
in
Section
II.
In
particular,
we
want
to
derive
a
form
of
(3.1)
in
which
the
complex
envelopes
of
input
and
output
signals
appear.
Consider
a
real
bandpass
signal
x(t)
and
the
analytic
signal
associated
with
it,
say
x(t).
Letting
the
symbol
rv
denote
complex
envelopes,
we
have
x(t)
=
.(t)
exp(-jcwot)
(3.2)
where
co0
is
the
center
frequency
of
x(t).
Since
x(t)
is
the
real
part
of
x(t),
we
can
write
x(t)
=
½/2
[x(t)
+
x*(t)]
=
1/2
[2x(t)
exp(jcot)
+
x"*(t)
exp(-jcoot).
(3.3)
Substitute
now
the
last
expression
in
(3.3)
for
x(t)
in
(3.1).
Every
product
in
the
right-hand
side
of
(3.1)
will
give
rise
to
a
sum
of
2'
products,
each
one
involv-
ing
l
factors
like
exp[±jcoo(t-
Tr)I.
Since
we
are
dealing
with
bandpass
systems,
we
can
assume
that
only
the
first-zone
spectral
components
of
the
output
signal
are
of
interest,
i.e.,
those
not
lying
near
the
center
frequency
w0
are
filtered
out
and
hence
will
be
neglected.
Incidentally,
this
is
the
assumption
that
allows
us
to
model
the
on-board
nonlinearity
as
in
(2.2).
In
conclusion,
among
the
products
just
described
we
shall
retain
only
those
ex-
hibiting
an
exponential
factor
exp(±jco0t)
and
disregard
those
with
exponential
factors
exp(±jswot),
s
an
integer
not
equal
to
1.
To
do
this,
we
consider
only
the
products
with
an
odd
number,
say
2k
+
1,
of
factors
exp[±1jw,(t-
Tr)]
and
such
that
the
numbers
of
exponentials
with
a
plus
sign
and
a
minus
sign
dif-
fer
exactly
by
1.
The
number
of
these
products
is
(2k+
1)
and
after
a
little
thought
it
is
not
difficult
to
see
that
all
of
them
give
the
same
values
for
the
corresponding
(2k
+
1)
-
fold
integral.
Here
the
hypothesis
that
the
Volterra
kernels
are
symmetrical
makes
its
appearance.
Denoting
now
as
y(t)
the
bandpass
signal
obtained
at
the
channel
output
after
retaining
only
its
first-zone
component,
we
can
write
y(t)
e
-jWI
O
[(2k+
)/22k+11
*f
_c
dTlf
Joo0dT2
...f
*
".
dT2k+l
*h2k+l
(Tl,
...,
T2k+1)
k
n,
(t
T,)
e
ijoTr
*
nfk+lx
(t
T
)
el/woTs
+
e+Jwt
k
[(2k+
)/2
2k+1
*f
Jo0
dT,
f
f0
dT2*f
J
odT2k+,
*h2k+l
(T1,
...,
T2k+l)
*
fl
X*(t
-
Tr)
e+i
woT
*n
2k+
2
X(t
-
Ts)
eiwoT,.
(3.4)
Compare
now
(3.3)
and
(3.4).
It
is
immediately
ap-
parent
that
the
complex
envelope
of
y(t)
is
given
by
y
(t)
=
k7=O[(2k-
)/2
2k]
*J
*-2k
dT2k++
*
h2k+1(Td,
...,
T2k+
l)
*
n
X
*(t
-
T,)
eJ,oT,
l2k+l
X(t
-
T)
ejw
CoTs.
(3.5)
Equation
(3.5)
relates
input
and
output
complex
envelopes,
as
required.
Anyway,
to
exploit
in
full
the
complex
envelope
representations
we
can
take
a
further
step
by
defining
low-pass
equivalent
Volterra
kernels.
To
this
purpose,
let
us
recall
the
hypothesis
that
the
channel
is
bandpass.
This
means
that
the
Fourier
transforms
of
its
Volterra
kernels,
i.e.,
H2k+1
((Vl,
(02,
...,
CO2k+1)
=
d-_
o
...
f
0J
_o
dT2k+l
*
h2k+l
(Tl,
...,
T2k+l)
exp[-j(w,l
T,
+
(02
T2
+
...
+
(02k+l
T2k+1)I
(3.6)
IEEE
TRANSACTIONS
ON
AEROSPACE
AND
ELECTRONIC
SYSTEMS
VOL.
AES-15,
NO.4
JULY
1979
496
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