BIOMATHEMATICS TRAINING PRDGRAM
ON THE RECONSTRUCTION OF THE COVARIANCE OF STATIONARY GAUSSIAN
PROCESSES OBSERVED THROUGH ZERO MEMORY NONLINEARITIES - PART II
Elias Masry*
and
Department of Applied Physics
and Information Science
University of California, San Diego
La Jolla, California 92093
Stamatis Cambanis t
Department of Statistics
University of North Carolina
Chapel Hill, North Carolina 27514
ABSTRACT
-e
We consider the problem of reconstructing the variance R(O) of a
zero mean stationary Gaussian process observed through a zero memory
nonlinearity f(x), from the knowledge of f and the first two moments of
the output process.
~
The reconstruction is shown to be feasible for
certain interval windows, convex nonlinearities
unimodal nonlinearities.
:a~
',<1;.,
discontinuous
The paper is the completion of the investiga-
tion begun in Cambanis and Masry (1978) where the reconstruction of the
normaZized covariance R(t)/R(O) was considered.
KEY WORDS AND PHRASES: RECONSTRUCTION OF STATfONARY GAUSSIAN COVARIANCES,
ZERO MEMORY NONLINEARITIES
*The work of this author was supported by the Office of Naval Research
under Contract N00014-75-C-0652.
t
The work of this author was supported by the Air Force Office of
Scientific Research under Grant AFOSR-75-2796.
1.
INTRODUCTION
Let X = {X , _00 < t < oo} be a real stationary Gaussian processes with zero
t
mean and continuous covariance function R(t). We observe the stationary process
Y = {f(X ),
t
line.
_00
_00
< t < oo}, where f is a real Borel measurable function on the real
We are concerned with the reconstruction of the covariance function R(t),
< t <
00,
from the knowledge of the nonlinearity f, the mean
and the correlation function
of the "output" process Y.
In Cambanis and Masry (1978), which will be referred to as CM(1978) in the
-~
following, we have presented procedures by which the
function
no~alized
covariance
:~~~ is reconstructed from the correlation function C(t) for broad
classes of nonlinearities f.
The purpose of this brief paper is to complete the
investigation of the problem under consideration by providing procedures for the
reconstruction of the varianae R(O) from the knowledge of f and the first two
moments
k = 1,2 ,
of the output process Y.
In Section 2 we study some properties of moments of functions of Gaussian
random variables, which form the basis for the reconstruction of RCO) given in
Section 3.
The reconstruction of R(O) is shown (Theorems I and 2, Section 3)
to be feasible for certain interval windows, convex or concave nonlinearities
~
as well as certain discontinuous unimodal nonlinearities.
Section 4 provides
2
a discussion relating the results of this paper with those of CM(1978) and
indicating the classes of nonlinearities for which the reconstruction of
R(t),
_00
< t < oo,from C(t),
2.
_00
< t <
00,
and
~l
is feasible.
SOME MOMENT PROPERTIES OF FUNCTIONS
OF GAUSSIAN RANDOM VARIABLES
In this section we study the moment of a function of a Gaussian random
variable with zero mean as a function of its variance, i.e.
o
where k is a positive integer and
variance
0
2
~
a < 00,
<
(1)
is a Gaussian random variable with zero mean,
and probability density function ¢(x;a).
f is a real (almost every-
where nonconstant) Borel measurable function on the real line satisfying
ePROPOSITION 1.
Let k be a posiHve 1:nl;euerl, 0 <
0
1
<
O
2
<
and leI; f(x)
00,
satisfy
k
(Cl)
2 k
f (x), x f (x)
(i)
€
Ll(¢(x;a)dx),
0
1
2a2
a .
If f(x) is of bounded variation on each finite interval, determines a
a-finite signed measure df on (-00,00) and is such that
lim
(C2)
Ixl-»>
= 0,
~oo Ix fk-l(x) I ¢(x;a) d!flex)
eC3)
then
x fk(x) ¢(x;a)
~(a)
is absolutely continuous on [0 1 ,0 2 ] and
0
<
00
,
<
1 -
a < a, ,
-
2
3
~'(a)
1<
(ii)
finite
k roo
= -a
J_.oo
-
x f
k-l
(x) ¢(x;a) df(x)
abso~ute~y
Iff(x) has a first derivative which is
interva~
we~~
and f(x) satisfies (C2) as
continuous on each
as
(C4)
lim
(CS)
Ixl-+oo
then
~(a)
Proof.
= 0,
¢(x;a)
is absolutely continuous on [a l ,a 2 ] and
We first note from ¢(x;a)
aax
-e
dfk(x)
dx
¢(x,a)
2 _k
= (2'ITa)
x
=- 2
2
2
2
exp( -x /2a ) that
(x;a),
(2)
a
and
(3)
Since a¢(x;a)/aa is continuous, it follows that ¢(x;a) is absolutely continuous
in a on any closed and finite subinterval of (0,00) and thus
¢(x;a)
= ¢lx;l)
+
f~ ap~~;u) du, a
>
°.
It follows that
ap(x;u) dU]dx .
au
Hence if for each a
l
2 a 2 a2
we have
(4)
~
then by Fubini's theorem
4
~ (a)
= ~K (1)
K
and thus
~(a)
+
fa [ roo fk(x) op(x;u) dx]du.
J _00
1
dU
is absolutely continuous on [01.a J and
2
It is clear from (3) and condition (CI) that (4) is indeed satisfied.
2
x -a
3
We' thus
2
</>(x;a)dx
a
l,ro
=~
J_ oo
2k
l,ro k
x f (x) </>(x;a)dx - a J_ oo f (x)</>(x;a)dx.(5)
a
(i)
For all _00 < a < b < 00 we have by (2) and integration by parts
fb
l-
a
03
= _
la
x 2 fk(x) </>(x·a)dx
= _
~
fba
•
[x fk(x) </>(x;a)]b
a
+
a
l
a
fb
a
x fk(x) ap(x;a) dx
ax
x fk-l(x) </>(x;a) df(x)
(6)
Under the conditions (C2) and (C3), letting a { _00. b t 00 and using the dominated
convergence theorem in (6), we obtain
r
k
k-l
1
roo
xf
(x)
J_
x 2 f k (x) </>(x;a)dx = oo
3"
a
-00
a
and the result follows by (5).
</>(x;a) df(x)
Note that if x fk-l(x) df(x) ~ 0 on (_00.00) (or 20)
then conditions (Cl) and (C2) imply (C3):
Let a { _00. b t 00 in (6) and use the
monotone convergence theorem.
(ii)
d
k
Since x dx f (x)
=k
x f
k-l
d
(x) dx f(x), it follows that condition (C4)
implies condition (C3) and thus under (ii) the conditions in (i) are satisfied so
that
~(a)
is absolutely continuous on [0 1 ,0 2 ] and a.e. on [0 1 ,0 2 ]
5
x fk-l(x) f' (x) cj>(x;o)dx
m (0) =~J'X)
o _00
k
=1-o Joo-00
= -0
x
r:-00
dfk(x)
dx
cj>(x;o)dx
dfk(x)
dx
ap(x;o) d
where the last equality follows by (2).
ax
(7)
x
As in part (i), integrating by parts
we have for all _00 < a < b < 00 ,
Jb
k
df (x)
a
dx
aep(x;o) dx
ax
2 k
k
=
b d f (x) ,j..
( . ) dx.
'I' X,O
2
a
dx
dfd(X) cj>(x;o)]b
x
a
Under the conditions (C4) and (CS), letting a
~
J
_00, b t 00
(8)
and using the
dominated convergence theorem we obtain
k
b df (x) aep(x;o) dx
Ja
-e
dx
ax
2
2 k
d f (x) ,j..(x'o)dx
i_ oo
2
'1"
,
dx
roo
Note again that if d f k(x)/dx ~ 0 a.e. on
2
and the result follows by (7).
(_00,00) (or
=_
2 k
2
0) then d f (x)/dx
L (cj>(x;o)dx) follows from the remaining
l
E
conditions and could thus be deleted from (C4):
use
2
Let a
~
_00, b t 00 in (8) and
o
the monotone convergence theorem.
REMARKS:
It is possible to delete some of the conditions (CI) - (CS) in
Proposition I under appropriate circumstances.
The following cases are of
particular interest.
1)
As was pointed out in the proof of Proposition 1, if
x f k-l (x) df(x) > 0 on (_00,00) (or < 0)
then (C3) follows from (CI) and (C2).
Similarly, if
2
d f k(X)
----::2"-'-
> 0 a.e. on (_00,00) (or
2
0)
dx
2
then the integrability condition d f k(x)/dx
(C4) .
2
E
LI(cj>(x;o)dx) may be deleted from
6
2)
unifo~m
The limiting conditions (C2) and (CS) may be deleted under appropriate
continuity assumption as follows.
We first note that if g(x)
and g(x) is uniformly continuous on (_00,00), then
lim
g(x) = O.
E
L (dx)
I
Thus, if
Ixl~
f(x) is uniformly continuous on (_00,00)
then (C2) follows from the integrability condition (CI).
Similarly, if
fk-l(x)fl(x) is uniformly continuous on (-00,00)
then (CS) follows from the first integrability condition in (C4).
3)
The limiting condition (C2) (respectively (CS)) may be deleted when
f(x) (respectively fl(x) has a polynomial growth OClx\n) or exponential growth
O(exp(alx,B)), a > 0, 0 < B < 2, as Ixl ~ 00
4)
All the conditions (CI) - (CS) are satisfied in case the nonlinearity
f(x) is a polynomial.
The case where f is an interval window function, f(x) = l(a,b)(x), is of
particular practical importance.
PROPOSITION 2.
(iJ
Let f(x)
We have,
= l(a,b)Cx),
_00 < a < b < 00,
m (a) is strictly monotonic on (0,00) if and only if the interval
l
(a, b) has anyone of the following forms
(iiJ
o
If _00
a)
(-00, b),
b
f- 0,
b < 00,
b)
(a, (0) ,
a f- 0,
-00 < a,
c)
(a, b),
< a < b <
00
_00 < a < 0 < b < 00 •
and r > 0, then miCa) is strictly monotonic for
< a < r if and only if a, 0 = b-a satisfy
where
e-
7
and a 3 (o,l) is the unique solution a (0)
2
(see Figure 1).
Proof·
o
~n(l + - )
a
=
0(0
= a 3 (o,l)
2a),
+
of the equation
a > 0, 0 > 0,
We also have that a 3 (o,r) is strictly decreasing in 0 with
When f = l(a,b) then
m (a) = 4>(b) - <p(a)
l
a
a
where <p is the distribution function of the standard normal density
-4It
¢(x)
= ¢(x;l).
(i) When _00 = a < b < 00, b
~
0, we have ml(a) = <p(b/a) and
m'(a) = _ b
I
a2
¢(b)
a
is strictly positive or negative on (0,00) as b < 0 or b > 0; and similarly
when _00 < a < b = +00, a 1
m' (a)
I
=_
b
2a
o.
When -00 < a < b <
,+,(b) + ~ <p(a)
't' a
2
a
a
If a < 0 < b then miCa) < 0 on (0,00).
we have
222
<pcb) (a e(b -a )/2a - b).
=.!..a
00
a
2
If 0 < a < b then
2
ml'(a) > 0 for all a such that 0 < a
<
b 2 _a 2
2
~n
and
miCa) < 0 for all
a such that
b 2 _a 2
b
2
~n
a
<
2
a
b
a
8
and similarly if a < b < O. (i) thus follows.
(ii)
It is clear from (i) that if a ~ 0 ~ b, i.e., a ~ 0, 0 ~ -a, then ml(O)
is strictly decreasing on (0,00).
If 0 < a < b, i.e., a > 0, 0 > 0, it is again clear
from (i) that ml(a) is strictly monotonic (increasing in fact) for 0 < a < r if
and only if
or equivalently
2r
2 '
0
~n(l + -)
a
< 0(0
+
2a) .
It is then clear that when a > 0, 0 > 0, ml(a) is strictly monotonic for
o
< 0 < r if and only if a
~
a (o,r).
3
,
Similarly. it is seen that if a < b < O.
i.e., a < O. 0 < -a. then ml(a) is strictly monotonic for 0 < a < r if and only
if a
2 a 4 (0.r) where a 4 (0.r) = -a 3 (0.r) -0. Thus. for a
strictly monotonic for 0 < a
2
r if and only if a
2
< O. 0 > U. ml(a) is
a 4 (0.r) or 0 > -a.
Finally.
the assertions contained in the last sentence of (ii) are established in a
straightforward way.
It may be worth noting that the solution of the equation
2
~n(l
+
i)
a
= 0(0 + 2a).
a > 0 •
which determines a (0.1). can be obtained parametrically as a function of x.
3
I < x < 00,
by
a(x)
3.
I
= 2(x-l)
o(x).
2
0
x+l
x-I
iVn - -
o
THE RECONSTRUCTION OF THE VARIANCE R(O)
Throughout this section X is a real stationary Gaussian process with zero
mean and continuous covariance function R(t).
We consider the problem of
4It'
9
determining the variance R(O) when we know the nonlinearity f and either the
first moment
Y
or the second moment
~l
= {f(Xt ), _00
< t < oo}.
~2
of the output process
We have
Since f is known, so is of course the function
knowing
~l
or
~2
~(o),O
> 0, and our problem is
to find conditions on f under which R(O) can be recovered.
Propositions 1 and 2 of Section 2 provide a number of such conditions on f.
It is convenient at this point to exclude those f's for which the output
moments
~k
provide no information whatever on the input variance.
Specifically,
we exclude the cases where
19)
-A
...
~k =
(in which case
case f(x)
= ax
+
1
2(a
+
b) which is independent of R(O)) and for k
b (in which case
~l
=
1 the
= b).
We also introduce the following terminology which will facilitate the
statement of the main Theorem.
A real function f on the real line is called
g-unimodal (g for generalized) if for some a
E
(_00,00), f is nondecreasing on
(-oo,a) and nonincreasing on (a,oo), or vice versa.
a mode of f.
The number a is then called
Note that the mode need not be an extremum of f; and that
unimodal functions in the usual sense (i.e., continuous functions which are
strictly increasing on (-oo,a] and strictly decreasing on [a,oo), or vice versa)
are g-unimodal.
Note also that f is g-unimodal with mode a if and only if f
has bounded variation on each finite interval and
lx - a) df(x) > 0
on (-00,00)
(or. 2 0
on (-00,00)).
10
A function f is called c-unimodal if it has a first derivative f' which is
absolutely continuous on each finite interval and f"(x)
or f"(x) < 0 a.e. on (-00,00).
or concave.
~
0 a.e. on (_00,00),
Here "c" stands for both continuous and convex
It is important to note that unlike c-unimodal functions, g-unimodal
functions may have discontinuities ljumps).
We can now state our first result
on the reconstruction of R(O).
THEOREM 1.
~k'
k
= 1,2,of
The variance R(O) can be reconstructed from the k-th moment
the output process Y
= {f(Xt)'
_00 < t < ooJ when the nonlinearity
f satisfies
E Ifk (X )
lC6)
t
EIX~ fk(X t ) I
< 00 ,
< 00
k
lim
(C7)
I
x fk(x) ¢(x; R 2(0)) = 0
Ixl~
and anyone of the following conditions
(iJ
fk(x) is g-unimodal with mode zero and is Lebesgue integrable in some
open neighborhood of zero.
(iiJ
fklX) is c-unimodal and satisfies
(C8)
(C9)
Elx t (fk) 'eXt)
lim
Ix\~
Proof.
d fk(x)
dx
I
< 00
1
<f>{x; R~(O)) = 0 .
We first note that if 0
rea~ number N
= N(01,02)
<
01 < 02
<
00
then there exists a positive
such that ¢lx;ol) < ¢(x;02) whenever Ixl > N.
Thus if
g(x) is a measurable function, then
f
Ix I >N
\g(x) I ¢(x;ol)dx ~
f
Ix I >N
\g(x) I ¢(x;02)dx .
It then follows that if g(x) is Lebesgue integrable on bounded intervals, then
(10)
~-
11
(i)
Since fk is g-unimodal with mode zero and is Lebesgue integrable in
k
some open neighborhood of zero, both f (x) and x
on bounded intervals.
that (C2) is satisfied for all 0 <
k
with mode zero we have x df (x)
° < Rk2(0).
° ~ R (0).
~2
=k
x f
k-l
Also, condition (C7) implies
Finally, since f
k
is g-unimodal
(x) df(x) > 0 on (_00,00), or < 0 on
Thus conditions (Cl) - (C3) are satisfied for all
~
Let 00 be such that R 2(0)
(Cl) - (C3) are satisfied for alIa <
~'(o)
K
-e
t
° ~ R-~lO).
Remark (1) following Proposition 1, condition (C3) is implied
by lCl) and (C2).
a <
k
f (x) are Lebesgue integrable
It then follows from (10) that condition (C6) implies that
condition (Cl) is satisfied for all 0 <
(_00,00), and by
2
~ 00 <
° ~ 00'
00 and such that conditions
'lben by Proposition lei) we have
roo
= -k0 _
1_.00
x f
= !° 1_
roo00
x ¢lx;o) d~(x)
k-l
lX) ¢lx;oJ dflx)
a.e. on (0'00] .
Now since fk is g-unimodal with mode zero, then x dfk (x)
~
a on (_00,00) (or
~
a
on (-00,00)) and since fk (x) is not of the form (9), it follows that
m{(o)
Thus
llk
~(o)
k
= ~ [R
(ii)
2
a (or < 0)
>
a.e. on (0'00] .
is strictly monotonic on (0'00] and R(O) can be reconstructed from
(0) ] .
is shown similarly so that under the assumptions (C6) - lC9) and the
c-unimodality of fk we have that Proposition l(ii) is applicable and
mk(o)
Now since f
k
d 2 fk( )
2 x ¢(x;o)dx
= ° f~oo
is c-unimodal then
dx
d 2f k(x)
dx
Leb
{x:
2
> 0 (or
a.e. on (0'00]
~
0) on (-00,00).
If
[fk(x)]" > O} > a
(11)
or
Leb
{x: [fk (x)] " <
a}
>
a
12
~(0)
then
> 0 a.e. on (0,0 ] so that ~(0) is strictly monotonic on (0'00] and
0
~k =
thus R(O) can be recovered from
!<
mk [R 2 (0)].
Finally, if (11) is not
satisfied then necessarily [l(x)]" = 0 a.e. on (_00,00) in which case
f(x) = ax
+
2
b for k = 1 and f (x) = b for k = 2.
These cases have been excluded. 0
It should be noted that conditions (C6) - (C9) have the somewhat unsatisfactory but quite natural feature that they are expressed in terms of the unknown
variance R(O).
This can be eliminated by assuming that the conditions hold for
all 0
This is certainly satisfied in practical situations where fk(x)
< 0 < 00.
has a polynomial growth o(lxl m) or exponential growth O(exp(alxI S)), a > 0,
o
< S < 2 as Ixl < 00.
In case the nonlinearity f is an interval window function, flx) = l(a,b)(x),
we have the following theorem on the reconstruction of R(O), which follows
directly from Proposition 2.
THEOREM 2.
(i)
Let f(x)
= lCa,b)lx),
_00
2
a < b
2
e-
00 •
~l
The variance R(O) can be reconstructed from the first moment
of
the output process Y for the following interval windows
(ii)
he Y'(j b
f(x) = 1 (_oo,b/x)
b)
f(x) = 1 ( a,oo
c)
f(x) = 1 (a, b) (x) or f(x) = 1 - l(a,b)(x) where
lx)
If an upper bound r
2
1,)
where a ., O.
on the variance R(O) is
R(O) can be reconstructed from the first moment
ever the interval window
pair
., O.
a)
~l
known~
(See Figure 1).
2
2
r , then
of the output process when-
f(x) = l(a,b)(x), _00 < a < b < 00, is such that the
(a, 8 = b-a) satisfies one of the following conditions
2
a)
a
b)
a < 0
c)
R(O)
-oo<a~O~b<oo.
a 4 (8,r)
and
8 > -a
13
When the first order distribution of the output process Y is known, rather
than the first two moments, we have the following result where B(f) denotes the
a-field of Borel sets generated by f.
COROLLARY.
(i)
The variance R(O) can be reconstructed trom the first
order distribution of the output process Y when f is such that B(f) contains an
interval (a,b) of the form
(ii)
a)
(-oo;b)
b .,
o.
b)
(a,oo)
a F
o.
c)
(a,b),
If an upper bound r
2
wherc-oo < a < 0 < b <
00
and a" b.
on the variance R(O) is known, R(O)
2
2
r ,
then
R(O) can be reconstructed from the first order distribution of the output process
Y when f is such that B(f) contains an interval la,b) such that the pair
-e
(a, 0 = b-a) satisfies one of the following conditions,
Froof.
Let (a,b)
E
2 a 4 (<S,r)
a)
a
b)
a < 0
c)
a
~
a
and
3
0 > -a
(<S,r) .
The result follows by Theorem 2 and the following observation.
B(f).
Then there is a Borel set B such that (a,b) = f- 1 (B) and thus
o
We conclude this section by noting that aside from the interval windows
of Theorem 2, the classes of non1inearities f for which RlO) can be recovered,
from the first two moments of the output process, are:
e
is g-unimodal with mode zero.
a)
f
b)
f2 is g-unimodal with mode zero.
c)
f
d)
f2 is c-unimodal.
is c-unimodal.
14
We note that continuous convex functions which are symmetric around some point
are included in (c), whereas discontinuous symmetric "unimodal" functions are
included in (a).
Classes (b) and (d) contain certain odd nonlinearities, possi-
bly discontinuous.
4.
For instance, all odd quantizers belong to class (b).
THE RECONSTRUCTION OF THE COVARIANCE R(t).
Throughout this section X is a real stationary Gaussian process with zero
mean and continuous covaraince function R(t).
Combining the results of Section
3 which provide the reconstruction of the variance R(O), with the results of CM(1978)
which provide the reconstruction of the normalized convariance function
R(t)/R(O), we obtain the reconstruction of the covariance function R(t) of the
input process from the knowledge of the nonlinearity f and the first moment and
correlation function of the output process.
following classes of nonlinearities.
This is accomplished for the
Here, for simplicity, we omit the integra-
bilityand the asymptotic conditions on the nonlinearity f lsuch as (C6) and (C7));
e-
these can be easily traced through the indicated theorems of this paper and the
earlier one CM(1978).
1.
Monotonic nonlinearities.
The relevant theorems are Theorem 2 of
CM(1978) and Theorem I of this paper. When f(x) is monotonic (possibly discon2
tinuous) such that f (x) is either g-unimodal with mode zero or c-unimodal, then
every
covariance function R(t),
_00
< t <
00
,
can be reconstructed from the
correlation function C(t) of the output process Y
=
{f(X ), _00 < t < oo}
t
Included here are smooth limiters, half-wave vth-law devices with v > 0, full-
wave odd vth-law devices with v > 0 (for definitions see Thomas (1969)), odd
quantizers and companders (see Taub and Schilling (1971)).
2.
Even nonlinearities.
The relevant theorems here are Theorem 5 of
CM(1978) and Theorem I of this paper.
zero and satisfies
When f(x) is even, g-unimodal with mode
4It
15
(12)
then covariance functions Rlt), satisfying the smoothness condition at their
zero crossings stated in Theorem 5 of CM(1978), can be reconstructed from the
correlation function C(t) of the output process.
Included here are the full-wave
even vth-law devices with v > 0 and the symmetric windows f(x) = l(
f(x) = 1 - l(
-a,a
lex) for 0 < a <
00
-a,a
lex) and
Condition (12) says that in the series
expansion of f in terms of the even Hermite polynomials f(x) =
I:=o
a 2n H2n (x;R(O))
This condition can be weakened to "the term H2k for
the term H should be present.
2
some k>l should be present" as discussed in the remark preceeding Theorem 6 of CM(1978).
3.
General nonlinearities.
The relevant theorems here are Theorem 6 of
CM(1978) and Theorem 1 of this paper.
When f(x) satisfies the condition stated
in Theorem 6 of CM(1978) (which is the analogue of (12) for non-even functions)
and for k=l or 2, fk(x) is either g-unimodal with mode zero or c-unimodal, then
covariance functions R(t) satisfying the smoothness conditions at certain negative level crossings stated in Theorem 6 of CM(1978), can be reconstructed from
the correlation function
C(t) of the output process when k=2, and from the
output correlation function and its first moment when k=l.
4.
Interval windows.
The relevant theorems here are Theorem 1 of CM(1978)
and Theorem 2 of this paper.
Arbitrary covariance functions R(t) can be recon-
structed from the output correlation function C(t) for the following infinite
interval window nonlinearities:
_00
< a <
00
l(_oo,b)' b ; 0,
_00
< b <
00,
and l(a,oo)' a ; 0,
For finite interval window nonlinearities lla,b) we have the
following results.
(i)
If an upper bound r
2
2
on the input power is known, R(O) = EX~ ~ r ,
then arbitrary covariance functions R(t) can be reconstructed from the output
~
correlation function provided (a, o=b-a) belongs to area (Ar ) of Figure 1.
16
(Similar results hold, and the corresponding areas in the (a,o) half plane are
(ii)
2
2
< R(O) or that 0 < ri < R(O) < r < co).
2
3
1
If (a,o) belong to area (B) of Figure 1, - - 0 < a .2 - 4" 0, and if R
4 -
easily determined, when we know that 0 < r
satisfies the conditions in Corollary 6 of CM(1978), then R can be reconstructed
from the output correlation C.
For i ; 1,2,3,4 and r > 0 the function a. (o,r) is given by
1
a.(o,r) ; r a. (~, 1)
1
1
r
where for i ; 1.2. a. (0,1) is identical to the function a. (0) defined in Proposi1
1
tion 3 of CM(1978); for i ; 3,4, a. (0,1) is given in Proposition 2(ii) of this
1
paper.
In Figure 1 the functions a.(o,r), i ; 1.2,3,4 and the area A are
r
1
plotted for r ; 1.
We finally note that for interval windows f(x) ; 1 (a,b) (x) for which the
pair (a, o;b-a) falls in the shaded area of Figure 1, we do not have as yet
reconstruction procedures for both R(O) and R(t)/R(O).
S.
(i)
Using the second order distribution of the output we can construct
any covariance R, if the a-field B(f) generated by f contains an
interval of the form (_co,b), b
(ii)
F 0.
covariances R with R(O)
2
or (a,co). a F 0;
2
r • if B(f) contains an interval (a.b) such
that (a, o;b-a) is in area (A ) of Figure 1;
r
(iii)
covariances Rsatisfying the conditions in Corollary 6 of CM(1978). if
B(f) contains an interval with (a.o) in area (B) of Figure 1.
6.
Finally if Ret)
~
0, or if R has a rational spectral density, or if it
is bandlimited, and if fk(x) is g-unimodal with mode zero or c-unimodal, for
k ; 1 or 2, then R can be reconstructed from the output correlation C and. for
k ; 1. its first moment.
e-
17
REFERENCES
Cambanis, S. and Masry, E. (1978), On the reconstruction of the covariance of
stationary Gaussian processes observed through zero memory non1inearities,
IEEE Trans. Information Theory IT-24, No.4, in press.
Taub, H. and Schilling, D.L. (1971), "Principles of Communication Systems",
McGraw Hill, New York.
Thomas, J.B. (1969), "An Introduction to Statistical Communication Theory",
Wiley, New York.
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UNCLASSIFIED
SECURI"ry CLASSIFI('/4.110N OF" THI~ PAr .. F (Whfltt f>a'"l':"(f'I,,,tl)
1<1o:AD INSTI<UCTIONS
UEFOI<E COMPLETI:'<<J FORM
REPORT DOCUMENTATION PAGE
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-
11FPORT NUMBER
_ _I
ACC [SSION NO.
TITLE (,"'" .sublll/~)
4
On the Reconstruction of the Covariance of Station
ary Gaussian Processes Observed Through Zero
Memory Nonlineari ties - Part II
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(.OVT
TECHNICAL
6.
I-'ERFORMING O'G. REPORT NuMBER
Mimco Series No. 1176
8.
Au THORf.)
Stamatis Cambanis and Elias Masry
AFOSR-7S-2796
>-----_._._._- ._-_._----- .. - .. ----------------9
PERFORMING c,r"GAtIiZATION NAMI-: AND ADIlRESS
Department of Statistics
University of North Carolina
Chapel Hill, North Carolina 27514
f----I 1.
C0NTROLLING OFFICE NAME AND ADDRESS
Air Force Office of Scientific Research
Bolling AFB, Washington, D.C.
~4
MONIToRING A(,ENCY NAME Ii
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REPORT DATE
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20
ABSTRACT (Contlnll" nn teVer •• • Id" IIl1ec«t .... aty .nd
---
IIthl ' 'rl1 I II,,· by block numhfl'r)
Reconstruction of stationary Gaussian covariance, zero memory nonlineari ties
--
Ilf'llCl",'PUlfy
-
Id~ntlfy
by blorlc number)
We consider the problem of reconstructing the variance R(O) of a zero mean
stationary Gaussian process observed through a zero memory nonlinearity f(x),
from the knOWledge of f and the first two moments of the output process. The
reconstruction is shown to be feasible for certain interval windows, convex
nonlinearities and discontinuous unimodal nonlinearities. The paper is the
completion of the investigation begun in Cambanis and Masry (1978) where the
reconstruction of the normalized covariance R(t)/R(O) was considered.
DO
FORM
1 JAN 73
1473
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