•
SERIES REPRESENTATIONS FOR
GENERALIZED STOCHASTIC PROCESSES
by
Muhammad K. Habib
•
Department of Biostatistics
University of North Carolina at Chapel Hill
Institute of Statistics Mimeo Series No. 1801
March 1986
SERIES REPRESENTATIONS FOR
GENERALIZED STOCHASTIC PROCESSES
•
by
Muhammad K. Habib
Department of Biostatistics
The University of North Carolina at Chapel Hill
Chapel Hill, North Carolina 27514 (USA)
Abbreviated Title: SERIES REPRESENTATIONS FOR PROCESSES
•
AMS 1980 Subject Classification:
60G55, 60K99, 62F99
KEY WORDS AND PHRASES: Communication theory, generalized functions,
generalized stochastic processes, information theory, sampling.
Research supported by the Office of Naval Research under contract
number NOOO-14-K-0387
Abstract
Series representations are derived for bandlimited generalized
functions and generalized stochastic processes.
This work extends
existing results concerning sampling representations of bandlimited functions
and stochastic processes.
The merit of such representations lies in the
fact that a function (or process) may be exactly reconstructed using
only a countable number of its values (or samples).
These types of
representations have found many applications in several areas of communication
and information theory such as digital audio and visual recording, and
satellite communications.
In addition, random distributions have also been
employed in a host of applied areas such as statistical mechanics,
.~
chemical reaction
kinetics and neurophysiology.
1.
Introduction.
This paper is concerned with sampling representations for
generalized functions (or distributions) and generalized stochastic processes
(or random distributions).
The terms distribution and generalized function
will be used interchangeably throughout the text.
The need to consider
distributions (beyond classical functions) arises from the fact that in many
physical situations it may be impossible to observe the instantaneous values
f(t) (of a physical phenomenon) at the various values of t.
For instance,
if t represents time or a point in space, any measuring instrument would
merely record the effect that f produces on it over non-vanishing intervals
of time I:
!
I
f(t)~(t)dt,
where
~
is a IIsmoothll function representing the
measuring instrument, i.e. the physical phenomenon is specified as a
functional rather than a function.
Furthermore, it is becoming exceedingly
clear that the tools and techniques of the theory of distributions are useful
.~
in investigating certain problems in many applied areas such as statistical
mechanics (Holley and Strook, 1978), Chemical reaction kinetics (Kole1enez,
1982), and neurophysiology (Kallianpur and Wolpert, 1984a,b and Christenson, 1985).
It is thus of interest to consider distributions beyond functions.
The sampling representations (expansion)
(1.1)
f(t)
=
n sin 1T(2W - n)
f(2W)
n=-oo
1T(2Wt - t)
1
tdR ,
L
was originated by E.T. Whittaker (1915).
J.M. Whittaker (1929, 1935),
Kote1nikov (1933), Shannon (1949), and others studied extensively the
sampling theorem and its extensions in developing communication and
information theory.
~
(1977).
For a review of the sampling theorem, see Jerri
A function f which can be represented, for some W
o>
0, by
2
t tt:: IR1
(1. 2)
is called Ll-bandlimited to Wo if Ft::L 1[-WO,woJt and is called conventionally
or L2-bandlimited to Wo if Ft::L 2[-W Ot WoJ. In both cases the sampling representation
(1.1) is valid for all W~ W00 The series in (1.1) converges uniformly on compact
sets for L1-bandlimited functions t and for conventionally band1imited functions
it converges in L2 (IR 1 ) as well as uniformly on IR 1.
However, a function need not be bandlimited in the above sense to exhibit a sampling
expansion of the form (1.1).
Zakai (1965) extended the concept of "bandlimitedness"
to a broader class in which functions need not be in the form (1.2).
For a
non-negative integer k, let L2 (~~) be the class of all complex valued functions
defined on IR l that are square integrable with respect to the measure
d~k(t) =
dt 2 k'
(1 + t )
~~
If fEL2(~k)' then f defines a tempered generalized function
(or tempered distribution) (denoted also by f) on the class S of rapidly decreasing
functi ons by
f(9)
=
co
f
f(t)e(t)dt t 6ES
-co
(See Section 2 for relevant definitions.)
The distributional Fourier transform
A
of f is the tempered distribution f defined by f(e) = f( e L et::s.
A
of f is the support of f.
For k
= 0,1,2, ...
and W
o
The spectrum
0, Bk(WO) is the class of
all continuous functions ft::L2(~k) whose (distributional) spectrum is contained
>
in [-WotWOJt and is called the class of Wo-bandlimited functions in L2(~k)' It
is clear that BO(W O) is the class of WO-bandlimited functions in L2(IR 1)t and
Bk(WO) CBk+l(W O)' A1so t BO(W O) is dense in Bk(WO) for every positive integer k
(see Lee, 1976 ).
.'
3
Zakai obtained a sampling representation for functions in Bl (W O). Cambanis
and Masry (1976) characterized Zarkai·s class Bl(WO) and as a consequence sharpened
Zakai's sampling expansion.
It was shown that if feB 1 (W O) and W> W then f has
o
a sampling representation of the form (1.1). Lee (1977) extended Zakai·s result
to functions in Bk(WO). He showed that if feBk(W O)' W> W
O' 0 < S
is an arbitrary but fixed CCO -function with support in [-1,1] and
l
co
<
W-W O and
~
w(t)dt = 1, then
-co
(1 .3)
f( t)
=
f(2Wn) sin n(2Wt - n)
n(2Wt - n)
n=-co
co
[
and the series converges uniformly on compact sets.
the presence of the (damping) factor ~ in
-~
It should be noted that
(1.3) cannot be eliminated, as (1.1)
is not valid for feB k , k > 2. As a counter example consider f(t)=t (fEBZ(W O));
then f(2~) = 2~ and the series in (1.1) does not converge.
Campbell (1968) derived sampling expansions for the Fourier transforms
(as functions)of tempered generalized functions with compact supports.
If a
tempered generalized function F has a compact support and eu(t) = e2nitu, then
co
F(e u) is well defined, since eueC for all ueIR 1 . In this case the Fourier
transforms F of F may be thought of as a function defined on IR l by
F(u) = F(e )' udR l (see Section 2). Campbell showed that if F is a tempered
u
generalized function with compact support and with Fourier transform f as a function
on IR 1 , i.e. f(t) = F(e t ), teIR 1 , Wis a test function such that ~(u) = 1
on some open set containing supp(F),
if W> 0 is such that the translates
{supp(~)
+ 2nW}, niO, are disjoint from supp(F), then
4
(1 .4)
where K(t)
= ~ fe2~itu~(u)du,
1
and the series converges for every teIR .
Sampling expansions for functions which are Fourier transforms of
generalized functions with compact support have also been considered by
Hoskias and De Sousa Pinto (1984a,b).
Sampling representations of the types discussed above have also been established
for stochastic processes. Let X={X(t), teIR1} be a measurable stochastic processes
with covariance function R(k,s) = E[t)X(s)], t seIR 1 , which satisfies
(1 .5)
f= R(t,t)d~k(t) <
-=
=
k
>
O.
e-
The process X was defined by Lee (1976) to be bandlimited if almost every sample
path of X was bandlimited, or equivalently, if the function R(t,.) was
bandlimited.
Let BPk(W O) be the class of mean square continuous second order
stochastic processes whose covariance functions satisfy (1.5). Zakai (1965)
established a series representation similar to (1.1) for stochastic processes
in BP1(W O) (see also Cambanis and Masry, 1976). Indeed, it was shown that if
X = {X(t),teIR 1} belongs to BP1(W O)' then for any W> Wo and teIR 1
=
(l .6)
X(t) =
1:
n=-=
X(~W)
sin 1T(2W-n)
k(2W-n)
5
where the series converges in the mean square uniformly on compact sets
Lee (1976) established the following representations which similar to
(1.3), for processes X X={X(t), tEIR 1} in BPk(W O)' k>l
<Xl
X(t)=
(1. 7)
X(n ) sin n(2Wt-n)
n{2W-n)
n=-oo 2W
~
A
n
1jJ(e(t- 2W)), tdR
1
for any W> Wo and a < e < W- Wo and where 1jJ is defined as in (1.3). The
series in (1.7) convergences in the mean square uniformly on compact sets.
See also Lee (1977) and Piranashvili (1967) for similar results.
Campbell
(1968) established a sampling representation similar to (1.7) for weakly
stationary stochastic processes whose covariance functions are Fourier
transforms of generalized functions with compact support.
In this case
the series converges in mean-square uniformly on compact set.
In this paper, series representations are derived for generalized
functions and generalized stochastic processes which extend the sampling
representations, of ordinary functions and stochastic processes, discussed
above.
In Section 2, notations and basic definitions needed in the sequel
are given.
In Section 3, the sampling representation (1.1) valid for
functions in BO(W O) and Bl(W O) and the representation (1.3) which is valid
for functions in Bk(WO)' k > 2 are extended to bandlimited generalized
functions (Theorem 3.1).
Examples which show how sampling representations
of 1I0rdinaryll functions are recovered from Theorem 3.1 are given.
.
In
Section 4, series representations for bandlimited generalized stochastic
processes are derived.
These results extend sampling representation
(1.7) for 1I0rdinaryll bandlimited stochastic process in Bk(W ), k:: O.
O
6
Theorem 4.1 derives sampling presentations for stochastic processes with
sample paths which have symmetric spectrums as well as spectrums which
are just compact sets in IR 1. Examples are also given to show how the
classic results may be recovered from the ones presented in this section.
7
2.
Notation and basic definitions.
Let c~ = c~ (IR 1) be the class of all
infinitely differentiable functions with compact support.
A topology
T
is
introduced on the linear space c~c which makes it into
. a complete space; that is
a sequence {~n} in c~ converges to zero in T if there exists a compact AEIR l which
contains the support of every ~n' and for every non-negative integer k, ~n (k)(t)
uniformly as n +~.
c~
called test functions.
with the topology
T
+
is denoted by D, and its elements are
The members of the dual D' of D are called distributions,
and the value of a distribution feD' at a test function
(weak-star) topology on D' is defined by the seminorms
~eD
is denoted by
If(~)I,
over all elements of D; thus for a sequence {fn} in D': f n
fn(~) + 0 for all ~eD.
+
feD', as
f(~).
~
A
varies
0 weakly whenever
The class S of rapidly decreasing functions consists of all infinitely
differentiable functons (~eC~) for which
Itm~(k)(t)1 <
C
, -~ < t < ~
- m,k
for all non-negative integers m,k.
I I~I 1m k =
,
A topology on
sup
sup 1 {(l+ltl)kl~(n)(t)l}
O<n<m teIR
S
is defined by the seminorms
, m,k = 0,1,2, ... ,
i.e., a sequence {~n}~n=l is of functions in S is said to converge in S,
if for every set of non-negative integers, the sequence {(l+ltl )m~n (k)(t)}~=l
converges uniformly on IR 1 • S is complete, and the dual S' of S is called the
class of tempered distributions.
on S' by the seminorms
f n converges in S' if
If(~)I,
fn(~)
Similarly, a (weak-star) topology is defined
feS', as ~ varies over all elements of
converges for all
~eS.
sa, i.e.,
The space D'(S') is (weak-star)
sequentially complete, that is, if {f}
is a sequence in D'(S') such that
n n
{fn(~)}n is a Cauchy sequence for every ~eD(S), then there exists a distributiJr
0
8
feD' (S') such that f n
f in D' (S).
-+
Finally, the space
e~
with the topology defined by the seminorms
E
supl,(n)(t)1
"ee~,
Q<n<m teA
where A ranges over all compact sets in IR l and m over all non-negative integers,
Pm A(')
,
=
is denoted by E.
The Fourier transform F(F(,)
mapping from S onto itself.
by f(,)
..
= f(,),
= ;"eS) is a one-to-one biocontinuous
..
If feS', the Fourier transform f of f is defined
,eS, and is a tempered distribution.
If feS' and ,eS, the
convolution f*, is defined as a function on IRlby
where ~(t)
f*,ee~has
= ,(-t)
and the shift operator Tt is defined by (Tt,)(U) = ,(u-t).
a polynomial growth and thus determines a tempered distribution.
Suppose feD', f is said to vanish in an open set UCIRlif f(,)
every ,eD with supp(,) cu.
= a for
Let V be the union of all open sets UdR l in which
f vanishes. The complement of V is the support of f.
compact supports are tempered distributions.
Distributions with
Now, if f is a distribution with
compact support (i.e., feS'), then f extends uniquely to a continuous linear
= 1 on some open set containing supp(f),
then ~f = f, i.e. (~f)(,) = f(~,) = f(,) for all ,eS, but since et(u) = e2nitu
..
is a em-function, f(e t ) = f(~et) exists, and the distribution f is generated by
functional on E. If
~eV
is such that
the function f(t) defined on IR l by
(2. 1)
Indeed,
~(u)
9
A
(2.2)
and
f
(~f)
= (~
f)
~
A
.
A
(and therefore f) is generated by the C -function
which has a polynomial growth (see Rudin, 1973, p.l79).
A
that
~
= ~,
A
(f*~)(t)
By choosing
~£s
such
we have
A
= f(et~) = f(~et) = f(e t )
and from (2.2), (2.1) is justified.
Hence the Fourier transform of a
distribution with compact support may be thought of as a function defined by
IR l by (2.1).
Let (Q,F,P) be a probability space.
~4It
A random distribution (or a
generalized stochastic process) is a continuous linear operator from D (or s)
into a topological vector space of random variables.
Specifically, a second
order random distribution is a continuous linear operator from D (or s) onto
L2(Q) = L2 (Q,F,P), the Hilbert space of all finite second moment random
variables. For example, let {X(t),t£IR1} be a measurable second order
zero-mean stochastic process with covariance
function R(t,s)
= E[X(t)X(s)].
Assume that R is locally integrable (i.e., R is integrable over every compact
subset of IR 2). The process defined by
X(~)
= Jm
X(t) ~(t) dt , ~£D
is a generalized stochastic process, i.e. X defines a continuous linear
mapping from D to L2 (Q).
Let R be the covariance functional of
10
the generalized process Xdefined on D x D
given
by R(~~w)
=
f~ f~
-~
-~
R(t,s)
~(t) ~(s)dt
by R(~,~)
ds.
= E[X(~)X(w)]. R is
11
3.
Sampling representations forband1imited distributions.
In this section a
sampling theorem for tempered distributions whose Fourier transforms have
compact supports is established.
W> 0, if supp(f) c (-W,W).
denoted by Bd (W).
A distribution feB' is said to be W-band1imited,
The class of all W-band1imited distributions will be
Let D[-W,W], W> 0, be the class of all C~-functions ~ with supp(~) C
[-W,W], and define Z(W) ~ D[-W,W]
= {~eB:
~eD[-W,W]}.
Pfaffe1huber (1971)
stated that if HeBd(W) and h is its Fourier transform (defined as a function
on I R1 ), then
(3.l)
h( t)
~(2Wt - n)
= E h( 2Wn) sin ~(2Wt
- n)
n=-~
and the series converges absolutely in ZI (W) (the dual of Z(W)).
(3.1) means precisely that, for every
f~ h(t)~(t)dt = ~
_~
n=-~
~eZ(W),
h(~) f~ sin ~(2Wt - n) ~(t)dt
2W _~
and the series converges absolutely.
Equation
~(2Wt
- n)
~
,
Campbell (1968) had already noted that
(3.1) does not hold pointwise for arbitrary band1imited distributions.
Though
(3.1) is correct, the arguments presented in its proof are not convincing.
The following lemma is a modification of Lemma 1 of Pfaffe1huber (1971)
and will be needed in the proof of theorem 3.1.
Lemma 3.1.
Let feB' be such that f has compact support.
closed set properly containing supp(f), and
E and
by
w= 1 on
wany
some open set containing supp(f).
Let E be a
test function with support
Then f is uniquely determined
its restriction to O(E), i.e., the values f(S),SeD(f),
by
12
(3.2)
The shift operator tl is defined on D'(S'), for every ltIR 1, by
A distribution fcD'(S') is said to be periodic with period T > 0, if
(3.3)
(tTf)(~)
•
f(~)
, for every cjltD(S)
,
and T is the smallest positive number for which (3.3) holds.
THEOREM 3.1.
f has
Let f£S' be a tempered distribution such that
compact
support, and let the closed set E and W> 0 be such that supp(f)c E and the
translates {E+2nW}, n ~ 0, are disjoint from supp(f).
functions such that
"
containing supp(f).
~
has support E. and a • 1.
~
Let
= 1 each
a
~
and ~ be any test
on some open set
Then
011
Kw(ep) • r
_011
Kw(t)~(t)dt.
epts.
~.
13
<Xl
where GW(t) = Si~1T~~Wt , and GW(4)) = f GW(t)4>(t)dt , 4>c:5.
_<Xl
SN =
PROOF.
N
~n=-n
It will first be shown that the sequence of partial sums
'"
L -2nW f , N > 1, converges in 51. For any 4>c:5,
N
'"
~
(t -2nW f)( 4»
N
'"
f(-r2nH
n=-N
=
~
n=-N
4»
N
=
f( ~
n=-N
=
(3.6)
-e
•
where
L2nW
4»
L2nW
4»
N
f(~ ~
n=-N
is a test function such that ;(t) = 1 on some open set containing
'"
N
supp(f). It will be shown that the sequence 4>N(t) = ;(t)~
4>(t - 2nW),
n=-N
N ~ 1, converges in S. Since 4>c:5, there exists a constant B > 0 such that
14>(t)1 < B(1+t 2)-1 for all tc:IR 1 , and thus
;c:D
14> (t-2nW) I
Since
A > O.
<
B
2B(1+t 2)
1+(2nW)2
<
1+(t-2nW)2
it follows that supp(;) c [-C,C] for some C > 0 and 1~(t)1 ~ A for some
It then follows that for all tc:IR l and non-negative integers m,
;c:D,
N
(3.7)
(l+ltl)ml~(t)1 ~
n=-N
14>(t-2nW)!
~ 2AB(1+C)m(1+C 2) ~
n=-<Xl
1
1+(2n~J)
2
<
<Xl ,
14
i.e., the sequence of partial sums on the left hand side of (3.7) converges
uniformly on IR1• Hence the sequence (l+ltl)m~N(t), N ~ 1, converges uniformly
on IR 1 for every m ~ O. Similarly, it can be shown that for every m,k ~ 0,
the sequence (l+ltl)m~N(k)(t), N ~ 1, converges uniformly on IR 1, i.e. {~N}'
N ~ 1, converges in 5, and since 5 is complete, its limi,t
~N ~ ep
in 5.
ep
belongs to 5, and
It follows from (3.6) that
and since s' is (weak-star) sequentially complete, then there exists a tempered
distribution FtS' such that SN
Therefore, F •
with period 2W.
~
+
F in S'.
aD
...
SN = t n__ T-2 nWf is a periodic tempered distribution
aD
It follows that F has the Schwartz-Fourier series (Zemanian,
1965, p. 332)
...
aD
(3.8)
where et(u)
aD
F = t T-2 nW f - t a n en' in
n--aD
n=-aD ~ ~
= e2~1tu
5' ,
, and
a n = ~ F(Ue n )
2W
-2'W
where UtU 2W is a unitary function (Zemanian, 1965, p. 315), i.e. UeVand
t~._ClD U(t-2nW) • 1 for all teIR 1• From (3.8) it follows that
15
=
2W a
n
'"
(t -2mW f) (Ue
1:
m=-CIl
n )
- 2W
2W
CIl
=
•
f([L2mw U]e
1:
m=-CIl
n )
- 2W
'"
Since f has a compact support and UeD, then there is only a finite number of
non-zero terms in the last summation, and hence
'"
2W a n = f([
2~J
CIl
n
)
'"
= f(ae
n
)
'"
= f(.
n a).
- 2W
- 2W
- 2vl
-e
n)
- 2W
m=-CIl
'"
= f(e
(3.9)
L-2mW U]e
1:
From (3.8) and (3.9) it follows that
00
'"
f(e)
(3.10)
CXl
1T
=
i
whe re e n (e)
2W
=
(3.11)
f(e) - f(e)
and by Lemma
f e
1
1:
n=-CIl
2W f(.
'"
n a)e n (e) ,
- 2W
2W
eeD(E),
n
Wue(u)du = e"'( - 2W'
n)
Thus
-00
A
A
=
~
1
A
A
n
A
A
2W f(. n a)e(2W) , eeD(E) ,
n=-oo
2W
1:
3. 1 it fo 11 ows that for every epeB
16
(3.12)
v
(since ~*~ = (~;)A
D(E)). But
€
= 2W
f=
-=
= 2W(.
(3.13)
n
KW(t - 2~)$(t)dt
KW)(~)
,
~ES,
2W
and (3.4)
follows from (3.12) and (3.13).
To prove (3.5) notice that when eED[-W,W],
e
(e) =
n
2W
W 'Trl. -n U
feW e(u)du
-W
"
e(t)dt
= 2W(.
"
n Gw)(e) .
-2W
" "
It follows from (3.10) that for eED[-W,W],
A
f(e)
and (3.5) follows
~
A
= f(e) =
by Le~ma
r
n=-=
3.1.
A
A
f(. n a)(. n Gw)(e) ,
2W
2W
o
17
Theorem 3.1 shows that a tempered generalized function f with compact
spectrum can be reconstructed via (3.4) from its values (samples) evaluated
at the translates of an arbitrary, but fixed test function a which equals one
A
A
on some open set containing supp(f).
On the other hand, if we denote f(e t ) by f(t)
n
then from (3.9) if follows that f(. n a) = f(e n) = f(2W) , and (3.4) reads
2W
2W
A
f(~) =
r
n=-co
A
f(2W)('
n
KW)(~)
~£S
2W
so that a tempered distribution f with compact spectrum can be reconstructed
A
using the samples of the function f(t)
= f(e t ).
Now it is shown that the sampling theorem for tempered generalized functions
-e
with compact spectrum includes as special cases the sampling theorems for
conventionally bandlimited functions (Example 3.1) as well as for bandlimited
functions in L2(~k) (Example 3.2).
Let f€L 2 (IR 1 ) be a
EXAMPLE 3.1. (Conventionally bandlimited functions).
A
continuous function such that f has compact support E. Then f determines a
tempered generalized function:
(3.14)
f(~)
=
fco f(t)~(t)dt ,~€s
,
....
and its distributional Fourier transform (denoted also by f) is defined by
f(~)
= f($), ~€S, or equivalently by
f(~)
= i f(u)~(u)du , ~€s .
18
...
f (as a tempered generalized function) is supported by E. Hence (3.4) applies
and if W> 0 is defined as in Theorem 3.1, we have from (3.14)
...
f( T n a)
2W
For v
>
...
= f(e
n)
2W
=
n
W ...
j
f(u)e
'lTi Wu
du
=
-W
f(2~) .
0, define the function
c- 1
v
ep)t)
-1
exp --------for I~I
v
1-(t/v)2
=
for 111
v
0
where Cv =
j
00
-
<
exp{
-1
2}dt.
For each v
0, ep ED and
>
l-(t/v)
_00
1 ,
v
/"'ep v (t)dt
-co
and for each continuous function g and every tEIR 1
/Xlg(U)ep (t-u)du .. g(t) as v+O.
_00
v
tdR 1 and v
(3.15)
>
From (3.4) -it follows
that for each
0
00
jOOf(u)cpv(t-u)dt
_00
= :
f(2~)
jOOKw(u- 2~)epv(t-u)dt.
n--oo-oo
Since f and KWare uniformly continuous, we have for each fixed tEIR 1
and ndN
jOOf(u)cp v (t-u)du .. f(t) as v+O
_00
,
=
1
19
Now by Theorem 24 of Lighthil1 (1958, p.64), if for any sequence
00
{b } which is O(n) as n
n
~
E
ba
is absolutely convergent
n=-oo n n,\I
00,
and tends to a finite limit as \I
1im
\I~O
E
n=-oo
0, then
00
00
(3.16)
~
an,\I
=
E
n=-oo
liman\l
'
\I~O
But, for each fixed tEIR 1 ,
00
f
_00
KW(u - 2~)<P)t-u)dul
4
since f is bounded, Ibn'
~
Bini, and for k
>
1,
It follows that the right hand side of (3.15) satisfies the conditions
leading
to (3.16), and hence by letting \ItO, we obtain
(3.17)
which is the sampling theorem for a conventionally bandlimited function
with compact spectrum.
20
Example 3.2.
(Bandlimited functions in L2(~k))'
be a continuous function.
(3.14).
Let f£L2(~k)' k ~ 0,
Then f determines a tempered distribution by
If its distributional Fourier transform f"
has a compact support, then
(3.4) applies and we have
Since f is a C~-function and !f(t)j ~ Ck(l+ltlk,
for C > 0 (Lee, 1977), then (3.15) holds and following the arguments used
k
in Example 3.1, one obtains (3.17) which is similar to (2.3) and is identical
to (2.4).
It should be noted, though, that (3.4) cannot be obtained from
CampbellLs result (1.4), since the convergence in (2.4) is not uniform on
compact sets.
21
4.
Series expansions for random distributions.
In this section sampling
expansions for stationary random distributions are derived.
Let
X = (X($), $€S} be a second order random distribution. X is said to be
weakly stationary, if for every h
>
a and
$,~eS,
..
If X is a weakly stationary random distribution (WSRD), then there exists a
unique tempered distribution peS' such that for every
$,~ES,
(4.1)
where ~(t)
•
A
= ~(-t) (Ito, 1954) and
p
has the spectral representation
(4.2)
-eo
where ~ is a non-negative measure on IR l such that
for some integer k.
d~(u)
<
eo
In this case X is said to be of type k, and
~
feo
-eo
(1 +u 2) k
is called
the spectral measure of X.
Let B* be the set of all Borel sets with finite
~-measure.
An
L2 (n)-valued function Z defined on.B* is called a random measure with
respect to
~
if
22
and Z(B 1) ~ Z(B 2) if B1 and B2 are disjoint. Since
~ is a-additive, then Z(B) = E~=l Z(B ), whenever B ,B 2 , ... are disjoint sets
n
1
Hence E(Z2(b))
= ~(B),
in B* with U~:lBn
= B. It follows by (4.1) and (4.2) that there exists a
I
random measure Z with respect to
00
X(~)
=
~
.
such that
,.
~(u)dZ(u)
f
~eS
,
•
_00
If H(X) is the linear subspace of L2(n) generated by {X(~),~eS}, then H(X) and
L2(J..t} are isometrically isomorphic under the correspondence X(4))I-~ ;, 4>eS.
A WSRD X is said to be W
O-band1imited, W
o
THEOREM 4.1.
aeD and
~eD[-W,WJ
>
0, if ~{[-Wo'WoJc}
= O.
(a) If X = {X(4)),
'
4>eS}is a W
O-band1imited WSRD, W> W
O
with a(t) = 1 = ~(t) on [-WO'WOJ, then for every 4>eS~
00
(4.3)
X(4))
E X(t n ~)(t n Gw
)(;*4»
=
N
n=-oo
=
in a mean-square, where Gw(4))
(b)
foo
2W
s~~~~ Wt 4>(t)dt.
-co
Let X = {X(4)), 4>eS} be a WSRD with spectral measure
compact support.
Let the closed set E and W>
a be
the translates {E+2nW} , n 1 0, are disjoint from
~
any test functions such that
has support E, and a(t)
=:n--oo X(t2W ~)(t 2W
00
(4.4)
X(~)
n
w)(4)) , 4>eS,
n K
C
Let a and
~
=1 =
Then
which has
supp(~)
such that
supp(~).
~
~(t)
1
on
E and
be
supp(~).
23
in mean-square, where KW(t) = 2~ ! $(u)e2TIitudu.
E
To prove (a), first let ~€s be such that ;€s[-W,W].
oo
Then ~(u) = r _ oo$(u+2nW) is a COO-function which is periodic with period 2W
n-and has the Fourier series
Proof.
•
(4.5)
~(u)
1
co
n
n
= r '2W ~ (2W)e
n=-oo
which converges uniformly on IR1. Since
n; W u
, udR
1
,
A
~€D[-W,WJ,
1 W TIi _.n u
= 2W _~ e
W ~(u)du
Consider the mean square error
.
There exists a constant M> 0 such that for all Nand u€IR 1 ,
24
N
..
IIjl (U )
Since, by (4.5),
N
1:
-
n=-N
1
N 9(2W) e
1Tl
n
WU
<
. n
WU
n
1
.
n
1Tl
M
...
converges to Ijl(u) on [-Wo,W o]'
n=_N 2W 1jl(2W)e
by the dominated convergence theorem, e~(Ijl) ~ 0 as N ~ =.
1:
A
Thus for every IjltD[-W,W], we have
(4.6)
Now for every IjltS and
~
as in part (a) of the statement of the theorem,
it follows
W ..
= f o ljl(u)dZ(u) =
-Wo
v
where ~*Ijl
= (~;)AIjlD
[-W,W], and (4.3) follows from (4.6) and (4.7).
wi~h
of part (b) is similar to that of (a)
the obvious modification and hence
o
is omitted.
= 1 on [-WO,WO],
It should be noted that, since a
1./
=
Defi ne
The proof
f
"0
-W O
•
e
-1Tl
n
W
U
dZ(u) ,ntIN •
25
Wo
f
e
-2nitu
dz(u) , teIR
1
,
-Wo
then {x(t), teIR 1} is a weakly stationary W
O-band1imited stochastic process,
X(. n ~) = x(2W)' and (4.3) reads
2W
(4.8)
x(~)
=
i.e., the random distribution X is reconstructed using the samples of the
ordinary stochastic process x.
Hence there is a one-to-one correspondence
between W
O-band1imited weakly stationary random distributions X and W
OA
W
O-band1imited weakly stationary processes x determined by X(~) = ,f ~(u)dZ(u)
W
-wO
and x(t) =
fO e2nitudZ(u) and satisfying (4.8).
-wO
Now it is shown that the sampling theorem for band1imited weakly stationary
random distributions includes as a particular case the sampling theorem
for band1imited weakly stationary processes.
Example 4.1. Let x = {x(t), teIR 1} be a measurable, mean-square
continuous, weakly stationary process which is W-band1imited, i.e.,
o
(4.9)
where Zis a random measure with respect to the spectral measure
~
of
Then x determines a W
O-band1imited WSRD by
26
=
W
o
f
-wo
A
~ ( u ) dZ( u ) ,
~ eS ,
which can also be written as
x(~)
OD
W
= f o ( f e-2nitU~(t)dt)dZ(U)
-Wo _OD
OD
= f
_OD
A
x(t)~(t)dt
where the latter integral exists both with probability one as well
Then by (4.4) if follows that for each teIR 1
as in quadratic mean.
and \)
>
0,
OD
...
OD
(4.10)
in quadratic meaD.
f
_OD
KW(u - 2~)~\)(t-u)dU
As in example 3.1,
OD
f x(u)~ \) (t-u)du ~
_OD
OD
in quadratic mean,
f
_OD
x(t)
as x+O
KW(u - 2~)~\)(t-u)du ~ KW(t - 2~) as \)+0, and the
right hand side of (4.10) converges in quadratic mean to E~=_ODX(2~)Kw(t - 2~)'
It follows that
x( t) =
in quadratic mean.
I
i
BIBLIOGRAPHY
Cambanis, S. and Masry, E. (1976). Zakai's class of band1imited functions
and processes: Its characterization and properties, SIAM J.
of ~. Math., lQ. No. 10.,.20.
--Campbell, L.L. (1968). Sampling theorem for the Fourier transform of a
distribution of bounded support, SIAM Journal of ~. Math.
16 626-636.
Christenson, S.K. (1985). Linear stochastic differential equations on the
dual of a countab1y Hilbert nuclear space with applications to
neurophysiology. Dissertation. Tech. report #104, Center for
Stochastic Processes. The University of North Carolina at
Chapel Hill.
Holly, R. and Stroock, D. (1978). Generalized Ornstein-Uh1enbeck and
infinite particle Brownian motions. Publications RIMS Ii
Kyoto University.
Hoskins, R. F. and De Sousa Pinto, J. (1984 a). Sampling expansions for
functions band-limited in the distributional sense. ~IAM~.~.
Math., 44 605-610.
Hoskins, R. F. and De Sousa Pinto, J. (1984 b). Generalized sampling
expansions in the sense of Papou1is. SIAM~.~. Math., 44
611-617 •
...
Ito, K. (1953). Stationary random distributions, Memorial Collection of
-Science, University of Kyoto, 28 209-223.
Jerri, A.J. (1977). The Shannon sampling theorem - its various extensions
and applications: A tutorial review, Proceedings lEEE, 65 15651596.
Ka11ianpur, G. and Wolpert, R. (1984). Infinite dimensional stochastic
differential equation models for spatially distributed neurons,
8m?J... Math. Optimize 11. 125-172.
Kallianpur, G. and Wolpert, R. (1984). Weak convergence of solutions
to stochastic differential equations with applications to
non-linear neuronal models. Tech. report #60, Center for
Stochastic Processes, The University of North Carolina at
Chape1 Hi ll.
Kotel'nikov, V.A. (1933). On the transmission capacity of "ether" and
wire in e1ectrocommunications (material for the first all-union
conference on questions of communications) Izd. Red. YEr. Svyazi
RKKA (Moscow).
Lee, A. (1976). Characterization of bandlimited functions and processes,
Inform'. Cont., li, No. 3 258-271.
ii
Lee, A. (1977) •. Approximate interoo1ation and the sampling theorem,
SIAM Journal of ~ Math., 32 731-744.
Lighthi11, M.J. (1958). Introduction to Fourier Analysis and Generalized
Functions, Cambridge University Press, London ..
Pfaffelhuber, E. (1971). Sampling series for band1imited generalized
functions, IEEE Transactions on Information and Control, IT-17,
No • .§., 650-65"4.
-Piranashvi1i, A. (1967). On the problem of interpolation of stochastic
processes, Theor. Probab. ~.,]~ 647-657.
Rudin, W. (1974).
Functional Analysis, McGraw-Hill, New York.
Shannon, C.E. (1949). Communication in the presence of noise, Proceedings
2i the Institute of Radio Engineers, R 10-21 •
...
Treves, F. (1967). Topological Vector Spaces, Distributions and Kernels,
Academic Press, New York.
Whittaker, E.T. (1915). On the functions which are represented by the
expansion of the interpoloation theory, Proceedings of the E~
Society, Edinburgh, 35 181-194.
Whittaker, J.M. (1929). The Fourier theory of the cardinal functions,
Proceedings, Mathematical Society, fdinburgh, 1 169-176.
Whittaker, J.M. (1935). Interpolutory Function Theory, Cambridge University
Press (Cambridge tracts in Mathematics and Mathematical Physics)
33.
Zakai, M. (1965). Bandlimited functions and the sampoing theorem,
Infor~
Cont., ~ 143-158, MR 30 #4607.
Zemanian, A.H. (1967). Distribution Theory and Transform Analysis,
McGraw-Hill, New York.
--
I
Department of Biostatistics
The University of North Carolina at Chapel H~
Chapel Hill, NC 27514 (USA)
...
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Series Representations for Generalized Stochastic Processes (Unclassified)
•
PERSONAL AUTHOR(S)
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~uhammad
I'JbF~OMTI"ElJ85__
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18 SUBJECT TERMS (ContInue on reve'" if nec.sury and idpntlfy by block number)
COSA TI CODES
FIELD
SUB·GROup
GROUP
. 19 ABSTRACT
(Continu~
on
r,v~rs,
Stochastic Process, Genera 1i zed Function, Sampling
if nNeSSi!ry and id.ntify by bloclc number)
Series representations are derived for bandlimited generalized functions and generaliz d
stochastic processes. This work extends existing results concerning sampling representations of bandlimited functions and stochastic processes. The merit of such representation
lies in the fact that a function (or process) may be exactly reconstructed using only a
countable number of its values (or samples). These types of representations have found
many applications in several areas of communication and information theory such as digital
audio and visual recording, and satellite communications. In addition, random
distributions have also been employed in a host of applied areas such as statistical
mechanics, chemical reaction kinetics and neurophysiology.
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