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SERIES REPRESENTATIONS FORGENERALIZED STOCHASTIC PROCESSES

by

Muhammad K. Habib

Department of BiostatisticsUniversity of North Carolina at Chapel Hill

Institute of Statistics Mimeo Series No. 1801

March 1986

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•

SERIES REPRESENTATIONS FORGENERALIZED STOCHASTIC PROCESSES

by

Muhammad K. Habib

Department of BiostatisticsThe 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 contractnumber 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: ! f(t)~(t)dt, where ~ is a IIsmoothll function representing theI

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) = Ln=-oo

n sin 1T(2W - n)f(2W)

1T(2Wt - t)1tdR ,

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. For a review of the sampling theorem, see Jerri

~ (1977). A function f which can be represented, for some Wo > 0, by

(1. 2)

2

1t tt:: IR

is called Ll-bandlimited to Wo if Ft::L 1[-WO,woJt and is called conventionally

or L2-bandlimited to Wo if Ft::L2[-WOt 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(IR1) as well as uniformly on IR1.

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 a2non-negative integer k, let L (~~) be the class of all complex valued functions

defined on IRl that are square integrable with respect to the measure ~~

d~k(t) = dt 2 k' If fEL2(~k)' then f defines a tempered generalized function(1 + t )

(or tempered distribution) (denoted also by f) on the class S of rapidly decreasing

functi ons by

co

f(9) = f f(t)e(t)dt t 6ES-co

(See Section 2 for relevant definitions.) The distributional Fourier transform

of f is the tempered distribution f defined by f(e) =A

f( eL et::s. The spectrumA

of f is the support of f. For k = 0,1,2, ... and Wo > 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(WO) is the class of WO-bandlimited functions in L2(IR1)t and

Bk(WO) CBk+l(WO)' A1so t BO(WO) is dense in Bk(WO) for every positive integer k

(see Lee, 1976 ).

.'

3

Zakai obtained a sampling representation for functions in Bl (WO). 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 (WO) and W> Wo then f has

a sampling representation of the form (1.1). Lee (1977) extended Zakai·s result

to functions in Bk(WO). He showed that if feBk(WO)' W> WO' 0 < S < W-WO and ~

is an arbitrary but fixed CCO -function with support in [-1,1] and

lco

w(t)dt = 1, then-co

(1 .3) f( t) = co[

n=-cof(2Wn) sin n(2Wt - n)

n(2Wt - n)

and the series converges uniformly on compact sets. It should be noted that

the presence of the (damping) factor ~ in (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(WO));

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

F(eu) is well defined, since eueCco

for all ueIR1. In this case the Fourier

transforms Fof F may be thought of as a function defined on IRl by

F(u) = F(eu)' udRl (see Section 2). Campbell showed that if F is a tempered

generalized function with compact support and with Fourier transform f as a function

on IR1, i.e. f(t) = F(et ), teIR1, 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, and the series converges for every teIR1.

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 seIR1, 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(WO) 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(WO) (see also Cambanis and Masry, 1976). Indeed, it was shown that if

1 1X={X(t),teIR } belongs to BP1(WO)' then for any W> Wo and teIR

(l .6)=

X(t) = 1: X(~W)n=-=

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), tEIR1} in BPk(WO)' k>l

(1. 7) X(t)=<Xl

~

n=-ooX(n ) sin n(2Wt-n)

2W n{2W-n)A n 11jJ(e(t- 2W)), tdR

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(WO) and Bl(WO) 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(WO), k:: 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 IR1. 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~ (IR1) be the class of all

infinitely differentiable functions with compact support. A topology T is

introduced on the linear space c~ which makes it into a complete space; that isc .

a sequence {~n} in c~ converges to zero in T if there exists a compact AEIRl which

contains the support of every ~n' and for every non-negative integer k, ~n (k)(t) + 0

uniformly as n +~. c~ with the topology T is denoted by D, and its elements are

called test functions. The members of the dual D' of D are called distributions,

and the value of a distribution feD' at a test function ~eD is denoted by f(~). A

(weak-star) topology on D' is defined by the seminorms If(~)I, feD', as ~ varies

over all elements of D; thus for a sequence {fn} in D': fn + 0 weakly whenever

fn(~) + 0 for all ~eD.

The class S of rapidly decreasing functions consists of all infinitelydifferentiable functons (~eC~) for which

Itm~(k)(t)1 < C , -~ < t < ~- m,k

for all non-negative integers m,k. A topology on S is defined by the seminorms

I I~I 1m k = sup sup 1 {(l+ltl)kl~(n)(t)l} , m,k = 0,1,2, ... ,, O<n<m teIR

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 IR1• S is complete, and the dual S' of S is called the

class of tempered distributions. Similarly, a (weak-star) topology is defined

on S' by the seminorms If(~)I, feS', as ~ varies over all elements of sa, i.e.,

fn converges in S' if fn(~) converges for all ~eS. The space D'(S') is (weak-star)

sequentially complete, that is, if {f} is a sequence in D'(S') such thatn n

{fn(~)}n is a Cauchy sequence for every ~eD(S), then there exists a distributiJr

8

feD' (S') such that fn -+ f in D' (S).

Finally, the space e~ with the topology defined by the seminorms

Pm A(') = E supl,(n)(t)1 "ee~,, Q<n<m teA

where A ranges over all compact sets in IRl and mover all non-negative integers,

is denoted by E.

The Fourier transform F(F(,) = ;"eS) is a one-to-one biocontinuous..

mapping from S onto itself. If feS', the Fourier transform f of f is defined..

by f(,) = f(,), ,eS, and is a tempered distribution. If feS' and ,eS, the

convolution f*, is defined as a function on IRlby

where ~(t) =,(-t) and the shift operator Tt is defined by (Tt,)(U) = ,(u-t).

f*,ee~has a polynomial growth and thus determines a tempered distribution.

Suppose feD', f is said to vanish in an open set UCIRlif f(,) =a for

every ,eD with supp(,) cu. Let V be the union of all open sets UdRl in which

f vanishes. The complement of V is the support of f. Distributions with

compact supports are tempered distributions. Now, if f is a distribution with

compact support (i.e., feS'), then f extends uniquely to a continuous linear

functional on E. If ~eV is such that ~(u) = 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

the function f(t) defined on IRl by

(2. 1)

Indeed,

(2.2)

and (~f)

9

A

f = (~ f)

A ~ . A A

(and therefore f) is generated by the C -function (f*~)(t)

which has a polynomial growth (see Rudin, 1973, p.l79). By choosing ~£s suchA

that ~ =~, 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

IRl by (2.1).

Let (Q,F,P) be a probability space. A random distribution (or a

~4It 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 IR2). 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 by R(~,~) = E[X(~)X(w)]. R is

given by R(~~w) = f~ f~ R(t,s) ~(t) ~(s)dt ds.-~ -~

11

3. Sampling representations forband1imited distributions. In this section a

sampling theorem for tempered distributions whose Fourier transforms have

compact supports is established. A distribution feB' is said to be W-band1imited,

W> 0, if supp(f) c (-W,W). The class of all W-band1imited distributions will be

denoted by Bd(W).

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 IR1 ), then

(3.l) h( t) = En=-~

h( n) sin ~(2Wt - n)2W ~(2Wt - n)

and the series converges absolutely in ZI (W) (the dual of Z(W)). Equation

(3.1) means precisely that, for every ~eZ(W),

f~ h(t)~(t)dt = ~ h(~) f~ sin ~(2Wt - n) ~(t)dt_~ n=-~ 2W _~ ~(2Wt - n) ~ ,

and the series converges absolutely. 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. Let E be a

closed set properly containing supp(f), and wany test function with support

E and w= 1 on some open set containing supp(f). Then f is uniquely determined

by 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 ltIR1, 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. Let f£S' be a tempered distribution such that f has 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). Let a and ~ be any test ~.

functions such that ~ has support E. and a • 1. ~ =1 each on some open set"containing supp(f). Then

011

Kw(ep) • r Kw(t)~(t)dt. epts._011

13

<Xlwhere GW(t) = Si~1T~~Wt , and GW(4)) = f GW(t)4>(t)dt , 4>c:5.

_<Xl

PROOF. It will first be shown that the sequence of partial sums

N '"SN = ~n=-n L -2nW f , N > 1, converges in 51. For any 4>c:5,

(3.6)

=

=

=

N '"~ (t -2nW f)( 4»

n=-N

N '"~ f(-r2nH 4»

n=-NN

f( ~ L2nW 4»n=-NN

f(~ ~ L2nW 4»n=-N

-e•

where ;c:D 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+t2)-1 for all tc:IR1, and thus

14> (t-2nW) I < B <

1+(t-2nW)22B(1+t2)1+(2nW)2

Since ;c:D, it follows that supp(;) c [-C,C] for some C > 0 and 1~(t)1 ~ A for some

A > O. It then follows that for all tc:IRl and non-negative integers m,

(3.7)N

(l+ltl)ml~(t)1 ~ 14>(t-2nW)!n=-N

~ 2AB(1+C)m(1+C2) ~ 1 2 < <Xl ,n=-<Xl 1+(2n~J)

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 IR1 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 IR1, i.e. {~N}'

N ~ 1, converges in 5, and since 5 is complete, its limi,t ep belongs to 5, and

~N ~ ep in 5. It follows from (3.6) that

and since s' is (weak-star) sequentially complete, then there exists a tempered

distribution FtS' such that SN + F in S'.aD ...

Therefore, F • ~ SN = t n__aD

T-2nWf is a periodic tempered distribution

with period 2W. It follows that F has the Schwartz-Fourier series (Zemanian,

1965, p. 332)

aDaD ...F = t T-2nW f - tan en' in 5' ,

n--aD n=-aD ~ ~

where et(u) =e2~1tu , and

(3.8)

a n =~ F(Ue n )2W -2'W

where UtU2W is a unitary function (Zemanian, 1965, p. 315), i.e. UeVand

t~._ClD U(t-2nW) • 1 for all teIR1• From (3.8) it follows that

2W a = 1:n m=-CIl

2W

15

'"(t -2mW f) (Ue n )

- 2W

CIl= 1: f([L2mw U]e

m=-CIln )

- 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

'" CIl2W a n = f([ 1: L-2mW U]e n)

2~J m=-CIl - 2W

-e From (3.8) and (3.9) it follows that

(3.9) '"= f(e )n- 2vl

'"= f(ae )n

- 2W

'"= f(. n a).- 2W

00

(3.10)'"f(e) 1= 1: 2W f(.

n=-CIl

'"n a)e n (e) , eeD(E),

- 2W 2W

whe re e n (e) =

2W

CXl 1T if e

-00

n

Wue(u)du = "'( n)e - 2W' Thus

(3.11)A A ~ 1 A A n A A

f(e) - f(e) = 1: 2W f(. n a)e(2W) , eeD(E) ,n=-oo 2W

and by Lemma 3. 1 it fo 11 ows that for every epeB

(3.12)

16

v

(since ~*~ = (~;)A € D(E)). But

= 2W f= KW(t - 2~)$(t)dt-=

(3.13) = 2W(. n 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 . n'Trl - U

= feW e(u)du-W

"= 2W(. n Gw)(e) .-2W

"e(t)dt

" "It follows from (3.10) that for eED[-W,W],

A A ~ A A

f(e) = f(e) = r f(. n a)(. n Gw)(e) ,n=-= 2W 2W

and (3.5) follows by Le~ma 3.1. o

-e

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 oneA A

on some open set containing supp(f). On the other hand, if we denote f(e t ) by f(t)A A n

then from (3.9) if follows that f(. n a) = f(e n) = f(2W) , and (3.4) reads2W 2W

f(~) = r f(2W)(' KW)(~) ~£Sn=-co n

2W

so that a tempered distribution f with compact spectrum can be reconstructedA

using the samples of the function f(t) = f(e t ).

Now it is shown that the sampling theorem for tempered generalized functions

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).

EXAMPLE 3.1. (Conventionally bandlimited functions). Let f€L2(IR1) be aA

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) = f(e n)

2W 2W

For v > 0, define the function

nW ... 'lTi Wu

= j f(u)e du =-W

f(2~) .

c-1 -1 I~Iexp --------- for <v 1-(t/v)2 v -ep)t) =

0 for 111 1 ,v

00 -1where Cv

= j exp{ 2}dt. For each v > 0, ep ED and_00 l-(t/v) v

and for each continuous function g and every tEIR1

/"'ep (t)dt = 1v

-co

/Xlg(U)ep (t-u)du .. g(t) as v+O. From (3.4) -it follows that for eachv

_00

tdR1 and v > 0

(3.15) 00

jOOf(u)cpv(t-u)dt = : f(2~) jOOKw(u- 2~)epv(t-u)dt._00 n--oo-oo

Since f and KWare uniformly continuous, we have for each fixed tEIR1

and ndN

jOOf(u)cp (t-u)du .. f(t) as v+O ,v

_00

19

Now by Theorem 24 of Lighthil1 (1958, p.64), if for any sequence

{bn} which is O(n) as n ~ 00,

00

E b a is absolutely convergentn n,\In=-oo

and tends to a finite limit as \I ~ 0, then

(3.16)00

1im E\I~O n=-oo

an,\I =00

E liman\ln=-oo \I~O '

But, for each fixed tEIR1,

00

f KW(u - 2~)<P)t-u)dul_00

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))' Let f£L2(~k)' k ~ 0,

be a continuous function. Then f determines a tempered distribution by"(3.14). 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 Ck > 0 (Lee, 1977), then (3.15) holds and following the arguments used

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)

A

where ~(t) = ~(-t) (Ito, 1954) and p has the spectral representation

• (4.2)-eo

where ~ is a non-negative measure on IRl such that feo d~(u)-eo (1 +u2)k

< eo

for some integer k. In this case X is said to be of type k, and ~ 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

Hence E(Z2(b)) = ~(B), and Z(B1) ~ Z(B2) if B1 and B2 are disjoint. Since

~ is a-additive, then Z(B) = E~=l Z(Bn), whenever B1,B2, ... are disjoint sets

in B* with U~:lBn = B. It follows by (4.1) and (4.2) that there exists aI .

random measure Z with respect to ~ such that

00 ,.

X(~) = f ~(u)dZ(u) , ~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 WO-band1imited, Wo > 0, if ~{[-Wo'WoJc} = O.

THEOREM 4.1. (a) If X = {X(4)), 4>eS}is a WO-band1imited WSRD, W> WO'

aeD and ~eD[-W,WJ with a(t) =1 =~(t) on [-WO'WOJ, then for every 4>eS~

(4.3)00

X(4)) = E X(t n ~)(t n Gw)(;*4»n=-oo N 2W

in a mean-square, where Gw(4)) = foo s~~~~ Wt 4>(t)dt.-co

(b) Let X= {X(4)), 4>eS} be a WSRD with spectral measure ~ which has

compact support. Let the closed set E and W> a be such that supp(~) C E and1

the translates {E+2nW} , n 1 0, are disjoint from supp(~). Let a and ~ be

any test functions such that ~ has support E, and a(t) =1 = ~(t) on supp(~).

Then

(4.4)00

X(~) =: X(t n ~)(t n Kw)(4)) , 4>eS,n--oo 2W 2W

•

23

in mean-square, where KW(t) = 2~ ! $(u)e2TIitudu.E

Proof. To prove (a), first let ~€s be such that ;€s[-W,W].

Then ~(u) = roo_ $(u+2nW) is a COO-function which is periodic with period 2Wn--oo

and has the Fourier series

(4.5) ~(u) =n

co 1 n n; Wu 1r '2W ~ (2W)e , udR ,

n=-oo

which converges uniformly on IR1.A

Since ~€D[-W,WJ,

n1 W TIi _. u= 2W _~ e W ~(u)du

Consider the mean square error

.. There exists a constant M> 0 such that for all Nand u€IR1,

24

N . n... 1 n 1Tl WU

IIjl (U ) - 1: N 9(2W) e < Mn=-N

. nN 1 n 1Tl WU ...

Since, by (4.5), 1:n=_N 2W 1jl(2W)e converges to Ijl(u) on [-Wo,Wo]'

by the dominated convergence theorem, e~(Ijl) ~ 0 as N ~ =.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

Wo ...= f ljl(u)dZ(u) =

-Wo

v

where ~*Ijl = (~;)AIjlD [-W,W], and (4.3) follows from (4.6) and (4.7). The proof

of part (b) is similar to that of (a) wi~h the obvious modification and hence

is omitted.

It should be noted that, since a = 1 on [-WO,WO],

o

1./ • n"0 -1Tl WU= f e dZ(u) ,ntIN •

-WO

Defi ne

25

Wo -2nitu 1f e dz(u) , teIR ,

-Wo

then {x(t), teIR1} is a weakly stationary WO-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 WO-band1imited weakly stationary random distributions X andWOA

WO-band1imited weakly stationary processes x determined by X(~) = ,f ~(u)dZ(u)

W -wOand 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), teIR1} be a measurable, mean-square

continuous, weakly stationary process which is Wo-band1imited, i.e.,

(4.9)

where Zis a random measure with respect to the spectral measure ~ of

Then x determines a WO-band1imited WSRD by

26

Wo A

= f ~ (u ) dZ(u ) , ~eS ,-wo

which can also be written as

Wo ODe-2nitU~(t)dt)dZ(U)x(~) = f ( f

-Wo _OD

ODA

= f x(t)~(t)dt_OD

where the latter integral exists both with probability one as well

as in quadratic mean. Then by (4.4) if follows that for each teIR1

and \) > 0,

(4.10)OD OD ...

f KW(u - 2~)~\)(t-u)dU_OD

in quadratic meaD. As in example 3.1,

ODf x(u)~ (t-u)du ~ x(t) as x+O

\)_OD

ODin quadratic mean, f KW(u - 2~)~\)(t-u)du ~ KW(t - 2~) as \)+0, and the_OD

right hand side of (4.10) converges in quadratic mean to E~=_ODX(2~)Kw(t - 2~)'

It follows that

I

x( t) =

in quadratic mean.

i

BIBLIOGRAPHY

Cambanis, S. and Masry, E. (1976). Zakai's class of band1imited functionsand processes: Its characterization and properties, SIAM J.of ~. Math., lQ. No. 10.,.20. ---

Campbell, L.L. (1968). Sampling theorem for the Fourier transform of adistribution of bounded support, SIAM Journal of ~. Math.16 626-636.

Christenson, S.K. (1985). Linear stochastic differential equations on thedual of a countab1y Hilbert nuclear space with applications toneurophysiology. Dissertation. Tech. report #104, Center forStochastic Processes. The University of North Carolina atChapel Hill.

Holly, R. and Stroock, D. (1978). Generalized Ornstein-Uh1enbeck andinfinite particle Brownian motions. Publications RIMS IiKyoto University.

Hoskins, R. F. and De Sousa Pinto, J. (1984 a). Sampling expansions forfunctions band-limited in the distributional sense. ~IAM~.~.

Math., 44 605-610.

Hoskins, R. F. and De Sousa Pinto, J. (1984 b). Generalized samplingexpansions in the sense of Papou1is. SIAM~.~. Math., 44611-617 •

...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 extensionsand applications: A tutorial review, Proceedings lEEE, 65 1565­1596.

Ka11ianpur, G. and Wolpert, R. (1984). Infinite dimensional stochasticdifferential equation models for spatially distributed neurons,8m?J... Math. Optimize 11. 125-172.

Kallianpur, G. and Wolpert, R. (1984). Weak convergence of solutionsto stochastic differential equations with applications tonon-linear neuronal models. Tech. report #60, Center forStochastic Processes, The University of North Carolina atChape1 Hi ll.

Kotel'nikov, V.A. (1933). On the transmission capacity of "ether" andwire in e1ectrocommunications (material for the first all-unionconference on questions of communications) Izd. Red. YEr. SvyaziRKKA (Moscow).

Lee, A. (1976). Characterization of bandlimited functions and processes,Inform'. Cont., li, No. 3 258-271.

i i

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 GeneralizedFunctions, Cambridge University Press, London ..

Pfaffelhuber, E. (1971). Sampling series for band1imited generalizedfunctions, IEEE Transactions on Information and Control, IT-17,No • .§., 650-65"4. - --

Piranashvi1i, A. (1967). On the problem of interpolation of stochasticprocesses, Theor. Probab. ~.,]~ 647-657.

Rudin, W. (1974). Functional Analysis, McGraw-Hill, New York.

Shannon, C.E. (1949). Communication in the presence of noise, Proceedings2i 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 theexpansion 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 UniversityPress (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. --

Department of BiostatisticsThe 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)~uhammad K. Habib

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17. COSATI CODES 18 SUBJECT TERMS (ContInue on reve'" if nec.sury and idpntlfy by block number)-FIELD GROUP SUB·GROup- Stochastic Process, Genera1i zed Function, Sampling

. 19 ABSTRACT (Continu~ on r,v~rs, if nNeSSi!ry and id.ntify by bloclc number)Series representations are derived for bandlimited generalized functions and generaliz

stochastic processes. This work extends existing results concerning sampling represent-ations of bandlimited functions and stochastic processes. The merit of such representationlies in the fact that a function (or process) may be exactly reconstructed using only acountable number of its values (or samples). These types of representations have foundmany applications in several areas of communication and information theory such as digitalaudio and visual recording, and satellite communications. In addition, randomdistributions have also been employed in a host of applied areas such as statisticalmechanics, chemical reaction kinetics and neurophysiology.

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