SERIES REPRESENTATIONS FOR GENERALIZED STOCHASTIC PROCESSES Muhammad K. Habib...
Transcript of SERIES REPRESENTATIONS FOR GENERALIZED STOCHASTIC PROCESSES Muhammad K. Habib...
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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.
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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.
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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
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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.-~ -~
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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.
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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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