MUHAMMAD KArvlliL HABIB.
Sampling Representations and Approximations
for Certain Functions and Stochastic Processes.
(Under the direction
of STAMATIS CAMBANIS.)
•
Sampling representations and approximations of deterministic
and random signals are important in communication and information
theory.
Finite sampling approximations are derived for certain
functions and processes which are not bandlimited, as well as bounds
on the approximation errors.
Also derived are sampling representa-
tions for bounded linear operators acting on certain classes of bandlimited functions and processes; these representations enable one to
reconstruct the image of a function (or a process) under a bounded
•
linear operator using the samples of the function (or the process)
rather than the samples of the image.
Finally, sampling representa-
tions for distributions and random distributions ar,e obtained.
ACKNCWLEDGEMENTS
I wish to sincerely thank my advisor, Professor Stamatis
Cambanis, who suggested the central thesis of my research.
His
insistence on rigor, simplicity, and elegance was a great educational experience for me.
I wish to thank the members of my examination committee,
Professors Walter L. Smith,
~1alcolm
R. Leadbetter, Charles R.
Baker, Joseph A. Cima and Loren W. Nolte for their review· of the
manuscript.
My
thanks to Ms. June Maxwell for her careful and
excellent typing of this manusc.ript.
I thank my colleagues and friends in Chapel Hill for their
•
support, in particular my dear friend Dr. Joseph C. Gardiner for
our late-night intellectual (and non-intellectual!) exchanges.
It is a pleasure to thank my mother and my wife.
Their faith
and confidence in me has been a valuable source of inspiration.
Finally, I am thankful to the Egyptian people for supporting
me throughout this study. To them I will always be grateful.
)I
TABLE OF CONTENTS
CHAPTER I:
Introduction----------------------------------------- l.
1.1.
Rationa1e- -- - - -- -- - --- --- -- - - -- -- -- -,. -- -- - --- -- ---.-- - - l.
1.2.
Sampling
1.3.
StmIII1ary----------------------------------------------- 3
1.4.
Notation----- - - -- - - - - - - -- ---,. ,.,.-.-,. -,. -- -,. . . - -,.- -,.- --,.- - - 4
CHAPTER II:
•
Representations~~-----------------~--------~-
Sampling Approximation for Non-Bandlimited
Functions and Processes------------------------~------
2
5
2.1.
Introduction-----------~-----------~-,.-----,.---------5
2.2.
Sampling Approximations for Non-Stationary
Stochastic Processes---- --- ------ -.;----- -- - - ~- - ------- 6
2.3.
The Rate of Convergence in the Finite Sampling
Approximation for Time-Limited Stochastic Processes---12
2.4.
Finite Sampling Approximations for Non-Band1imited
Functions and Stochastic Processes------------------- 17
2.5.
Sampling Approximation Using Walsh Functions----- n -- 35
CHAPTER III: Sampling Expansions for Operators Acting on
Certain Classes of Functions and Processes----------- 46
3.1.
Introduction- - - - - - - - -,. - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 46
3.2.
The Band1imited Case--------------------------------- 47
3.3.
Bandlimited in Lloyd's
CHAPTER IV:
Sensen--,.-------n--------n-~O
Sampling Expansions for Distributions and Random
Distributions----------------------------------------74
4.1.
Introduction----------------------------------------- 74
4.2.
Notation and Basic Definitions-----------------------·74
iii
4.3.
Sampling Expansions for Certain Distributions------77
4.4.
Sampling Expansions for Random Distributions-------86
BIBLIOGRAPHY--------------------------------------------------92
CHAPTER I
Introduction
1.1.
Rationale.
In many areas of communication engineering, such as radio and
television broadcasting and satellite communications, one is concerned with the transmission of several signals (functions) through
a "channel." Since it is inefficient to transmit one signal at a
time over the channel, it is natural to ask if it is possible to
"reconstruct" a continuous-time signal from its samples under cer--
•
tain conditions on the signal and the sampling scheme.
If the answer
is in the affirmative, one may transmit only the samples of the
signal, thus occupying the channel only at the instants of sampling.
Between these instants the samples of other signals can be transmitted.
Another interesting applicatioR of "sampling" is in the field of
sound recording (see, e.g., Vitushkin, 1974).
The most widely used
technique is the analogue method where the signal is recorded without
any preceding transformations.
However, the signals recorded by this
method suffer distortion due to the defects of recording and reproduction devices.
A more promising method of recording is called the
digital recording technique.
By this method, the signal is first
transformed into a discrete code.
In other words, the signal is
sampled and the samples are coded; then the code of the signal is
recorded, and finally the discrete code is read off the recording and
is transformed into its continuou5 -time form in order to reproduce
the signal.
More precisely, the signal is reconstructed from the
2
(decoded) samples read off the recording.
This technique has been
recently employed with remarkable results.
Clearly, sampling representations and approximations aTe very
significant in communication and information theory, especially in
the era of digital computers.
1.2.
Sampling Representations.
The sampling representation (expansion, theorem)
(1.2.1)
f(t)
=
~ f(--!!..)sin rr(2Wt-n)
n=-oo 2W
l.
rr(2Wt~n)
was originated by E.T. Whittaker (1915).
tElR 1 ,
J.M. Whittaker (1929, 1935),
Kotelnikov (1933), Shannon (1949), and others have 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 W > 0, by
o
W
o
(1.2.2)
f(t) = J e 2rritUp(u)du, tElR 1 ,
-Wo
is called Ll-bandlimited to Wo if FELl[-WO'WoJ, and is called conventionally or L2-bandlimited to Wo if FEL 2[-WO'Wo]' In both cases the
sampling representation (1.2.1) is valid for all W ~ WOo The series
in (1.2.1) converges uniformly on compact sets for Ll-bandlimited
functions, and for conventionally bandlimited functions it converges
in L2(lR 1) as well as uniformly on lR 1.
In reconstructing a function (signal) f from a periodic set of
samples (sampling at a constant rate), errors of the following types
may arise:
3
(1)
f is bandlimited but it is observed only over a finite
interval, and hence only a finite munber of samples can be used for
its reconstruction.
•
This type of error is called a truncation error .
f is bandlimited, but there are observation errors, so
n .but f(2W)
n + En' where {En}
that the observed samples are not f(2W)
(2)
are random variables.
(3)
f is not bandlimited (or not bandlimited to the frequency
it is sampled at), and yet a reconstruction of the type of the
sampling theorem is attempted.
1.3.
Summary.
In this study the samples are assumed to be error free, and only
errors of type (1) and (3) (possibly combined) are considered.
•
Further-
more, the area of inquiry is limited to constant rate (or uniform)
sampling schemes.
It should be mentioned, though, that non-uniform
sampling schemes, as well as random sampling schemes, are of considerable interest in communication and information processing.
Chapter II deals with sampling approximations as well as error
estimates of functions and stochastic processes which are not bandlimited.
In Section 2.2 a sampling approximation is derived for
processes which are not necessarily weakly stationary.
In Section 2.3
the rate of convergence in the finite sampling approximation for time
limited processes is estimated; the convergence holds both in the
mean square sense and with probability one.
Finite sampling approxi.ma-
tions for functions which are Fourier transforms of finite measures
are derived in Section 2.4, along with error estimates under various
conditions.
These results are then extended to various types of
4
stochastic processes.
In Section 2.5 Walsh functions are used to
derive sampling approximations for functions which are not necessarily
continuous and for processes which are not necessarily mean square
continuous, as well as error estimates under further conditions.
In Chapter I I I we turn to the problem of sampling expansions of
bounded linear operators acting on various classes of bandlimited
functions and stochastic processes.
The merit of these representations
lies in the fact that the image of a function under the operator is
expressed or represented in terms of the samples of the function
rather than the samples of its image.
Section 3.2 deals with bounded
linear operators acting on classes of functions bandlimited in the sense
of Zakai (1965) and of Lee (1976a), and Section 3.3 considers bounded
linear operators acting on classes of functions with wandering spectra
in Lloyd's sense (Lloyd, 1959).
Finally, in Chapter IV, distributions and random distributions are
considered.
Section 4.3 deals with sampling representations for dis-
tributions with compact spectra and shows that a distribution with
compact spectrum can be reconstructed using samples of its Fourier
transform (regarded as a function).
In Section 4.4 similar results
are derived for certain types of random distributions.
Notation.
1.4.
The n-dimensional Euclidean space is denoted by IR n , n ~ 1,
,and the complex numbers by C.
The class of all absolutely integrable
or square integrable functions on IR n , with respect to the measure ~
is denoted by Ll(~) or L2(~), and when the measure is Lebesgue
1
n
2
n
measure by L (IR ) or L (lR ).
by AC .
The complement of a set A
.
IS
denoted
Finally, the symbol 0 is used to signal the end of each proof.
•
CHAPTER II
Samp1 ing Approximati on for Non-Band1 imi ted
Functions and Processes
2.1.
Introduction.
In this chapter we consider the problem of deriving sampling
approximations and their rate of convergence for functions and stochastic processes which are not necessarily bandlimited.
The merit of
these approximations lies in the fact that in many practical engineering systems, such as causal systems and time-limited systems, the
•
signals under consideration are not bandlimited.
It is thus of
interest to consider non-bandlimited signals .
In Section 2.2 we derive a sampling approximation for stochastic
processes which are not necessarily stationary or handlimited.
Section 2.3 deals with the rate of convergence in the finite sampling,
approximation for time-limited stochastic processes.
In Section 2.4
a finite sampling approximation is derived for functions which are
the Fourier transforms of finite signed measures, as well as the rate
of convergence.
This result is extended to weakly stationary,
harmonizable, and certain stable processes.
In Section 2.5 a sampling
approximation using Walsh functions is derived for functions which are
not necessarily continuous and for processes which are not necessarily
.,;,
mean-square continuous ..
6
2.2.
Sampling Approxi.mations for Non-Stationary Stochastic Processes.
In this section we prove the stochastic process analogue of the
sampling approximation theorem proved for (deterministic)
limited functions of one variable by Brown (1967).
non~band­
The n-variable
version of Brown's result is stated below as Theorem 2.2.1 •
· d tEll'
Tn 1 and W >0, denote by ew21Titu t he 2W -perlo
. d'lC
For f 1Xe
extension of the function e2TIitu, -W < u ~ W, to the real line.
Fourier series is given in the following well known result.
For any fixed tdR 1 and W > 0
.n
. _ \ eTIl -W uSln
. TI (2Wt-n)
2TIltU
(a.e. (u))
ew - n=-oo
L
TI(2Wt-n)
Lennna 2. 2.1.
00
and the Fourier series converges boundedly.
Theorem 2.2.1.
(Brown,1967).
If FELl (IR n), and f(t) =
Tnn , were
h
tu = tlul+ ... +tnun , and
J F(u)e 2TIitud]Jn(u), t Ell'
IRn
d]Jn (u) = du1 ... du,
n
then for each tErn. n and W ::> 0,
Its
7
where A(W) = {udR n : lu·1 > W; j=l, 2, •.• ,n}.
J
'lhus for each tE:rn. n ,
f(t) = lim fW(t) .
W+oo
If, in addition, for some W > 0, F(u) = 0 for almost all UEIR n with
IUjl
>
W, j=1,2, ... ,n, then we have the classical sampling expansion
f(t) = fW(t) .
The following will be needed in stating the stochastic process
analogue of Theorem (2.2.1). Let {x(t), tE:rn. l } be a second order
mean square continuous stochastic process with correlation function
. 1 2
A·
11
R(t,s) = E[x(t)x(s)]. Assume that REL (IR ) and Rt(o) ~ R(t,o)EL (IR ).
•
The Fourier transform of R is denoted by
R(u,v) =
Jf
R(t,s)e- 2TIi (tu+sv)dtds
U, VE:rn.
1
.
-00
Since R is continuous and Lebesgue integrable on :rn.
2
it is also
Riemann integrable on lR 2, and thus the quadratic mean integral
00
.
y(u) = R J x(t)e-2TIitudt , uEIR l ,
-00
exists and defines a mean square continuous stochastic process (since
E[y(u)y(v)] = R(u,-v)is continuous in both u and v). Since RtELl(IR l )
1
for all tEIR,
its Foutier transform
is well defined for all tEIR l .
In fact, Rt(v) = E[x(t)y(-v)] for all
t,VEIRl, and since both x(t) and y(v) are mean square continuous, then
8
A
1 2
Now we also asstune that REL (m ),
A
Rt(v) is continuous in both t,v.
.
I I I
A.
and R( 0, v)EL
ern. )
for all VE rn. .
I'>
Since Rt(v) is continuous in both
t and v, it follows that
for all t, VE m l . Hence
A
J
A
IRt(v)ld~l(v) ~
for every tE m l , i. e.
ff IR(u,v)ldudv
<
00
,
Rt (v)
is a continuous function in both t, v, and
is integrable with respect to v for every tErn. l .
Remark. It should be noted that if REL l (rn. 2) and R(t,s) ~ o for
.
1
all t,sEIR l , then the condition RtELl(JRl) is satisfied for all tErn. .
Indeed, the two conditions, REL l cm 2) and R is continuous onlR 2 ,
imply that the Riemann integral RfJR(t,s) exists and is finite.
follows that the Riemann mean integral
RfE[~.x(s)]ds
It
exists and is
finite, and
E[F,;oRjx(s)ds]
Taking F,;
=
RfE[F,;ox(s) ]ds .
= x(t) gives that for each tEm l , the Riemann integral
RfR(t,s)ds exists and is finite.
Since R(t,·) ~. 0 for all tEm l ,
it follows that the integral exists as a Lebesgue integral as well,
Le.
jIR(t,s) Ids <
00
for all tErn. l .
We now establish the analogue of Brown's result (Theorem 2.2.1)
for a non-stationary (non-bandlimited) second order stochastic process.
9
Theorem 2.2.2.
Let {x(t), t€JR I } be a second order mean square
continuous stochastic process with correlation function R. Assume
that R€L I (JR2), R(t,o)€LI(JRI) for aU tdR I , and R€L I (IR 2),
All
R(· ,v) €L
ern. )
xw(t): . =
(2.2.1)
1
.
for all v€ ill..
.
Then for each fixed t€ JR
1
and W > 0,
~ x(..B..)sin TI(2Wt-n = Rf e.~.TI. itu y(u)du
l.
2W
n=-oo
TI(2Wt-nw
.
where the equality is a.s., the series converges in quadratic mean,
and y(u) is defined by the quadratic mean integral
y(u) =
RJ
U€ill. l .
e-2TIitux (t)dt,
Also for each t€ill. 1 and W > 0, the error
•
(2.2.2)
2
~(t): = Elx(t)-xw(t) 1 ~ 4
.
.
J
J IR(u,v)ldu
lui >W Ivl >W
.
dv ,
and thus for each t€ ill. 1 ,
x(t) = lim xW(t)
(2.2.3)
W-+oo
is quadratic
A
mean~
If, in addition, R(u,v) = 0 for almost all
lul,lvl > W, for some W > 0, then x(t) =xwCt), which is the classical sampling theorem for non-stationary bandlimited processes.
Proof.
Let us denote
sin TI(2Wt-n)
TI(2Wt-n) . by
~(t;W).
It has already
been noted that the quadratic mean integral y(u) defines a mean
.
square continuous stochastic process with correlation function
E[y(u),y(vJJ = R(u,-v). Also, the quadratic mean integral
RJ~TIitUy(u)du, t€IR l , exists since the integral
10
II
~TIitu ~2TIitvE[Y(U)Y(V)]du dv ,
exists (and is finite) as a Riemann and as a Lebesgue integral (for
R(y, v) is continuous and Lebesgue integrable on
~(t;W):
= EI
I
n=-N
=
N
X(2~)~(t;W) ~ RI ~TIitUY(U)duI2
n
N
I
I
n=-N m=-N
- N
I
n=-N
N
- I
n=-N
:m 2) . Let
m
R(2W' 2W)~(t;W)iin(t;W)
~(t;W)
gn(t;W)
RI
RI
2 "t
~ TIl ~[x(2~)·y(u)]dU
~.
.
.
~TIltU E[x(2~)Y(u)] du
(2.2.4)
We notice that
EIRf ~TIitUy(u)duI2 = Rff ~TIitU ~2TIitvR(u,_v)du dv
= Rif ~TIitu ~TIitv R(u,v) du dv .
A
Since R is continuous and Lebesgue integrable, then by Theorem (2.2.1)
N
N
\'
\' R(2W'
n 2W)~(t;W)~(t;W)->
m
L
L
n=-N m=-N
N-+oo
Since
Rn
II
2"
2·
eWTIltUe TI 1 tv""R(u,v)du dv .
is continuous and integrable, we have
2W
.
N
.
\
' 1 ew-2TIltV
L
gn(t;W)R
n=-N
=
N
I
nE[x(2W)y(vJjdv
2 .
~(t;W)I e- TIltVR (-v)dv
n=-N .
2~
.
•
11
On
N
= JJ [L e
WU
'IT1
27T1°t
gn(t;W)].ew v R(u,v)du dv
A
n=-N
_>
JJ
A
·)du dv,
ew27Titu ew27TitV R u,v
(
where we used Lennna (2.2.1) and the dominated convergence theorem.
Similarly the same is true for the third term in expression (2.2.4),
and
2
thus~(t;W)
+
0 as N
+
00
for all tErn. 1 ,W > 0, proving (2.2.1).
To prove (2.2.2), we have
~(t} = EIx(t) - RJ ~7Tituy(u)duI2
= R(t,t) - RJ ew-27Tit~
. b[x{t)y(u)]du
•
. J ew27Titv
-R
. E[x(t)y(v)]dv
+
(2.2.5)
RJJ
~7Titu ~27Titv R(u,-v)du dv .
But
RJ ~27Tit~[x(t)y(u)]dU = J ~27Titu
Rt(-u)du
= JJ ~27Titu e 27Titv R u,v)du. dv ,
A
(
and similarly for the third term in (2.2.5).
2.
fJ( 27Titu 27Titv
eWCt) =
e
e
+
..
=
(2.2.6)
ew27Titue27Titv
.
Hence
- e
27Titu 27Titv
ew
~7TitU~7TitV) R(u,v)du dv
II.
lul,lvl>w
(e27Titu _~7Titu)(e27Titv _ 27Titv) R(u,v)du dv
.
ew
· ow the lnequa lOt
1 y Ie 27Titu - e 27Titvl <2, we 0 b taln
N
0
•
0
12
2
ew(t)
~ 4
f
f
lui, Ivl >W
IR(u,v)ldu
A
dv ,
proving (2.2.2), and (2.2.3) follows from the fact that R€L 2(m 2).
o
Remark. Theorem 2.2.2 holds formu1tiparameter processes
{x(t), t€m n}, n ~ 1.
Remark.
The bound in (2.2.2) may be written in a different
form as follows.
From (2.2.6) we have
00
=4
The tenn k
=
0
=
I
k,j =-00
j is always zero, so that only the integral of
over the remaining squares enters into the
2 n
n
n
n, ew(2W) = 0 and thus x(2W) = xW(2W)·
2.3.
..
Isin 27TkWtl·lsin 27TjWtl·IR(u,v)ldu dv
SlUll.
IRI
Also, for any integer
The Rate of Convergence in the Finite Sampling Approximation
for Time-Limited Stochastic Processes.
Butzer and Sp1ettstosser (1977) derived a sampling approximation
for time-limited functions which, in fact, is a special case of
Brown's result (Theorem 2.2.1), and determined the rate of convergence
of the approximating series under certain conditions.
These results
are summarized in the following theorem.
Theorem 2.3.1.
(Butzer arid Sp1ettstosser, 1977).
Let f be a
continuous function defined on lR 1 such that, for some (fixed) T > 0,
f(t) =0 for all It I > Tand f€L 1 (R 1). Then
•
13
f(t) = lim NfW) f(..E.) sin 'IT(2Wt~n) ,
W400 n=-N(W) 2W
'IT(2Wt~n)
(2.3.1)
te: m. 1 ,
where N(W) = [2Wf], the largest integer less than or equal to 2Wf.
If, in addition, f(r)e:Lip a, 0 < a
r+l
with constant Lr , then for W> --2--
S
1 for some (fixed) re:{1,2, .•. J,
If(t) - NfW) f(2Wn) sin 'IT(2Wt-n) I
n=-N(W)
'IT (2Wt-n)
(2.3.2)
Here Lip a, 0
•
S
(T+l)L
r-l r
2 (r+a-l)
<
a
S
1
1
r+a-l ' te:m.
W
1, is the Lipschitz class of continuous
functions f on lR 1 for which there exists a constant 0 < L <
that sup If(t+h)-f(t) I s Llhl a , he:lR l .
00
such
te:lR 1
The following result is the analogue of Theorem (2.3.1) for second
order processes which are not necessarily stationary.
duce some notation.
First we intro-
Let C (m. 2) be the class of all continuous functions
on m. 2 , and IIRII = sup
IR(t,s) I. LetLip (2) a, adO,l], be the class
t,se:lR 1
·
..
of all functions Re:C(lR 2) for which there exists a constant 0 < L < 00
such that
where
,gR(t,s) = R(t+h, s+g) - R(t+h,s) - R(t,s+g) + R(t,s).
first modulus of continuity is defined by
~h
WI (o,A;R)
= sup{ II~h,gRII: Ihl so, Igi
where 0> 0, A > O.
S
The
A}
Similarly, the r,.th modulus of continuity of R, r is
a positive integer, is defined by
14
wr(o,A;R) = sup{IIL\~,gRII:
where
L\~ R(t,s) =
r
Ihl~o,lgl~A},
r
L (-l)k+t(~)(~)R(t+kh, s+tg) •
k=O t=O
,g
It should be noted that for Cl.E(O,l] ,
Lip(2)CI. = {REC(IR 2): WI (o,A;R) ~ LOCl.ACI.;o > 0, A >O}.
l
Let x = {x(t), tElR I} be a second order mean ~square
Theorem 2. 3. 2.
continuous stochastic process with correlation function R such that,
for some (fixed) T
1
A
> 0,
2
process) and REL (IR ).
e.~Ct):
w
Then for all tEJR
1
>
T (time limited
and W> 0,
= Elx(t) _ NfW) x(""'!!') sin n(2Wt-n) 1 2
n=-N(W) 2W
n(2Wt-n)
~ 4J
(2.3.3)
R(t,t) = 0 for all It I
J
lui, Ivl >W
IR(u,v) Idu dv ,
•
and thus
N.cW)
•
x(t) = lim
L xC""'!!') SIn n(2Wt-n)
W+oo n=-N(W) 2W
n(2Wt-n)
(2.3.4)
in the mean square sense, where N(W) = [2Wf].
If, in addition, for
some positive integer r and some Cl.E(O,l] ,
with constant Lr , then for every tEJR 1 and W >
~(t)
(2.3.5)
Proof.
(2.2.3).
$
°we have
Lr
r+l 2
1
2
2 (r+CI.-I)(r+CI._I)2 (2T + 2W) W2(r+CI.-I)
(2.3.3) and (2.3.4) are special cases of (2.2.2) and
To show (2.3.5) notice that, for all u,v
r 0,
we have
.
15
)
(-1 ) n+Jru:!(
K u,v =
IJ R(t
n s + 2v
m) e -27fi(tu+sv)dt ds ,
+ 2u'
and thus for all u,v > W,
T
=
J
r r (_l)n+m(r+l)(r+l)
T
r+l r+l
J
-T- r+l -T- r+l n=O m=O
2W
2W
n
m
• R(t + ....!!.
+....!!!.);.. 27fi (tu+sv) dt d .
2u' s 2v e
s
which yields the inequality
•
Z2(r+l) IR(u,v) I <
<
2r
Now since a R(t,s)
ar as r
positive integer j,
€
T
JJ
T
r+l
I~ I
IR(t,s)ldt ds
-T- r+l -T- r+l 2U' 2v
.
. 2W
2W
r+l Z
I
1
(ZT + ZW) wr+I(Zn ' Zv;R) .
Lip(Z)a, it can be easily shown that, for any
. wr+Jo (~'
'R) -< U~r,/\ r w ' (~'
u, /\,
U, 1\ ,
J
2r
R(t,s))
r
at as r
a
and hence for all u,v > W ,
(Z.3.6)
IR(u,v) I ~ Z-2(r+I)L (2T + r+I)Z
I
1.
r
ZW (Zu) r+a (Zv) r+a .
From (Z. 3.3) and (2.3.6) we have
16
e~(t) ~ 4 J
J
lui, Ivl >W
IR(u,v)ldu dv
Lr
r+1 Z
1
= ZZ(Zr+a-1) (r+a-1)Z (ZT + ZW) WZ(r+a-1) ,
o
proving (Z.3.5).
Under the conditions of Theorem ,Z.3.Z the approximating
sequencexW(t), in fact, converges to x(t) with probability one for
. each fixed tE lR 1.·
The rate of convergence is given in the following
corollary.
Let x be as in TheoremZ.3.2 and assume that
Corollary Z.3.l.
r
~
1
when l/Z
<
a
<
and that r
1
~
Z when
a<
a ::; liZ.
Then for a
separable version of x and each tE lR 1,
w6
(Z.3.7)
where
a<y<
NiW)
n
sup x(t) L x(ZW)
W>WO
n=-N(W)
I
r+a - 23 .
Consider a separable version of x. For W ~ 1 put W= -u1
and for each fixed tElR l , define~, uE[O,l], by
Proof.
X(t)
,u
Xu = { xl(t),
=
a<
a
U ::;
1 ,
u
\,N(W)
n sin rr(ZWt-n)
where Xl (t) = xW(t) = Ln=-N(W)x(ZW)
rr(ZWt-n)
. Then X is separable
u
17
(in u).
From (2.3.5), we have
where C = 2-2(r+u-l)C2T+l)2 L (r+u-l)-2 and S
r
= 2(r+u) - 3
> O.
Thus,
by Kolmogorov's theorem (see Neveu (1965, p. 97)),
1
-- sup IXo-xul
hY· O<u<h
+
and (2.3.7) follows by putting h
2.4.
0 a.s. as h
+0
.
,0 < y <
2S '
o
= wI .
o
Finite Sampling Approximations
and Stochastic Processes.
forNon-Bandlimit~d
i
Functions
: .•
Theorem 2.2.1
(the case n = 1) states that, if f is the Fourier
transform of an L1 (IR l )-function, then the infinite sum
I
f(-"!!') sin 7T(2Wt"n)
7T(2Wt-n)
n=-oo 2W
converges to f(t) pointwise everywhere as W+
00.
It is oE practical
interest to investigate the possibility that a finite sum of the
form
Ni W)
(2.4.l)
.
L f(-"!!') sin 7T(2Wt-n)
n=-N(W)
2W
7T(2Wt-n)
.
where N(W) is a positive integer-valued function of W > 0, would
converge to f(t) under suitable conditions on N(W), and also to
determine the speed of convergence.
Such a result is obtained in
Theorem 2.4.3·, and its analogue for an appropriate modification
of the finite sum (2.4.1) is obtained in Theorems
2.4.1 and 2.4.2 ,
18
and then extended to weakly stationary, hannonizable, and certain
stable processes.
In all these results N(W) is required to tend to
infinity, as the sampling rate Wtends to infinity, fast enough so
that N~W)
+
00.
Theorem 2.4.1.
If f is the Fourier transfonn of a finite
signed (or complex) measure
~
on the Borel sets of the real line,
i. e, ,
.00
(2.4.2)
f(t) =
•
J e2rrltud~(u)
, t€IR l ,
-00
and if N(W) is a positive integer valued function of W> 0 such that
N(W) + as W+
then for each t€IR l
W
'
00
00
and the convergence is unifonn on compact sets.
Proof.
Consider the error
I
ew(t) : = f(t) -
NfW)
~
sin rr(2Wt-n) I
L (1 - N(~+1)f(2~) rr(2Wt-n)
n=-N(W)
e2rrltud~(u)
NiW)
00.
= I_L
-
n=~N(W) (1
Inl. -
-
rri!!. u
W d~(u) J
mWr+r) 1_ ~oo e
00
• sinrr(2Wt-n)
rr(2Wt-n)
~
J
00
-00
I
12'
I"'n'
rri-wnu
e rrltu - NfW) (1 ~)
N(W)+l
e
I
n=-N(W)
(2.4.4)
• sin rr(2Wt-n)
rr(2Wt-n)
I dl ~ I(u)
19
v-u
. W"
. N(W) k
-2rrin(-)
e rrltu 1
\.
\
2W ]d
2W
e 2rrltv[NeW)
+1 l
l e v
-W
k=O n=-k
001 2 .
=f
-00
-If
Idl J-l
1(.)
u
-_ fool e 2rritu -00
.
1
U
-'2--
e
-
-
-
I
fu2We 4rriWtx~(W) (x)dx d IJ-l I (u)
-~--
2W
h
were
\k
-2rrinx_ 1 [sin rr(N+1)x] 2 l"sthe
- ( ) - 1 \N
K
N x - N+1 lk=O In=",ke
- N+1sin rrx
Fejer kernel.
Since IJ-l/(JR 1) <
00
(see Rudin (1974) , p. -126J, given
E > 0, there exists an a = a(E)E(O,OO) such that if A = [-a,a], then
- IJ-li (Ac) <
!.
Since K ~ 0 is periodic with period 1 and
N
1:
J~(x)dx
= 1, (2.4.4) can be written as
-~
where
1
U
-'2--
=
sup
lul::;a
/1
2W 4rriWtx
fe
1
u
--'2--
2W
KN(W) (x)dx
I
20
By writing
~-y
J
-~-y
-~
~
= J
-~-y
QN(t,W) ~ ·1 -
I
~-y
J
+
+
-~
L '
(y = 2W) , we obtain
2
Jl.:~e47TiWtx KN(W) (X)dxl
- 2
l.:+~
2
4W
+ sup
I J.
lul~a l.:-~
e47TiWtx KN(W) (x) dx
I
4W
2
_~+Iul-u
4W 47TiWtx
+ sup J
e
KN(W) (x)dx
lui ~a _l.:Jul +u
2
4W
I
(2.4.7)
I
Let us denote by Ik(t,W), k = 1,2,3, the three terms on the right
hand side of (2.4.7) in the order they appear.
For IZ(t,W) we have
7T 2 is a constant
Z C Z, where C(z 4)
7T (N+1)x
(see Zygmund (1959), p. 90) and thus for W> a
But for 0
<
x
~ ~
, KN(x)
~
1
( W)
IZ t ,
---=--=-2C~_
- 7T 2[N (W) +1]
<
'2
1
J 2" dx
l.:-.....§!. X
2 2W
.
e
21
2C
2a
_. 2
(W-a)
TI [N(W) +1]
+
a
asW
+
00
a
as W +
00
•
S:irnilary ,
I (t W) <
. 2C
2a +
3 '
- TI 2 [N(W) +1] (W-a)
.
Now for II (t,W) , uSIng the fact that
II (t,W) = 2
f
~
a
fk:2K
a
N(x)dx
1
= 2'
we have
(1 - cos 4TIWtx)~(W)(x)dx •
a<
Choosing any 0(W) such· that
o(W) <
~
and o(W)
= a(NIW)) ,
we
obtain
.
I (t,W)
1
.
e
=
2
(O(W)
f
a
Since for all x, KN(x)
~
+
f~
0(W)
)
(1 - cos 4'lTWtx)KN(W) (x)dx •
.
.
2N+1 (see Zygmund (1959, p. 90), then
o(W)
f
a
(1 - cos 4TIWtx)~(W)(x)dx ~ 2[2N(W)+1]0(W)
+
a
as W +
Also
f
~ ...
000
.~.
(1 - cos 4TIWtx)KN(W) (x)dx
.
r
000
=2
2
sin 2TIWtx • ~(W) (x)dx
Notice that this is the only bound which depends on t and that the
dependence is linear in
It I.
I t follows that for W > a,
00.
22
(2.4.8)
QN(t,W)
<
8aC
+ 2[2N(W)+1] 0 (W) + 2CI tjW
- 1T2[N(W)+1](W-a)
N(W)+l '
and thus
~(t,W)
-)- 0 as W-)-
00.
Hence, for each fixed t and e: > 0,
we have by (2.4.5) that lim sup ew(t)
.
as W-)-
S E
which implies that ew(t) -)- 0
W+oo
It is clear from (2.4.8) that the convergence is uniform
00.
o
on compact sets.
The following theorem gives a more concrete bound on the error
committed by considering only a finite number of samples inreconstructing a function which has the representation (2.4.2).
Theorem 2.4.2.
(2.4.1).
It I
<
Let
f,~,W,N(W),
and ew(t) be as in Theorem
Then for an arbitrary (but fixed) W > 1 and every
N(W).
2W"
WI th t
.J.
T
n nE IN -- {0,
2W'
.!. 1, .!. 2,... } we have
(2.4.9)
Proof.
Fix W > 1 and t ~ 2~ , nE:N.
From (2.4.4) we have
00
(2.4.10)
ew(t) s
J l~w(t,u)ldl~l(u),
_00
,
where
(2.4.11)
~,W(t,u) = e21Titu
N{W)
L .. (1
n=-N(W)
~ 7Ti ~u
+ N(W +l)e
.
.
• sin 1T(2Wt-n)
7T(2Wt-n)
. If A(W) = [-W+1, W-1], then the inequality (2.4.10) may be written
as
23
(2.4.12)
ew(t) ~ ( f + f )]HN W(t,u) Idl~1 (u) .
A(W) AC(W)
,
. that IHN W(t,u) 1= I1 - f~- (u/2W) e 4TIiWtx~(W)(x)dx I ,
By (2.4.4) we notIce
-~-(u/2W)
,
and thus IHN,w(t,u) I
1
1
2 for all tE:rn. , UE:rn. , N ~ 1, and W> O.
~
From (2.4.12) we obtain
ew(t) ~
(2.4.13)
J
A(W)
I~ W(t,u) IdlJJI (u) + 21JJI (Ac(W))
,
= sup I~ WCt,u)I
uEA(W)
,
Now consider the integral
.
e
I
N,W'
2TIi
(1 _ 2WI z I )
f
(t u)= _1_
N(W)+l
. .
CN,W
e
2TIiuz
(z~t)sin
where CN,W = {zd:: I z I = 2~[N(W)+1]}.
2TIWz
dz
'
_ ~ e2TIitu
= (1 N(W)+l) sin 2TIWt
NiW)
+
L
n=-N(W)
(1 -
.-lE.L-)
.n
TIl Wu
e
N(W)+l (2~ -t)
and thus from (2.4.11)
_ ~ 2TIitu
.
. •
HN,w(t,u) - N(W)+l e
+ SIn 2TIWt IN,W(t,u),
Hence
N~~
From Cauchy's residue theorem,
we have
(2.4.14)
It I <
24
I~,W(t,u) I
(2.4.15)
2WI t I +
::;; N(W)+l
'
ISIn
·
. an entIre
.
f unc t'Ion
S1nce
e 27Tiuz IS
27TWt I •
I IN,w(t,u) I ,
0f·
exponent'la1
It I < N(W)
---zw- .
t ype 27T Iu I , t h en
from a result by Piranashvili (1967) we have for ue:A(W) that,
and fran (2.4.15), that for W > 1
If-L
-~,W
(t,u)
I :;
ri.n-
2Wltl +
2Wlsin 27TWtL
N(W)
7T(1-e-7T)[N(W)]2 (W-,uIJ
and hence for W > 1 and
I ~~
r It I < N~W
,ue:A(W), It I < N(wW)
2
'
P (t,W) ::;; 2Wltl + ~
N
~ 7T(l-e- 7T ) [N(W)]
(2.4.16)
.
e
Thus from (2.4.13), we obtain
for W > 1 and Inl
2W
Remark 2.4.1,
r
It I < N~W)
W •
o
ne::N •
We now connnent on the two bounds obtained in
Theorems 2.4.1 and 2.4.2 . From (2.4.5) and (2.4.8) in the proof of
Theorem
2.4.1
ew(t)
~
we have,
2IlJl{lul>a} + 21lJI
(IR1H~
.
+ [N (W) +1] 0 (W) }
+ ---:2:-._4....:..C_a
_
7T [N(W)+l](W-a)
25
1·,·
N(W)
4C
N(W)
a
= 2 I]1 I{Iu I>aJ + 2 I]11 (lR HC tIN(W)+l + 1T2 N(W)+l ·W(W~a)
+ N(W) [N (W) +11 0 (W) J""!"W
N(W)
=: Bl (t,W)
for all W > a and tE:rn. 1.
fixed W > 1 and t with
From Theorem (2.4.2) we have that for each
1~1 t
It I
<
N~~
,
=: B (t,W) .
2
2
1T
In the bomdB l , C could be taken to be if'""" and 0 (W) may be chosen as
small as desirable so that the fourth term in the expression ofB
-
e
l
may be omitted (since it can be made arbitrarily smaller than the
other three terms by an appropriate choice of 0(W)).
The bound Bl
1
has the advantage over the bound B2 of holding for all. tE lR . It
should be noted that even though for each fixed t, Bl(t,W)
+
Oas
.
I 1 N(W)+l ' ..
for t' s large relatlve to W, namely for It> 2C W
the bound B (through its second term) is larger than /]11 (lR l ) which
l
W +00
,
is an upper bound on
If(t) I.
Large as this bound on the error may
seem, there is no smaller bound available for such t's and, in fact,
it seems quite likely that at least for certain t's the approximation
error would be of the magnitude of
on the interval /t
I
If(t) I.
The bound B2 holds only
< N~~ excluding the sample points 2&
(at which
we know anyway that the approximation error ew(t) vanishes).
it has the following advantages over the bound B .
l
However,
It contains one
term which vanishes at the sampling points and is thus very small near
26
the sampling points; thus the bOlmd B2 is more sensitivethan BI
near sampling points, Also, the second tenn of B2 is always smaller
than the second term of BI , and putting a • W-I to equalize the
first tenn of both bounds, it is easily seen that the third tenn of
B2 is always smaller than the third tenn of BI , In conclusion, in
the restricted region where it holds the bound B2 is lower and more
sensitive to the location of the sampling points than the bound BI ,
while, of course, BI provides a bound even where BI does not hold,
A typical plot of the two bounds for fixed W > I (and a = W-I)
is as follows,
-
e
o
I
2
:3
-2W
-2W
-2W
•••
N (W)-I
N(W)
2W
2W
• ••
t
27
. Under further conditions a, function of the fonn (2.4.2) can be
approximated by the simpler finite sums in (2.4.1) rather than those
in (2.4.3).
Theorem 2.4.3.
Let f be the Fourier transfonn of a finite signed
(or complex) measure
on the Borel sets of the real line, which
~,
satisfies
00
f
(2.4.17)
~(u)dl~1
(u) <
00
,
-00
~
where
(0,00),
is a symmetric
and such that
non~negative
~{u) -l- 00
as u
integer valued function of W>
W > 1, and It I
<
N~~)
e..,(t) = f(t) .,
'I
-w
o.
function, strictly increasing on
-l- 00.
Let N(W) be a positive
Then for an arbitrary (but fixed)
such that t f 2~ , nE:IN, we have
NfW) f(2Wn) sin 7T(2Wt-n)
n=-N(W)
7T(2Wt-n)
(2.4.1$)
where ¢l(W) =
f~(u)dl~l(u).
.
.. lu/>W-1
( i)
(ii)
N(W)
W
-l-OO
as W-l-
00
'
If, in addition,
and
N(W) = o(e ctPCW - 1)) for sane c > 0 ,
then for alltdR 1, B(t,W)
-l-
0 and ew(t)
-+
0 as W-l-
00.
28
Proof.
r 2Wn
Fix W> 1 and t
' ndN. As in (2.4.4) we obtain
00
f
ew(t) ~
I~ W(t,u)Jdl~l(u)
-00
'
= (f
f ) I~ W(t,u) Idllli (u) ,
+
A(W)
AC(W}
,
where A(W) = [-W+1, W-1] and
(2.4.20)
~,W(t,u)
= e 2'ITitu
Also as in the proof of Theorem (2.4.1), we notice that
~- !:!....
2W 4'ITiWtx
I
I~,w(~'u) I ~ 1 - f u e
DN(W) (x)dx,
I
-~-2W
f IGN w(t,u)ldl~l(u)
AC(W)
,
~ 2L1+tn[N(W)+~]
(2.4.21)
cj> (W-1)
Now, for uEA(W) and 1~1
r
It I
<
f cj>(u)dlll/CU) .
AC(W)
N~W ' consider the integral
1
e 2'ITiuz
IN,W(t,u) = 2'ITi C f (z-t)sin .2'ITWz dz
N,W
where ~,W = {ZE~: Izi = 2~ [N(W)+~]}.
Proceeding as in the proof
29
of Theorem (2.4.2), we obtain
I~,w(t,u) I
.
= jsin 2~Wtl • IIN,W(t,U)I
"
il sin 2~Wtl_ •
$
~(l-e-~)N(W)
4 sin
<
(2.4.22)
-
~(l-e
2~Wt
-~
)
W, uEA(W) ,
W-TUT
2W
WIn]
, 2W 'f I t I
N(W)
and (2.4.8) "follows from (2.4.1) and (2.4.22).
conditions (i) and (ii) that B(t,W)
"1
fix tErn.
t
==
"
and let W-+
00.
-+
0 as W-+
ew(t)
~
As W-+
B(t,W)
-+
O.
Remark 2.4.2.
00
<
2W
N(W)
---zw-
It is clear from
00
for all tE:rn. 1 .
Now
Whenever Wis such that N(W) > 2Wltl and
2~ for some ndN, then we clearly have ew(t)
definition.
lEi 'f Itl < N(W)
= 0 from its very
along any other values (t 'f 2~ , nElN), then
It follows that ew(t)
-+
0 as W-+
00.
0
It is clear that the approximation of f(t) by a
fini te sum of the form (2.4.1) is a much more delicate problem than
its approximation by the modified finite sum of the form (2A.3).
Hence the additional assurnptionson the function f required in
Theorem 2.4. 3,
compared with Theorem .2 • 4.1.
As in Theorem 2.4.1, "
condition (i) of Theorem 2.4.3 puts a lower bound on the growth of
N(W) as the sampling rate tends to infinity. Such a condition is
quite natural and anticipated, as one intuitively expects that unless
enough terms are employed, the approximation may be inadequate.
Condition (ii) on the other hand puts an upper bound on the growth
of N(W) and in this sense it may seem somewhat counterintuitive.
Condition (ii) could be improved, I.e. the restriction on the growth
of N(W) could be weakened, if a better bound than that used in the
30
proof of Theorem 2. 4. 3· could be fOlDld for the flDlction
( U) = e
~ Wt,
,
2nitu _ -.!. fW 2nitvTL (u-v) d
2W
e
-N(W)· 2W v
-W
as a flDlction of Wfor fixed t,uelR l (or after sane algebra, if the
rate of growth of Ifn/2Du(x)dxl as u ~
o
00
can be fOlDld, instead of
the rate of growth of fn/2IDu(x)ldx which is used in the proof of
o
.
Theorem 2.4.3, where Du is the Kirichlet kernel Du(x) = s~~n~) .
It is not known at present whether some restriction on the growth
of N(W) is necessary for the approximation error to tend to zero as
W~
00,
or whether this result hoiLds with no upper limit on the growth
of N(W) as one may intuitively be tempted to expect, and condition
(ii) arises only because of the specific proof used here.
Also, it
is not known at present whether a bOlDld similar to Bl (see Remark
2.4.1) holds, in this case, for all teIR l .
The preceding results are now extended to weakly stationary
stochastic processes (Theorem 2.4.4),
(Theorem 2.4.5),
Theorem 2.4.4.
harmonizable processes
and to certain stable processes (Theorem 2.4.6).
If x = {x(t) , teIR l } is a mean square continuous
weakly stationary process, and if N(W) is a positive integer-valued
ftmction of W> 0 such that N~W) ~
(2.4.23)
00
as W.~
00,
then for each telR1
_ . N{W)
Inl. n sin n(2Wt-n)
x(t) - 11m L
(1 - NLW:fiT)x(2W)
n(2Wt.,.n)
,
W~ n=-N(W)
where the convergence is in the mean square sense lDlifonnly on compact
sets.
Furthennore, for any fixed W > 1 and It I < N~~
t ~ 2~ , ne IN, we have
such that
•
(2.4.24)
and
(2.4.25)
where ep and
¢
are as defined in Theorem 2.4.3.
N(W) and the spectral measure
~
If, in addition,
of the process satisfy the condition
in Theorem .2 . 4. 3, then for all tE lR 1
-
e
x(t) = lim NfW) x(~)sinTI(2Wt-n) ,
W-+oo n=-N(W) 2W 7T(2Wt-n)
(2.4.26)
is mean square.
·Proof.
Since x is mean square continuous weakly stationary, we
have for all tE lR 1
R(t) = E[x(t)XlOJ] =
fooe27TitAd~(A) ,
-00
and
-00
where
(the spectral measure of x) is a finite measure defined on
(lR 1,B(lR I)}, {Z (A), AElR I } is a process with orthogonal increments,
and for all -00 < a ~ b < 00, EIZ(b) - Z(a)1 2 = ~{(a,b]} .
~
Consider the mean square error
2 . _I
ew(t). - E x(t) -
~
_l.
n--N
~. n sin 7T(2Wt-n)\2
(1 - N(W)+1)x(2W) 7T(2Wt-n) .
32
I
N.{W)
7Ti n A
2
(1 _Inl ·)e W sin 7T(2Wt-n)] dZ (A)
n=-N(W)
NlWJ-iT
7T(2Wt-n)
_ L
~
J
00
NiW)
7Ti n A .
12
L (1 - ~
. ,n )e W SIn 7T(2Wt-n)] d]J(A)
n=-N(W)
N +1
7T(2Wt-n)· I
1
A
'2-2W 4 'Wt
/2
JA
e 7TI Ur~(W)(u)du d]J(A)
•
le27TltA -
-00
(2.4.27)
_L
00
=
1
1
-!.:-2
2W
Since this expression is similar to (2.4.4), (2.4.23) follows as in
the proof of Theorem 2.4.1.
The proofs of (2.4.24) and (2.4.25),
(2.4.26) are similar to those of Theorems
2.4.2 and 2.4.3 respect-
o
ively, and hence they are omitted.
A second order stochastic process x = {x(t) , tElR 1} is called a
harmonizable process if its correlation function R(t,s) = E[x(t)x(s)]
-
e
is of the form
oo
R(t,s)
=
Joo J e 27Ti (tu-sv)dJl.(U,V) , t,sE:m 1 ,
-00
-00
where]J (the spectral measure of x) is a complex measure on (IR 2 ,B(lR 2)).
Theorem 2.4.5.
If x = {x(t), tElR l } is a harmonizab1e process,
. N(W)
1
and if W ~ 00 as W~ 00, then for each tE lR (2.4.23) holds, where
the convergence is in the mean square sense uniformly on compact sets;
Furthermore, under the conditions of Theorems
2.4.2 and 2.4.3"
bounds on the mean square error similar to (2.4.24) and (2.4.25) hold
as well as the approximation (2.4.26).
Proof.
2
ew(t):
Consider the'mean square error
=
I
E x(t) -
N{W)
L (1 n=-N(W)
Inl·
n
NtWJ+'
P x(ZW)
sin 7T(2Wt-n) /2
7T(2Wt-n)
33
=
J J
00
-00
00
-00
2·t
N{W)
~ 7Ti
[e 7Tl U - L
(1 - N +l)e
n=-N (W)
• [e
•
- 27Titv
N{W)
In I
-7Ti
L
(1 - ~N(W)+l)e
n=-N(W)
-
1
U
'2--
I
WU sln~'
.
(2Wt)
~n]
2W 4 ·Wt
U e 7Tl U KN(W) (u)du
7T(2 t-n)
WV
.
(2Wt)
Sln 7T
-n ]dll(U v)
7T(2Wt-n)
.
,
I
-~-2W
V
1
'2--
• I11 -
I
I
2W
v e 41TiWtv KN(W) (v) dv d'1 11 I (u ,v)
-~-2W
By the familiar technique used in the proof of Theorem 2.4.1, we
have
where A = [-a,a], a
>
0, and the proof is completed as in Theorems
o
2.4.1 to 2.4.3.
We finally consider certain harmonizab1e, but non-stationary, stable
processes.
A random variable X is synnnetric a-stable (SaS), 0 < a < 2,
if its characteristic function is of the form E(e itX) :z:: expC-bx1tla)
for some positive constant bX. A stochastic process x = {x(t), t€IR l }
is called Sas, if every finite linear combination of its random
variables is SaS.
The following can be found in Shilder (1970).
If
X is a Sas random variable, then, for 1 < a < 2, the map X 1-> bi/a
defines a norm on a linear space of Sas random variables:
If the process {Z(A)):
A~O}
I IXI 10. = bi/a .
is SaS with independent increments, then
the function F defined on [0,00) by F(A) = IIZ(A)ll a , A ~ 0, is nona
34
decreasing and thus defines a Lebesgue-Stieltjes measure Pp on
the Borel sets of [0,(0). If the family of functions {ft(o), tEIR l }
belongs to L2(Pp), then the integral
00
x(t) =
f
o
•
ft(A)dZ(A) , tEIR l ,
.
1
defines an SaS process and for every tE lR ,
00
Ilx(t) II~ =
Theorem 2.4.6.
b Ift(A) ,adpp(A)
Let a stochastic process x be defined by
00
x(t) =
where {Z(A) ,
A~O}
J cos 2TItA dZ(A) , tEIR l ,
o
is an Sas process with independent increments and
finite measure Pp' and 1 < a < 2. If N~W) ~ 00 as W~ 00, then for
every tEIR l , (2.4.23) holds, where the convergence is in the
II II a-nonn.
Furthennore, under the conditions of Theorems
and 2.4.3,
boWlds
on the
.
0
11 11
0
2.4. 2
a -nonn error similar to (2.4.24)
and (2.4.25) hold as well as the approximation given in (2.4.26).
Proof.
Consider the a-mean error
~(t): =
II x (t )
-
NfW)~.
. TI (2Wt)
n
( n) Sln
-n II a
(1- N i of, )x 2W TI(2Wt-n)
a
n=-N(W)
.
00
=
f If N W(t,A) IadPF(A)
o
'
As in (2.4.4), we have
, say .
35
1
A
'2--.
fN,W(t,A)
=
cos 2'ITtA -
J
2W
A
cos 2'ITt(2Wy+A)~(W)(Y)dy
-~-2W
1
A
'2--
= Re [ e 2'ITitA - J 2WA e2'ITit(2Wy+A)~. (W). (y)dy]
-~-2W
and the proof is completed as in Theorems 2.4.1 to 2.4.3.
0
The same results hold for $aS processes
00
x(t) =
00
J cos 2'ITtA d2 1 (A) + J sin 2'ITtA d2 Z(A) , tEffi 1 ,
o
0
where 21 and 22 are independent processes as in Theorem 2.4.6.
These processes are the SaS (non-stationary) analogues (1 < a < 2)
-
e
of the real stationary Gaussian (a = 2) processes.
2.5.
Sampling Approximation Using Walsh Functions.
Recently, Walsh functions have been increasingly used in digital
communication systems: they are easily generated by semiconductor
devices, their pulse shape (1,-1) conforms with operations of digital
computers, and they play, for discontinuous signals, the role complex
exponential functions (and Fourier transforms) play for continuous
signals.
In addition, they have been used in experimental sequency-
multiplex systems, image coding and enhancement, general two-dimensiona1 filtering, etc.
In this section, using Walsh functions,we derive a sampling
approximation and error estimates for functions which are not necessari1y continuous, and for stochastic processes which are not necessarily
mean square continuous.
These results are the Walsh-analogues for
36
W-continuous functions of the Fourier results of Sections 2.2 and
2.3 for continuous functions.
Results similar to those of Section
2.4 should be feasible for W-continuous functions, but complete proofs
have not as yet been obtained.
The following notation and definitions will be used in the
sequel.
ffi. + = [0,00), :IN is the set of all integers, :IN + is the set
of all non-negative integers, and D+ is the set of all
dyadic rationals.
Each t >
non~negative
°has the dyadic expansion
~I.
t.2 -j ,t.dO,l }
j=-N(t) J
J
for all j, where N(t) is such that 2N(t) ~ t < 2N(t)+1, and we put
(2.5.1)
tj =
t =
°for j
<
-Net).
If tED+, there are two expansions and the
finite one is chosen so that expansion (2.5.1) becomes unique.
The
componentwise addition modulo 2 (dyadic addition) of t,SEffi.+ is defined
by t ~ s = \~
It.-s·12- j .
I.J=-oo
J
J
The Walsh functions can be defined in several ways.
The following
. definition is based on the system of Rademacher functions {~(t) }ndN '
+
tE[O,l), where
Ro(t) .= e7Ti [2t]
= RO(2n t)
~ (t)
, n ~ 1 ,
and by
7Tit
R(t) = e
-n
n+1 ' nE+
:IN ,tE [0 , 1)
...
The set of Walsh functions {¢ (t)}
n
nE
:IN
+
on [0,1) is defined for each
non-negative integer n = I~=_N(n)nj2-j by
(2.5.2)
N(n)
n .
N(n)+l
¢n(t) = IT (R.(t)) -J = exp{7Ti I nl_·t.},
j=l J
j=l
J J
37
-and is orthononnal and complete inL 2[0,1).
The Walsh functions are
extended to {\jJ (t)} t IR by
u
u, e +'
Niu) +1
\jJt(u) =\jJ (t) = exp{7Ti LUI_Jot.}, ~,ueIR+ '
_
u
j=-N(t) '. J
..
and they have the property that, for all ueIR+, whenever t e s ¢ D+,
A function f on IR + is called W-continuous if f is continuous on
IR+\D+ and right continuous on D+.
It is clear that the Walsh func'"
1
tions are W-continuous.
-
If feL (IR+) , then the Walsh-Fourier transfonn
(WFf) fW of f is defined by
00
-
e
(2.5.3)
fW(u) =
J f(t)\jJu(t)dt,
o
and fW is bounded and W-continuous.
ueIR+,
If f, fWeLl(IR+) and f is W-
continuous, then the WFT can be inverted to give
.
(2.5.4)
f(t) =
00
W
J f (u)\jJt(u)du, teIR+.
o
(See Butzer and Splettstosser, 1978.)
The Walsh modulus of continuity
of a function fe L1 (IR +) is defined by.
For a > 0 and a constant L > 0, the Lipschitz class LiPLa is defined
by
A function feLl(IR+) is ~aid to be dyadic differentiable if there
38
g is called the first strong dyadic derivative of f, and is denoted
by D[l]f.
For r'> 1, D[r]f is defined iteratively by D[r]f =
D[l] (D[r-l]f).
-"
If f and D[rJ f belong to Ll(IR+), there exists a
constant M such that w(f;o) ~ Morw(D[rJf;o), 0 >
o.
A complex function f on IR+ of the fonn
f(t) = f
(2.5.5)
2n
o
F(u)1)Jt(u)du, tEIR+
for some nElN and some FEL1 (0,2n ), is called sequency limited
to 2n . A sequency 1m ted function f which is W-continuous and in
L1 (IR +) has a sampling expansion of the fonn:
(2.5.6)
tEIR+
.
-
e
v
whereJ(v;t) = J 1)Jt(u)du, t,VEIR+ (Fine, 1950). As it was pointed out
o
by Kak (1970) and Butzer and Splettstosser (1978),
and thus (under the stated conditions) the functions that are sequency
limited to 2n are precisely the functions that are constant on each
interval [2- nk, 2- n (k+l)), a rather small class (Unlike the class of
baridlimited functions).
A (dyadic) sampling approximation for time-limited functions
(which are not necessarily continuous) was derived by Butzer and
Splettstosser (1978):
.
39
Theorem 2.5.1.
..
(Butzer and Sp1ettstosser, 1978).
Let f be a
W-continuous function on ffi.+ such that f(t) = 0 for alIt
some T
>
(2.5.7)
0, and f, f W€L 1 (lR +).
f(t) = lim
n+oo
where N(n) = [2Dr].
Nfn)
k=O
~
T, for
Then
f( ~) 1 ~n ~n
(t) , tdR+
2 [2 k,2 (k+l))
If, in addition, either (i) D[r]f exists and
D[r]f€LiPL/Nf or (ii) f€LipL(a+r), for same fixed a > 0 and r€{1,2, •.. };
then
(2.5.8)
NW
sup f(t) - L f(.-1)l.
(t)
t€ffi.+
k=O 2n [2-nk,2~n(k+l))
I s L2
~a
r+a~
1 2",n(r+a-l) •
--
We now derive a sampling approximation as well as error estimates
..
e
for functions which are the WFT of finite (or complex) measures and
thus not necessarily time-limited nor sequency limited.
Theorem 2.5.2.
~
If f is the WFT of a finite (or complex) measure
on the Borel sets of ffi.+, i.e.
00
f(t) = !1JJt (u)dp(u) , t€ffi.+
o
then for every t€ IR+ and n€:IN ,
00
(2.5.9)
fn(t) =
L f( ~)l
k=O
2
where for each (fixed) t€IR+,
00
-n -n
(t) = !
[2 k,2 (k+1))
0
1JJ~n](u) is the 2n -periodic extension
of the function 1JJ t (u), 0 s u < 2n to ffi.+,and
(2.5.10)
Thus for each t€ ffi. +'
1JJ~n](u)d~(u) ,
40
(2.5.11)
If, in addition,
~
is absolutely continuous with respect to Lebesgue
measure, and f€LipL(a+r) for some a > 0, L > 0, and r€{1,2, ... },
then for every t€ lR +
(2.5.12)
If(t) - fn(t)
I
L2 r +a - 1 -n(r+a-1)
~ r+a-1
2
.
Since ~in]€L1[0,2n) is periodic with period 2n , W-continuous, and of bounded variation on [0,2n), then the partial sums of its
Proof.
Walsh Fourier series converge everywhere to~in] (see Chrestenson,
1955, Theorem 2), i.e. for each t€IR+
00
(2.5.13)
~ [n] (u)
t
where
2n
n
an, k(t) = 2- f I/Jt(v)~k(2-Ily)dv
0
= La
k=O n,
k(t)~k(2-nu) , U€IR+
=
1
f ~t(2Ily)~k(v)dv
0
=
1
f ~v(2nt)~v(k)dv
0
=
1
f ~v(2nt e k)dv
0
= J (1 ; 2nt e k)
Let
•
41
Using (2.5.13) and the fact that for each nEIN and tEIR+,
for all UE IR + and K > 1, it follows that
EK(t;n)
«
-+-
0 as K -+-
proving (2.5.9).
for each tEIR+ and ndN ,
00,
Now
00
If(t)
- fn(t)1
=
00
If 1JJ t (u)dll(U)
-
o
J 1JJ~n] (U)dll(U) I
0
00
~ J l1JJ t (u) - 1JJ~n] (u) Idllll(u)
n
2
hence (2.5.10) and (2.5.11).
To prove (2.5.12), notice that if
d~~u) = F(u) and fELipL(r+a), then
(2.5.14)
IF(u)
I
~
r+a-1u -r-a ,u > 0 ,
L 2
(see, Butzer and Sp1ettstrosser (1978)).
From (2.5.10) and (2.5.14)
we have
~ L 2r +a - 1
00
J
1_ d
r+
u
n u a
2
r+a-1
_L2
2-n(r+a-1)
- (r+a-1)
.
o
42
We end this section by extending the results in Theorem 2.5.2
to stochastic processes which are not necessarily continuous.
first introduce the following notation.
A function R on
We
:JR.: =
[O,oo)x[O,oo) is called W2-continuous, if R is continuous on lR:\D:
and continuous from above on D: (in the sense of Neuhaus, 1971). If
2
REL 1 (IR+),
the first modulus of continuity is defined by
w(o,A;R) = sup{IILlh Rill
2' 0 ~ h < 0, 0 ~ g < A}, 0,1. >0,
,g L (IR+)
where Llh,gR(t,s) = R(t~h,
s~g)
-
R(t~h,s)
~
R(t,
s~g)
+ R(t,s). Also
the class LiP~2)a is defined by
The WrT of a function REL 1 (IR:) is defined by
w
00
R (u,v) =
00
J J R(t,s)ljJu(t)1/Jv(s)dt
o 0
ds , U,VEIR+ .
2
Finally, if R is W2-continuous and R,RWEL 1 (IR+),
then (as in (2.5.4)
(2.5.15)
R(t,s)
Theorem 2.5.3.
Let {x(t) , tElR+} be a second order stochastic
Assume that R is W2-continuous,
W 1
2
1
R,R EL (IR+), R(t, 0) L (IR+) for all tE]R+, and RW(o,v)EL 1 (IR+) for
process with correlation function R.
all VE]R + •
Then for each tE IR 1, nElN
00
(2.5.16)
xn(t):
k
= L x(ln)l -n
k=O
2
[2 k,2
00
-n
(k+1))
(t)
= RJ ljJIn](u)y(u)du,
0
where the equality is a.s., the series converges in quadratic mean,
and y is defined by the quadratic mean integral
43
00
y(u)
Rf
=
o
~
U
(t)x(t)dt , U€IR+ .
Also, for each te: IR + and ne: IN ,
~ 4
(2.5.17)
00 00
f 1. IRW(u,v) Idu
n
2
dv
n
2
and thus for each te: IR +
x(t) = lim xn(t)
n-+oo
(2.5.18)
If, in addition, R€LiPi 2)(r+a), for some a > 0
is quadratic mean.
and re:{1,2, ... }, then
(2.5.19)
-
e
Proof.
The proofs of (2.5.16), (2.5.17), and (2.5.18) are
similar to those in Theorem 2.2.2 and hence omitted.
To show
(2.5.19), notice that for any u > 0 we have the dyadic expansions
00
u =
u.2- j
I
j=-N(u)
J
and thus
-1
~u(u
) = exp{TIi
Nf u ) _ l
L
j=-N(u)
(u) . uj } = ·1 ,
J
(see Butzer and Sp1ettstosser, 1978, p. 102).
From (2.5.15), we
have that for any u > 0
00
=
00
f f R(t~u-1,s)~u(t)~v(s)dt
o0
Similarly for any v
>
0
ds , ve:IR+ ~
44
and for u,v > 0
00
RW(u,v) = J
f
00
o0
Hence for u,v
R(t$u- 1 , S$V-1)~u(t)~v(s)dt ds .
0
>
W
R (u, v) = l
00
ff
00
4 00
[R(teu- 1 , s~v-1) - R(t~u-1,s)
- R(t,s~v-1) + R(t,s)]~u(t)~v(s)dt ds
and
Now from (2.5.17) we have
2(t) ~ L 22(r+a)
en
f
00
)
(
2n u
1_ d
r+a u
2
= L 22(r+a) 2-2n(r+a-1) .
o
(r+a-1)
-
The following corollary shows that the approximating sequence
xn(t) converges to x(t) with probability one for each t€:rn. 1 , and
gives the rate of convergence.
Corollary 2.5.1.
r + a > 2.
Then for each tE rn. 1,
(2.5.20)
where 0
< y <
Proof.
Let x be as in Theorem (2.5.3) and assume that
2yn
2 0 sup Ix(t)-~(t)1
n>n
O
r+a
--2-- - 1.
+
For each fixed t, define
x = { x(t)
u
~(t)
0 a.s. as nO
~,
0
~
u
~
+
00
,
1, by
for u = 0
for
2~n
<
u
~
22(n:1)
,n
~
1,
e
45
where
~ (t)
k
\00
.
= I.k=Ox(n) 1 -n -n
(t) . Then X 1S separable
2 [2 k,2 (k+1))
in u and from (2.5.19), we have (with n such that 2-2n < u ~ 2-2(n-1) )
where C =
L 22(T+a)
2
and B = r+a-2 > O.
Thus, by Kolmogorov's
(r+a-l)
theorem (Neveu, 1965, p. 97),
1
1
Y sup Ix(t)-x (t)1 = -- sup IXo-~1 + 0 a.s. as h + 0 ,
h O<-l-<h
n
hY O<u<h
22n
-2n
and (2.5.20) follows by putting h = 2 0, nO + 00 •
o
CHAPTER III
Sampling Expansions for Operators Acting on
Certain Classes of Functions and Processes
3.1.
Introduction.
In this chapter the problem of reconstructing bounded linear
operators acting on classes of functions band1imited in the sense
of Zakai (1965) and of Lee (1976a) will be considered.
Sampling
expansions for bounded linear operators acting on classes of
func~
tions with wandering spectra will also be investigated.
.
e
Recall that a function of the form f(t) = fW o e2nitu~(u)du ,
-Wo
where Wo > 0 and fEL 2 [-WO'Wo] is called conventionally band1imited
to WOo The class of all such functions will be denoted by BO(WO)
and is a Hilbert subspace of L2(JR1). Every fEBO(W O) has the following sampling expansion and convolution representation
f(t) =
y f(~)
2W
n~-oo
00
= ~ooJ feu)
(3.1.1)
where W ~ Wo
' and
sin n(2Wt-n)
n(2Wt-n)
sin 2nW(t-u)
1
2nW(t-u)
du, tElR ,
the series in (3.1.1) converges uniformly on lR 1
and also in L2(JRI).
The functions
= sin n(2Wt-n)
n(2Wt-n)
n
= 0,
~
1,
~
2, ... ,
form a complete orthogonal set in BO(W) which is strictly larger
47
than BO(WO)' It is thus interesting to notice that when W > Wo
the partial sums of the series in (3.1.1) belong to BO(W) and yet
its pointwise (or L2(lR 1)) limit belongs to the smaller subspace
BO(WO)' Of course, when W= WO' (3.1.1) is the expansion of f
in terms of the basis {¢n(t;WO)} of BO(WO)'
Zakai (1965) extended the classical concept of conventional
bandlimitedness to a broader class in which the functions need not be
square integrable.
He also proved that if f€BO(W O) and W > WO' then
00
L
(3.1.2)
n=-oo
(-1)nf(2~)
A
and if, in addition, f(u)(l - e
. U
nl ~
0)
=0 ,
-1
1
belongs to L [-WO,WO], then
(3.1.2) is also valid for W = WOo
3.2.
.
The Bandlimited Case.
Kramer (1973) derived a sampling expansion for bOlmded linear
operators acting on conventionally bandlimited functions, and Mugler
(1976) derived a convolution representation for such operators.
Theorem 3.2.1.
(Kramer (1973) and Mugler (1976)).
If T is a
bounded linear operator on BO(WO), then for every f€BO(WO)
sin n(2W (e)-n)
I f(2~) T[ n(2W (~) -n) ](t)
n=-oo
0
0
00
[Tf](t)
=
00
(3.2.1)
-L
sin 2nWO(e-u)
feu) T[
2nW (e -u) ] (t)du,
O
When T is time invariant, i.e. [Tf(e-a))(t)
(3.2.1) takes the simpler form
1
t€R
.
= [Tf)(t-a),
e
48
00
[Tf](t) =
L
n=-oo
n
sin 2TI WO(e)
f(2W) T[ 2TIW (e)
] (t
0
0
- -2Wn)
O
sin 2TIW (e)
00
= J f(t-u) T[ 2TIW (e)O
0
-00
1
](u)du, tEffi
.
The significance of Kramer's expansion is that Tf may be reconstructed from samples of f itself rather than from samples of
Tf.
For instance, the differentiation operator [Df](t) = d~ f(t)
is a time invariant bounded linear operator on BO(WO)' hence from
(3.2.1) we have
00
f' (t) =
n
d sin TI(2Wot-n)
[ TI(2W t-n)
].
L f(2W)
dt
n=-oo
0
0
Kramer's sampling expansion (3.2.1) will be generalized to
-
e
broader classes of functions.
in the sequel.
The following notation will be used
For a non-negative integer k, L2(~k) is the class of
all complex valued functions defin~d on IR I that are square integrable
with respect to the measure d~k(t) =
d~ k' If fEL2(~k)' then f
(l+t )
defines a tempered distribution (denoted also by f) on the class S
of rapidly decreasing functions by
00
fee) = J f(t)e(t)dt , 8ES •
-00
(See Chapter 4 for relevant definitions.)
The distributional Fourier
transform of f is the tempered distribution
8ES.
'"
The spectrum of f is the support of f.
f
defined by fee) = fee),
For k = 0,1,2, ... and
Wo > 0, Bk(WO) is the class of all continuous functions fEL2(~k)
whose (distributional) spectrum is contained in [-WO,WO]' and is
called the class of WO-bandlimitedfunctions in L2(~k)' It is clear
49
that BO(WO) is the class of WO~band1imited functions in L2 (]R 1) , and
Bk(WO)cBk+1(WO)' Also, BO(WO) is dense in Bk(WO) for every positive
integer k (see Lee, 1976b).
Zakai (1965) obtained a sampling representation for functions
in B1 (WO)' Cambanis and Masry characterized Zakai's class B1 (WO)
and as a consequence sharpened Zakai' s sampling expansion (see
also Piranashvi1i (1967) and Lee (1976a)).
Theorem 3.2.2.
(Zakai (1965), and Cambanis and Masry (1976)).
If fEB 1 (WO) and W > WO' then
00
f(t) = \' f(2Wn) sin TI(2Wt-n)
L
TI(2Wt-n)
n=-oo
(3.2.2)
tEIR 1 ,
and the series converges uniformly on compact sets.
Thus, functions in B1 (WO) are reconstructed from their samples
using functions in BO(W) , W > WOo
Remark 3.2.1.
It should be noted that (3.2.2) holds for
W = Wo if the Fourier transform g of get) ~ [f(t)-f(O)]t- 1 is
.u
TIl
A
such that g(u) (1 + e
W- -1
0)
1 .
belongs to L [-WO,Wo].
Lee (1977) proved an analogue of Theorem 3.2.2 for functions
fEBk(W O)' k
~
O.
If fEBk(W O), W > WO' 0 < S < W-WO'
and ~ is an arbitrary but fixed COO-function with support in [-1,1]
Theorem 3.2.3.
(Lee, 1977).
00
and
J ~(t)dt = 1, then
-00
-
e
so
,tElR 1 ,
f(t) =
(3.2.3)
and the series converges unifonnly on compact sets.
sin 27fW(.)
27fW(o)
It should be pointed out that the function
belongs to BO(W+S), W:> WOo
Thus functions in
~(WO)
are
A
~(S(·))
reconstruc~
ted from their samples using functions in BO(W+S), W > Wo and
o<
S
<
W-wO.
It should also be noted that the presence of the
(damping) factor
not valid for
"-
~
fE~,
f(t) = t(fEB 2(WO));
in (3.2.3) cannot be eliminated, as (3.2.3) is
k
~
2.
As a counter example conSider
then f(2~) = 2~ and the series in (3.2.2) does'
not converge.
Campbell (1968) derived sampling expansions for the Fourier
transforms (as functions) of tempered distributions with compact
supports.
If a tempered distribution F has a compact support and
oo
eu(t) = e 27fitu , then F(eu) is well defined, since euEC for all uE:m. l •
In this case the Fourier transfonn F of F may be thought of as a func~
tion defined on :m. l by F(u) = F(eu) , uEIR l (see Section 4.2).
Theorem 3.2.4.
(Campbell, 1968).
Let F be a tempered
distribu~
tion with compact support and with Fourier transfonn f as a function
on :m. 1, i. e. f(t) = F(e t ), tE :m. l . Let ~ be a test function such
that
~(u)
= 1 on some open set containing supp(F) , and let W>
such that the translates
{supp(~)
+ 2nW}nrO are disjoint from supp(F).
Then
00
(3.2.4)
f(t)
=
I
n=-oo
0 be
f(2~)K(t - 2~)
51
where K(t)
= 2~ fe2TIitu~(u)du , and the series converges for every
tEIR I .
Campbell's result does not require the support of F to be
symmetric with respect to the origin and is more general than Lee's
(Theorem 3.2.3) as functions in
~(WO)
have (distributional) Fourier
transform with compact support in [-WO'WO]' Campbell considered also
a sampling expansion similar to (3.2.3) when supp(F)c(-WO'WO),
Wo > 0, (for specific~) and derived an upper bound for the truncation error which was recently made more explicit by Lee (1979).
To establish notation, let
in S whose Fourier transform
(i)
'"
~
~
be a fixed but arbitrary function
satisfies the conditions
~ is a symmetric test function supported by [-Wo-o,Wo+o],
o<
(ii)
(iii)
0 < WO'
~(t) = I for all tE[-WO'WO],
'"
~(t) < 1 for all t4[-W 'W ] .
O O
Then for all fEL2(~k)' the convolution f*~ exists and is a Coo-function
in L2(~k)'
If fE~(WO)' then
f(t)
=
(f*~) (t) , tElR 1 ,
(see Lee (1976a), also Cambanis and Masry (1976)).
characterization of
Lemma 3.2.1.
~
will be needed.
(Lee, 1976b).
the following are equivalent,
(a)
f(t)
=
The following
(f*~)(t) , tE IR 1,
If fEL2(~k) is continuous, then
52
(b)
f(t)
_ k-l
- I
j=O
C)
feb)
j
I
jo
t
k
t
I
+ ~ get) , t€IR , and some g€BO(WO),
k
o
(c)
the spectnun of f is contained in [-WO,WO] ,
(d)
f has an extension to an entire function satisfying
k 27TW IDnz I
, Z€C and some Ck > O.
If(z)1 ~ Ck(I+lzl) e O
The follawingresu1t is a generalization of Kramer's sampling
expansion (3.2.1), in the sense that (3.2.1) is established for bandlimited functions f€B 1 (WO)
expansion (3.2.2).
(~BO(WO))'
as well as of Zakai'ssampling
Theorem 3.2.5.
Let f€BI(WO) and W > WOo
linear operator T on BI(W)
Then for any bounded
-
e
00
[Tf](t) = L\
(3.2.5)
n=-oo
n
I
f(2W)[T<P
n](t) ,tdR ,
where <Pn ( t ;W)
Proof.
Fix fEB 1 (WO) and W > WOo We first show that the convergence in (3.2.2) is in L2(~I) as well. From (b) of Lemma (3.2.1) we
have that f(t)
f(O) + tg(t) for all t€IR I, and some g€BO(WO);
hence by (3.1.1) we have
'=
00
a = Jlg(t) -
(3.2.6)
~
\
L
n=-
00
n
12
g(2W)<P
n (t;W) dt
Jlf(t)-f(O)-f'(O)<PO(t;W)
2Wt[f(2~)-f(0)]
2
n7T(2Wt-n)
- sin 7T(2Wt-n)1 dt 2
nfO
l+t
-I
53
Since
1
= 1. + 2Wt-n
' by writing
net __
n)
n
_---'-t__
2W
t[f(2~)-f(0)]
- - - - n - sin n(2Wt-n) =
nn(t - 2W)
+ [f(2~)-f(0)] <Pn(t;W)
in (3.2.6) we obtain
o = Jlf(t)-f(O)-f'(O)<POt;W)
-
~
n 0
n
n f(2W) -f(O)
[(-1)
n/2W
<PO (t;W)
(3.2.7)
Now (3.2.2) applied to g gives
(3.2.8)
fICO) + L (_I)n
nrO
f( --!!) -f(O)
2W
n/2W
=0 .
-
e
Also from the classical sampling theorem (see Zakai, 1965) we have
00
L
<Pn ( t
n=-oo
; W)
=1
, tElR
1
•
N
N
For any positive integer N, let S_N(t) ~ L <Pn(t;W).
n=-N
any t ~ 0
Then for
(3.2.9)
Consider S~N(t) and let t be such that sin 2nWt ~ O.
The successive
tenns of S~N(t) can be written as
sin 2nWt
2nWt
sin 2nWt sin 2nWt
N sin 2nWt
n(2Wt+1) , n(2Wt+2) , ... ,(-1) n(2Wt+N)
and are thus alternating in sign and decreasing in magnitude.
It
54
a I .s. ,sin
follows that IS_N(t)
21TWt2TIWt,
l2Wt]
N
for So
and S[2Wt]+1'
oS.
1 for all t
~
Thus for all N > a and t
0, and similarly
~ 0,
IS_N(t)
N
I~
3.
The same result holds for t < O. Thus IS~N(t)-ll oS. 4, and since
S~N(t) ~ 1 for all tEIR l , and ~l is a finite measure, it follows by
the bomded convergence theorem that S~N(t) ~ 1 in L2(~1)'
It then
follows from (3.2.7) that
00
o = flf(t) - L f(2~)¢n(t;W)12d~1(t)
n=-oo
,
proving the convergence of (3.2.2) in L2(~1)' Since T is a bounded
linear operator on Bl(W) , and hence continuous, (3.2.5) follows with
the series converging in L2(~1)' Masry and Cambanis (1976) showed
-
e
that if hl ,h 2EB l (W), then
which implies that (2.5.2) converges unifonnly on compact sets.
0
Remark 3.2.2.
As in Remark 3.2.1, we notice that if the Fourier
transfonn of get) = [f(t)-f(O)]t- l satisfies the condition:
ou
TIIW -1 1
g(u) (1 + e o) EL [-WO,WO], then (3.2.8) and thus Theorem 3.2.5
g
A
.
remain true for W = WOo
For k
~
2, (3.2.5) is not valid, since it is not valid when T is
the identity operator (a counter example is f(t) -= t which was mentioned
earlier).
The following expansion for k
~
of f at zero and the functions tjEBo+l(W),
.
t
J
k-l sin 21TWt
.
2TIWt € Bk_l(W), so that functIOns
2 involves the derivatives
a~
j ~ k-2, and
f€~(WO)
are reconstructed
55
from their samples and the values of their derivatives at zero using
functions in Bj (W), 1
j
s;
k-l, W > WO' Le. in
s;
~-1 (W).
Let fEBk(W O), k ~ 2, and W > WOo
bounded linear operator on ~(W), then
Theorem 3.2.6.
If T is a
where
f
k-2
i
(t) = k 2 f(j)(O) t j , fk_l(t)
j=O
j!
=f
k-2
(t) + f(k-l) (0) t k- l sin 2nWt
(k-l)!
2nWt'
~n k(t) = (2Wt)k-l sin n(2Wt-n)
,
n(2Wt-n)'
n
and the series converges in L2(~k)' as well as uniformly on compact
-
e
sets.
Proof.
(3.2.11)
From (b) of Lemma 3.2.1 we have
f(t)
~~
If(t)
K
- t -1
- k;2 f(j) (0) t j )
j ~O . J !
l
=
kf(k-l) (0) + tg(t) ,
for some gEBO(WO)' Since F(O) = kf (k-l) (0), then by part (a) of Lemma
3.2.1 we have that FEBl(WO) and then by (3.2.5)
00
(3.2.12)
F(t) =
r F(2~Hn(t;W)
n=-oo
,tElR
1
,
where the series converges in L2(~1) and uniformly on compact sets.
Then from (3.2.11) and (3.2.12), it follows that for all tEIR l ,
(3.2.13)
f(t) = f k- l (t)
+
\ {
n
n }(n)
2Wt k-l ¢n(t;W),
f(2W)-f
- 2 (2W)
k
nfO
L
and the series converges in L2(~k) and uniformly on compact sets.
S6
We notice that t j - 1EB j (W), 1 ~ j ~ k, since tj-lEL2(~k) and for any
.
~
function
A
~(t)
whose Fourier transform
~ES
= 1 on [-W,W], and
A
~
is a test function with
< 1 for all tE[-W,W] we have
~(t)
We also notice that t k- 1<P n (t;W) belongs to ~(W) by (b) of Lennna 3.2.1
since <Pn(t;W) BO(W) for all n. Now since T is a bounded linear oper-
e
"
ator on
and hence continuous, (3.2.10) holds where the converg-
~(W),
ence is in L2(~k)' and the uniform convergence on compact sets follows
by (c) of Lenuna 2 of Masry and Cambanis (1976).
Example 3.2.1.
o),
fE~ (W
m, k
Proof.
~
The m-th derivative operator ID(m) f) Ct)
1, is a bOlID.ded 1inear operator on
~ CWO)
=
f(m) (t) ,
.
Since fEBk(W O), then by (a) of Lemma 3.2.1 we have
00
f(t) =
J f(u)~(t-u)du
-00
for any ~ESsuch that ~ is a symmetric test function supported by
[-W-o,W+o] for some 0 < 0 <WO'
A
~(t)
= 1 on I-WO'WO], and
on [-WO'WO] c . Hence, for all tElR 1 ,
00
[D(m)f](t) = J f(u)[D(m)~](t-u)du
-00
and
A
~(t)
<1
57
-00
-00
Since D(m)~ES, m~l, then
I [D(m)~](x) I ~
C
m,n ,for every n ~ 0,
(l+x2)n
and thus for n = k,
du dv
du dv
~
(3.2.14)
where
C~k =
,
C'
m,
f
k
2
00
-00
If[u) 1"'2
~
c; k J (l+x~) k ' m,k ~
du, tElR 1 ,
00
,
1.
From (3.2.14) we obtain
-00
(3.2.15)
Now we argue as in Lee (1973).
I (u)·
k
.
00
Letting
dt
(t+t )k(I+(t_u)2)k
= J ---"-,------.,,,..,2
-00
'
and using (l+u 2) ~ 2(I+t 2) (1+(t-u) 2) , we obtain
and thus
Ik(u)
(3.2.16)
~
2k-1
2 k-l II (u)
(l+u )
~
n2 k-l
(1+u 2)k
,
e
58
since Il(u) ~ ~ (Zakai, 1965).
l+u
it follows that
From (3.2.15) and (3.2.16),
o
From Example 3.2.1 we notice that if
W > WO' m
~
fE~(WO)'
then for any
1
f(m) (t) = fern) (t)
k-l
+
~n,k
where f k - l , f k - 2 , and
I {f(.2!.)
nrO 2W
1
- fk-2(2~) }¢n(m,~(t) , tElR,
are as defined in Theorem 3.2.6.
We now obtain a convolution representation (which is a variation
of part (a) of Lemma 3.2.1) and an alternative proof of the sampling
-
e
expansion (3.2.3) for functions in
~(WO)'
k
~
1.
the analogue of (3.1.1) for functions in Bk(WO), k
This result is
~
1.
Let fEBk(W O), W > WO' and 0 < 8 < W-WO. If ~
is an arbitrary (but fixed) test function with support (~)c[-l,l]
Theorem 3.2.7.
00
and
J ~(t)dt
= 1, then
-00
00
f(t) = J feu) sin 2nW(t-u) ~(8(t-u))du
2nW(t-u)
-00
00
= 'i'
L
n=-oo
f(.2!.) sin n(2Wt-n) 11.(Q(t- .2!.))
t JRl
2W n(2Wt-n)
't' I-'
2W ,E
,
and the series converges pointwise everywhere.
Proof.
Fix fcBk(W O), W >
sEJR l , f(·)~(S(s-.)) belongs to
WO' and 0 < S < (W-WO). Then for all
L2 (JRl). Indeed, for all ZEC,
59
~(BZ)
=
J1~(u)e-2niBzu
du ,
-1
and integrating by parts n times, we obtain
Thus for any n
2:
1 we have
1
2nSIlinzi 1 (n)
Izln~(Bz) ~ ------n e
f I~ (u) Idu ,
(2nB)
-1
and hence
(3.2.18)
I~(sz) I
2nS IImz I
C e
< n,B
- (l+lzl)n
It follows that for every SEIR 1 ,
-
e
(sillnce (1+t 2)k ~ 2k (1+lsl)2k(1+lt_sl)2k for all t,sEIR l ). Also, by
by (d) of Lemma 3.2.1 we have
and by (3.2.18)
k 2n(WO+B) IIrnzl
~ CkCk,S(l+lsl) e
for all sEIR l and ZE~
(since (l+lzl)k ~ (l+lsl)k(l+ls-zl)k).
Thus
60
by the Paley-Wiener Theorem, we have that for all sEIR l , f(o)$(S(s_o))
Since WO+S < WO+W-WO = W, by (3.1.1) we have
belongs to BO(WO+S).
•
that, for all t,sEffi 1 ,
00
f(t)1J!(f3Cs-t) ) = f feu) sin2rrW2rrWt-ut-u) ~(S(s-u))du
'I'
A
-00
(3.2.19)
and the series converges tmiformly on
s
=
3.3.
-00
<
t <
00.
Now putting
t in (3.2.19), the required representation (3.2.17) follows.
0
Bandlimited in Lloyd's Sense.
Our goal in this section is to obtain sampling expansions for
botmded linear operators acting on classes of ftmctions and stochastic
processes bandlimited in Lloyd's sense.
Lloyd (1959) extended the
concept of "bandlimitedness" by allowing a "bandlimited" ftmction to
have a wandering, rather than compact, spectnun. An open set VcIR 1
is called a wandering set if there exists a real number W > 0 such
that all its translates {V+2nW}, nEIN, are disjoint.
Lloyd derived
a sampling expansion for wide-sense stationary processes whose
spectral distributions F have wandering supports, and also proved
that if 2WO is the Lebesgue measure of supp(F) , then Wo s; W.
Theorem 3.3.1.
(Lloyd, 1959).
Let {x(t}, tEIR I} be a measurable,
second order, mean-square continuous, wide-sense stationary stochastic
process, and V be the support of its spectral distribution F.
If, for
some fixed number W > 0, the translates {V+2nW} of V are all disjoint
then
61
(3.3.1)
x(t) = lim
M
N
L
N*o n=-N
where K(t) = 2~
(1 - N~1)X(2~)K(t - 2~) ,tElR
J e2rritudu
V
1
,
, tElR 1 , and if, furthennore lim s11PltK(t) 1<
tdRl
00,
then
(3.3.2)
where the convergence in both (3.3.1) and (3.3.2) is in the mean square
sense.
Lee' (1978) extended Lloyd's result to functions in L2(~k)
with wandering (distributional) spectra, and to non-stationary processes
whose correlation functions have (distributional) spectra.
Before
stating Lee's results we introduce the following notation. Let
x = {x(t), tEIR l } be a measurable stochastic process with correlation
function R(t,s) = E[x(t)x(s)], t,SEIR 1 , which satisfies
00
J R(t,t)d~k(t)
(3.3.3)
<
00
,
k ~ 0 ,
-00
dt
.
(1+t 2)k
1
We may define an operator R on L2(Vk) by
00
[Rf](t) =
J R(t,s)f(s)d~k(s).
R is a trace class operator, with non-
-00
00
00
zero eigenvalues {A k}k=l and corresponding eigenvectors {fk }k=l .
Cambanis and Masry (1971) obtained the following representations for
x and R:
00
(3.3.4)
x(t) =
L fk(t)~k
k=l
' tEIR
l
,
where the series converges in the mean square sense and also in L2(~k)
a.s., and {~k}k fonns an orthogonal basis with EI~kI2 = Ak for the
•
62
Hilbert space H(x) generated (in the
mean~square
sense) by the random
variables of the process x, and
00
(3.3.5)
R(t,s) =
I
k=l
Akfk(t)fk(s) , t,sElR 1 ,
where the series converges absolutely and in L2(~k).
Remark 3.3.1.
00
ID(+
00
<,,1,2
J J ~ dt
-00
-00
Since R satisfies (3.3.3), then
(l+t +s )
ds <
00,
and hence the Fourier transform
R of
R
exists as a tempered distribution in the Sobolev space H2 ,-2k(]R2)
(see Treves, 1967).
If the support of ft is contained in an open set
V whose translates by (2nW,2nW), for some W > 0, are all disjoint,
A
A
then (Lee, 1978) for all k, supp(fk) = {y: (y,y)Esupp(R)}cVO =
{v: (V,V)EV}, and the translates of Vo by 2nWare all disjoint.
~
_
00
Let Uo be an open set such that supp(fk)cUOcUOcVO' let 1/J be a C-
function that equals 1 on Uo and 0 on ~, and 11/J(t) lsI for all tElR l ,
and let the function K be defined by
(3.3.6)
K(t) =
2~
Jooe2nitu1/J(u)du.
_00
Theorem 3.3.2.
(Lee, 1978).
Let fEL2(~k)' and suppose that there
exists an open set V::>supp(~) such that for some fixed W > 0 the translates {V+2nW}, ndN, of V are all disjoint.
Let U be an open set such
that supp(f)cUcUcV, 1/J be a Coo-fUnction that is 1 on U and 0 on VC ,
.
1 Joo 2nitu
and K be the functIon K(t) = 2W
e
1/J(u)du. Then
-00
00
(3.3.7)
f(t) =
I f(2~)K(t
n=-oo
- 2~) , tdR
where the series converges pointwise everywhere.
1
second order, mean-square continuous process with correlation function
R which satisfies (3.3.3) for some
non~negative
integer k.
Let V be
A
an open set such that supp(R)cV, and suppose that, for some W > 0,
the translates {V
+
(2nW,2nW)}, nElN, are all disjoint.
Then for each
tE m1 ,
00
(3.3.8)
x(t)
=
\
n
n
x(2W)K(t - 2W) ,
n=-oo
L
where the series converges in quadratic mean and almost surely, and
K is defined as in Remark 3.3.1.
We now generalize Kramer's expansion (3.2.1) to bandlimited functions with wandering spectra and Lee's expansion (3.3.7) (the case
k
=
.
0) to bounded linear operators acting on the space L (D ,V; W)
k
which is defined as follows.
Let U be an open set in
lR
1
such that
for some open set V i U and W > 0 all the translates {V + 2nW}, nEIN
2
are disjoint. The class of all functions fEL (~k) with supp(f)cU
A
is denoted by Lk(U,V;W) , and simply by Lk(U) when the set
required to have the properties stated above.
V
is not
Since the translates of
IT are disjoint, its Lebesgue measure is finite (Lloyd, 1959), for
k
= 0,
fEL l (lR 1)
n
L2 (m. l ) .
It follows that
00
(3.3.9)
f(t)
•
= J e2TIltu f(u)du , a.e. ,
-00
and thus every function in LO(UiV,W) has a continuous version.
Only
continuous versions will be considered in this section.
Theorem 3.3.4.
If T is a bounded linear operator in L2 (nt) such that T
maps LO(UO,VOiW) into LO(UO'VO;W), and K is defined as in Theorem 3.3.2,
then for every tEm 1 ,
e
64
ex>
(3.3.10)
\,n
[Tf]Ct) =
L.
n=-oo
n
f(2W)[TK(· - 2W) Jet) ,
and the series converges lUliform1y on m1 as well as in L2(m 1).
Proof.
Define the function F by
ex>
I
F(u) =
m=-oo
f(u-2mW) , uEIR 1 .
F is the 2W-periodic extension of
f,
J IF(u)1 2du =
and
I
W
I
2
If I 1 2 1
L (lR )
Thus from the L2-theory of Fourier series, F has
where I W= (-W,W).
the Fourier expansion
(3.3.11)
F(u) = lim
~
.
. n
N
u
-TIl -
I
Cne
W
n=-N
in L2(-W,W), where
e
.n
1
TIl
Cn = 2W J F(u)e
IW
1
= 2W
Wu
du
.n
I J _ f(u-2mW)e
TIl
~(u)e
TIl
A
WU
du
m IwnUrn
1
= 2W L J _
rn IWnUrn - 2mW
But the sets (IwnITrn) - 2rnW
lUlion is IT, so that
(3.2.12)
1
Cn = 2W
=
L f(u}e
U
du .
(I W - 2mW) n IT are disjoint and their
.n
A
.n
WU
TIl
Wu
1. n
du = 2W f(2W)
65
We notice that for any gELO(U,V;W) ,
(3.3.13)
where
lui
is the Lebesgue measure of U; i.e., the evaluation map
1
on LO(U;V,W) is bounded with norm ~ IUI~.
Now consider the error
N
I
eN(t): = I [Tf](t) -
: ; lui
~
n=-N
f(2~)[TK(
N
I
IITf -
n
0
-
2~) ](t) I
n
f(2W) [TK(o- 2W)] II 2 1
n=-N
L (IR )
looN
f If(t)
::; IUI~IITII {
-
-00
:;
k
lui 211TII {f
IT
N
A
Ifeu)
I f(..2!.)K(t
n=-N 2W
-
I
n=-N
2 N
2
. n
1
n -rrl W u 2
2W f(2W)e
I du
1·
fJl/J(u) I I I
2W f(2~)e
v-u
n=-N
. n
-rrl W u 2
+
I
1
dU}~
. n
-rrl - u
::; IUI~IITII {L!F(u) - I Cne
W 12du
U
n=-N
. n
N
-rrl - u 2
+ f_ I I C e
W 1 du} ~ .
V-U n=-N n
N
(3.3.14)
1
- ..2!.) I dt}~
2W
By (3.3.11) and noting that F(u) = 0 on v-IT, it follows that the
right hand side of the last inequality in (3.3.14) tends to zero
independently of t, then eN(t) converges to zero uniformly in t on
IR 1.
0
We now consider the stochastic analogue of Theorem 3.3.4 for
processes bandlimited in Lloyd's sense which are not necessarily
66
stationary.
To fix notation, let x = {x(t), tdR I} be a measurable,
second order, mean-square continuous process with correlation
tion R which satisfies (3.3.3) for k = O.
func~
Asstune that V is an open
set in JR2 such that, for some fixed W> 0, the translates {V+(ZnW,2nW)},
ndN, of V are all disjoint, and let U ~ V be a fixed open set in JR 2.
/\
If suppeR)
c
U,
then almost all sample paths of x belong to
LO(UO;VO'W), where Uo = {u: (u,u)cU} and Vo = {v: (v,v)cV} (Lee, 1978).
It then follows from Theorem 3.3.4 that if T is a bomded linear oper2
ator in L (JR1) such that T maps LO(UO'VO;W) into LO(UO'VO;W), then with
probability one
~oo
n
n
. 1
(3.3.15)
[Tx] (t) = Ln=-ooX(ZW) [Tk(
zw)] (t) ,tdR ,
0
-
where the series converges uniformly on JR 1 , as well as in L2 (IR 1) .
We show that under appropriate conditions on the operator T
(Theorem 3.3.5) or on the correlation function R of the process x
(Theorem 3.3.6) the expansion (3.3.15) converges also in quadratic
mean.
Let T be a bounded linear operator in LZ(IR l ) of integral type
with kernel ScL 2 (JR2):
00
(3.3.16)
[Tf](t) =
J S(t,u)f(u)du a.e., fcL 2(JRl) ,
-00
and asstune that the kernel S satisfies the following conditions:
(i)
(ii)
S(t,o)cL 2(JRl) for all tcJR l and t r>S(t,') is a continuous map from JR 1 into L2 (JR 1), so that each Tf has a
continuous version for which (3.3.16) holds for all tcJR l .
SdLO(UO;VO'W)
@
2
LO(UO;VO'W)] e [L (:JRl)
@
LO(U~)]; so
that T maps LO(UO;V,W) into LO(UO;V,W), (note that
~O(U~) = L~(UO ;VO,W)) .
67
00
(iii)
f
(1+u 2)IS(t,u)ldu <
for all t€IR I .
00
-00
Since the sample paths of x belong to LO(UO;V0' W) a. s., we have with
probability one
00
(3.3.17)
[Tx(·,w)](t) =
f
S(t,u)x(u,w)du,
for all t€JR l ,
-00
and we now show that tmder condition (iii) on the kernel S of T, the
series in (3.3.15) converges also in quadratic mean for each t€IR l .
Let x = {x(t), t€JR l } and T be as defined above.
Theorem 3.3.5.
Then
00
(3.3.18)
[TxJ(t) =
\
n
n
x(2W)[TK ( • - 2W) lCt)
n=-oo
L
where the series converges in the mean square sense for every t€ JR 1 ,
and K is as defined in Remark 3.3.1.
Proof.
Recall the expansion (3.3.4) of x where the series con-
verges in L2(JRl) a.s.
Since almost all sample paths of x as well as
all f k belong to LO(UO;V,W) , (Lee, 1978), and since T is a continuous
linear operator on LO(UO;VO'W) , it follows that with probability one,
00
(3.3.19)
[Tx](t) = L [Tfk](t);k
k=l
in L2(IR 1) and also pointwise everywhere by (3.3.13).
Thus, for
every t€ JR 1 , we have
00
= L
00
L I [TfklC t ) 1·1 [TfplC t ) IE(;k~ )
k=N+l p=N+l
00
=
L
A 1 [TfklCt) 1
k=N+l k
P
2
68
But by (3.3.13)
Thus for every tE IR 1 ,
and the series in (3.3.19) converges in the mean square sense, for
each tEIR 1.
Now consider the mean square error
2·
~(t): = E I [Tx](t) -
•
N
L
n
n
x (2W)[TK( • - 2W) Jet)
12
n=-N
,tElR
1
.
Making use of the convergence of the series in (3.3.19) in quadratic
mean for each tE IR 1 , we obtain
e~(t) =
N
I[Tx](t)!2 -
L E([Tx](t)X(2~))[TK(.
n=-N
N
_ L
n=-N
N
-
2~)](t)
E([Tx](t)x(2~) )[TK(· - 2~) Jet)
N
\'
\'
n-m
n
m
+ L
L E(x(2W)x(2W)) [TK(· - 2W)] (t) [TK(· - 2W)] (t)
n=-N m=-N
(3.3.20)
69
We have
00
n
~ J IS(t,u) I-Ifk(u) - I fk(zW)K(U - 2~)ldu
-00
n=-N
N
- I
n=-N
(LUa
. n
1Tl W V
fk(v)e
dv)K(u - 2~)ldU
A
.n
N
1Tl WV
- I e
K(u - 2W) Idv) du
n=-N
(3.3.21)
For every UE:rn. 1 , define <Pu(v) = I~=_001/J(v+2nW)e21Ti(v+2nW)u
(Lee,
1978) . Then for each UE lR 1, <P u is a periodic COO -function with Fourier
expansion
.n
00
n 1Tl W v
<Pu(v) = I K(u - 2W)e
n=-oo
where the series converges unifonn1y on -00 < v < 00.
.
1Tl.
N
(N(u): = s~ le21TlUV - I K(u - 2~)e
VEUa
n=-N
.
N
1Tl
~ sup I<p (v) - L K(u - 2~)e
VE:rn. 1 u
n=-N
=
sup 1 I L
VElR InT>N
1Tl.n
- V
K(u - 2~)e W I
We have
-n v
W I
n
Wv
I
•
70
~
I IK(u - ....!!.) I
Inl>N
2W
and since KES implies IK(u) I
C_ for some C > 0 ,
- 1+u 2
<
2
1
I
~
2C(1+u)
~
2C
2
l6W
N
( 1+u) ,
---=2
InT>N 1 + (2~)
UE:m.
1
._
From (3.3.21) we thus have,
00
EN k(t) ~
•
J
IS(t,u)IEN(u)du
'-00
2
J Ifk(v)ldv
o
IT
00
~ l6~ C IUol~ J (1+u2) IS(t,u)ldu .
-00
It follows from (3.3.20) that for all tER\
2
~(t) ~
IUOI (16W2C) 2
2
N
2 _ _
( I Ak) J (l+u ) IS(t,u)ldu + 0 as N+
k=l-oo
00
00
00
by property (iii) of S, and thus the series in (3.3.18) converges in
the mean square sense.
0
It should be noted that the integral type operator T was defined
on all fEL 2(JRl) since K ~ LO(UO;VO'W)'
However, one could take the
Coo-function ~ (of Remark 3.3.1) equal to zero on
A such that Uo ~ A ~ VO'
A
C
for some open set
In this case KEL O(A;VO,W) , and it would
suffice to define the integral type operator T on LO(A;VO,W), rather
than on L2(rn. 1) (also conditions (i) and (ii) would need the obvious
modifications).
71
Under further conditions on the process x, (3.3.15) converges
in quadratic mean for all bounded linear operators.
Let x = {x(t) , t€IR l } be as in Theorem 3.3.5,
Theorem 3.3.6.
and in addition assume that
00
f
(i)
(1+t 2)IR(t,t) dt <
00
-00
00
L
(ii)
IRe (n/2W) ,(nnW))
(1 + (2W) 4)
n=-oo
<
00
•
Then if T is any bounded linear operator in L2 (lR 1) which maps
LO(UO;VO'W) into LO(UO;VO'W) , (3.3.15) holds where the series converges
in quadratic mean for every t€ IR 1.
Proof.
Notice that by (3.3.20) we have
00
•
00
2
2
= IITI1 L Ak f dt{!fk(t) 1
k=l -00
It follows by (3.3.5) and monotone convergence that
L~=lAk
00
f
-00
00
Ifk(t)
I
2
00
dt =
f
R(t,t)dt.
By (3.3.4) we obtain
-00
00
m- - 2W)dt
m
- - 2W)dt
m
-m}
f R(t'2W)K(t
= E{f x(t)K(t
x(2W)
o
-00
-00
..
72
and thus (3.3.22) can be written as follows:
II TI1 2
~(t) : ;
00
f I [R(t,t)
L R(t,
Inl::;N
-
-00
-
L [R(..1!,t)
/nl::;N
2W
2~)K(t - 2~)]
\
n m
m
R(2W' 2W)K(t - 2W)]
Iml::;N
-
L
• K(t - 2~) Idt .
(3.2.23)
But by Theorem 3.3.3 we have for every t, SE:R 1
00
R(t ,s) =
•
\
n
n
R(t, 2W)K(s - 2W)
n=-oo
L
substituting in (3.3.23) we obtain
00
~(t).~ /ITI1 2{
f
-00
L IR(t, 2~)IIK(t
Inl>N
- 2~)ldt
00
L
+
L IR(2~'
Inl::;N Iml>N
2~) I
f
~OO
IK(t -
2W) 1·IK(t
- 2~) Idt} .
C
Since KES implies IK(t) I ::; (1+tk2)k for each k ~ 0 and some Ck> 0,
and jR(t,s) I ::; lR(t,t) • lR(s,s), we have that the first term on the
right hand side of (3.3.24) is less than
2C111 T 11 2C
t
-00
by (ii).
Cl+t 2) IR( t, t) dt)
l:
Inr
IR( (n!2W) ~ ('f2Wl)
>N 1 + (2W)
For the second term, notice that
->- 0
as N
->-
00,
73
00
f
-00
00
IK(t - 2Wn) 1·1 K(t - Zmw) Idt ~ BC1C2_ f
00
[l+(""'!!') 2] 2
2W
(1+t2)2
Thus the second term on the right hand side of (3.3.24) is less than
. ( I lR((m72~, ~m72W)))
Imj>N
1 +
+ 0
as
N +
00
,
(2W)
by (ii) and the result follows.
0
It should be noted that when R(t,t) is asymptotically monotonic
at ±oo then conditions (i) and (ii) of Theorem 3.3.6 may be replaced by
00
f
-00
(1+t 4)IR(t,t) dt <
00
•
•
CHAPTER IV
Sampling Expansions for Distributions and
Random Distributions
4.1.
Introduction.
This chapter is concerned with sampling representations for
distributions and random distributions.
The need to consider dis-
tributions (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: Jf(t)~(t)dt, where ~
I
is a "smooth" function representing the measuring instrument, Le.
the physical phenomenon is specified as a functional rather than a
function.
Furthern~re,
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.
It is thus of
interest to consider distributions beyond functions.
4.2.
Notation of Basic Definitions.
1
0000
Let Cc = Cc(IR ) be the class of all infinitely differentiable
functions with compact support. A topology T is introduced on
oo
the linear space Cc which makes it into a complete space: Coo~~
c n
I
in T if there exists a compact set AcIR which contains the
+
support of every ~n' and for every non-negative integer k, ~~k)(t)
+
a uniformly
as n
+
00.
C~ with the topology
T
is denoted
a
7S
by V, and its elements are called test functions.
The members of the
dual V' of V are called distributions, and the value of a distribution
fEV' at a test function ¢EV is denoted by f(¢).
logy on V' is defined by the seminorms
If (¢)I,
A (weak-star) topofEV', as ¢ varies
over all elements of V; thus, V'3fn~ 0 weakly whenever f n (¢)
all epEV.
~
0 for
The class S of rapidly decreasing functions consists of all
00
infinitely differentiable functions (epEC ) for which
for all non-negative integers m,k.
A topology on S is defined by the
seminorms
Ilepllm k = sup sup {(1+lt/)k1ep(n)(t)l} , m,k = 0,1,2, ... ,
,
O;5;n:SJl1 tE R 1
00
i. e. , a sequence {epn}n=l is of functions in S is said to converge in S,
if for every set of non-negative integers, the sequence
m (k)
{(l+ I t 1) epn
00
(t)}n=l converges uniformly on IR
1
S is complete, and
the dual S' of S is called the class of tempered distributions.
Simi-
larly, a (weak-star) topology is defined on S' by the seminorms If(ep) I,
fES', as ¢ varies over all elements of S, i.e., f n converges in S'
if f n (¢) converges for all epES. The space V'(S') is (weak-star)
sequentially complete, that is, if {fn}n is a sequence in V'eS') such
that {fn (¢)} n
is a Cauchy sequence for every ¢EV(S) , then there exists
a distribution fEV'(S') such that f n ~ f in V'(S').
Finally, the spaae C with the topology defined by the seminorms
00
Pm A(¢) = I
supl¢(n)(t)!
,
O;5;n;5;m tEA
where A ranges over all compact sets in IR I and m over all non-negative
•
76
integers, is denoted by E.
A
The Fourier transfonn F(F(<P) = <p, <PES) is a one-to-one bicontinuous mapping from S onto itself.
If fES', the Fourier trans-
fonn ~. of f is defined by f(<P) = f(¢),
<PES, and is a tempered dis-
If fES' and <PES, the convolution f*<p is defined as a
function on ill. 1 by
tribution.
, tE ill. 1 ,
where ¢(t)
<P(-t) and the shift operator Tt is defined by (Tt<P)(u) =
oo
f*<PEC has a polynomial growth and thus determines a tempered
<P(u-t).
=
distribution.
Suppose fEV', f is said to vanish in an open set UeIR 1 if f(<P) = 0
for every <PEV with supp (<P) cU. Let V be the union of all open sets
UEill. l 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.
such that
~(u) =
If
~EV
1 on some open set containing supp(f) , then
is
~f =
f
i.e. (~f) (<P) = f(~<P) = f(<P) for all <PES, but since et(u) = e27Titu
is a Coo-function, feet) = f(~et) exists, and the distribution f is
generated by the function f(t) defined on IR I by
(4.2.1)
Indeed
A
(4.2.2)
and (~f)
f
A
=
A
A
(~f)
A
A
(and therefore f) is generated by the Coo-function (f*~)(t)
77
which has a polynomial growth (see Rudin, 1973, p. 179). By choosing
A
cj>ES
such that cj> = l/J, we have
(f*~) (t) = (~*¢) (t) = f(Ttcj» = fCCTtcj»")
= f(e t ¢) = f(l/Je t ) = feet) ,
and from (4.2.2), (4.2.1) is justified.
Hence the Fourier transform
of a distribution with compact support may be thought of as a function
defined on IR I by (4.2.1).
Let (n,F,p) be a probability space.
A random distribution is a
continuous linear operator from V (or S) into a topological vector
space of random variables.
Specifically, a second order random
distribution is a continuous linear operator from V (or S) onto
L2 (n) = L2(n,F,p), the Hilbert space of all finite second moment
random variables.
4.3.
Sampling Expansions for Certain Distributions.
In this section we establish a sampling theorem for tempered
distributions whose Fourier transforms have compact supports.
A
A
distribution
(-W,W).
fES' is said to be W-bandlimited, W > 0, if supp(f)
c
The class of all W-bandlimited distributions will be denoted
by Bd(W).
00
Let V[-W,W] , W> 0, be the class of all C -functions cj> with
supp(cj»
A
C
A
[-W,W], and define Zew) ~ V[.-W,W] = {cj>ES: cj>EV[-W,W]}.
Pfaffelhuber (1971) stated that if HEBd(W) and h is its Fourier transform (defined as a function on lR l ), then
00
(4.3.1)
n) sin 7T(ZWt-n)
h(t) = \' h(zw
L
7T(ZWt-n)
n=-oo
78
and the series converges absolutely in Z'(W) (the dual of Z(W)).
(4.3.1) means precisely that, for every
00
f
h(t)¢(t)dt
-00
=
~€Z(W),
~ h(~) foosin TI(2Wt-n) ~(t)dt ,
/..
n=-oo
2W
-00
and the series converges absolutely.
TI(2Wt-n)
Campbell (1968) had already
noted that (4.3.1) does not hold pointwise for arbitrary band1imited
distributions.
Though (4.3.1) 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 4.3.1.
Lemma 4.3.1.
Let f€S' be such that '"f has compact support.
Let
A
E
be a closed set properly containing supp(f) , and
with support E and ~
=
~
any test function
1 on some open set containing supp(f). Then
'"
f is uniquely determined by its restriction to V(E)
, i.e., the values
fce), e€V(f), by
(4.3.2)
The shift operator T~ is defined on V'CS'), for every ~€IR1,
by
A ,distribution fEV'(S') is said to be periodic with period T > 0, if
(4.3.3)
(TTf)(~)
= f(~) , for every
~€V(S)
,
and T is the smallest positive number for which (4.3.3) holds.
We now state and prove our result •.
79
Theorem 4.3.1.
Let fES' be a tempered distribution such that
~ has compact support, and let the closed set E and W >
that E
J
;t
supp(t) and the translates {E+ZnW}, n
from supp(f).
r 0,
° be such
are disjoint
Let a and ljJ be any test functions such that ljJ has
support E, and a = 1, ljJ = 1 each on some open set containing
supp(f).
Then
00
(4.3.4)
f(¢) =
where Kw(t) = Z~
L f(T ~)(T J<W)(¢)
n=-oo
.2!.
ZW
¢ES ,
_
ZW
J e Z7TituljJ(u)du and Kw(¢)
00
=
E
J KW(t)¢(t)dt, ¢ES.
-00
If fEBd(W), then
00
f(¢) = ~ f(T n~)(T nGw)($*¢) , ¢ES
oo
n-ZW
ZW
(4.3.5)
where Gw(t) =
sin Z7TWt
Z7TWt
.
00
and Gw(¢) =
J Gw(t)¢(t)dt
, ¢ES.
-00
Proof.
N
We first show that the sequence of partial sums
A
~ = Ln=-NT-ZnWf , N ~ 1, converges in
~(¢) =
N
S'.
For any ¢ES,
A
L (T_ZnWf)(¢)
n=-N
A
= f(
N
L
n=-N
TZnW¢)
(4.3.6)
where
~EV
is a test function such that
containing supp(t).
~(t)
= 1 on some open set
We now show that the sequence
e
80
~N(t) = ~(t)I~=_N ¢(t-2nW), N ~ 1, converges in S. Since ¢€S,
there exists a constant B > 0 such that 1¢(t)1 s B(1+t 2)-1 for all
tE lR 1 , and thus
IcH t - 2nW) I
2
B
< 2B
l+t
1+(t - 2nW) 2 1+(2nW) 2
s
Since ~EV, we have supp(~)
[-C,C] for some C > 0 and 1~(t)1 sA
It then follows that for all tElR 1 and non-negative
for some A > O.
c
integers m,
(4.3.7)
(l+ltj)ml~(t)
I
N
I IcHt-2nW) I
n=-N
s 2AB(1+C)m(1+C 2)
•
00
I
1 2
n=-oo 1+(2nW)
<00,
i.e., the sequence of partial sums on the left hand side of (4.3.7)
converges unifonnly on lR 1. Hence the sequence (1+1 t I)m~N(t), N ~ 1,
converges uniformly on lR 1 for every m ~ o. Similarly, it can be
shown that for every m,k ~ 0, the sequence (l+ltl)m~~k)(t), N ~ 1,
l
converges uniformly on lR , i.e. {~N}' N~l, converges in S, and since S
is complete, its limit
~
belongs to S, and
~N + ~
in S.
It follows
from (4.3.6) that
and since S' is (weak-star) sequentially complete, then there exists
a tempered distribution FES' such that ~ + F in S'.
oo
Therefore, F = lim ~_ oo T - 2nW~ is a periodic tempered
-N = l n-N-+oo
distribution with period 2W.
It follows that F has the Schwartz-
Fourier series (Zemanian , 1965, p. 332)
. 81. ,
00
(4.3.8)
F
where et(u)
00
= L T_znwf = Lan en' in S' ,
n= -00
zw "2W
n= -00
= e ZTIitu , and
a n= -ZW1 F(Ue
ZW
where UEU ZW is a unitary function (Zemanian, 1965, p. 315), i.e.
oo
UEV and l n-_
U(t-ZnW) = 1 for all tdR 1 . From (4.3.8) we have
oo
/\
Since f has a compact support and UEV, then there is only a finite
number of non-zero terms in the last summation, and hence
/\
/\
= fee
(4.3.9)
n)
- ZW
= f(ae
n)
- ZW
=
f(T
From (4.3.8) and (4.3.9), we have that
(4.3.10)
/\
1
00
fee)
= I ZW f(T
n=-oo
n&)e nee) ,
- ZW zw
.n
00
where e nee)
zw
(4.3.11)
=Je
TIl
WU
e(u)du
eEV(E) ,
n
/\
= e(- 1W). Thus
-00
/\
fee)
=
/\
fee)
00
=
\
~
n--oo
1
/\ /\
n
zw f(T na)e(ZW)
ZW
/\
/\
eEV(E)
•
82
and by Lenuna 4.3.1 we have that for every ¢e:S
(4.3.12)
1
A
f(¢) = f(l/J*¢) = 2W
00
\
A
A
n
f(T nex)(l/J*¢) (2W)
n=-oo 2W
L
v
(since ~*¢ = (l/J¢)A
E
VeE)).
But
00
(~*¢)(2~) =
~(2~ - t)¢(t)dt
J
-00
00
= 2W
J Kw(t
- 2~)¢(t)dt
-00
(4.3.13)
= 2W(T nKw) (¢),
2W
¢e:S,
and (4.3.4) follows from (4.3.12) and (4.3.13).
To prove (4.3.5) notice that when 8e:V[-W,W] ,
~~~~L...
=
= 2W(T
A
8(t)dt
nGw) (8)
- 2W
It follows from (4.3.10) that for ee:V[-W,W] ,
00
fee) =£(8) =
L
n=-oo
f(T na)(T n-W
rL) (8) ,
2W
"2W
and (4.3.5) follows by Lenuna 4.3.1.
o
Theorem 4.3.1 shows that a tempered distribution f with compact
spectrum can be reconstructed via (4.3.4) from its values (samples)
at the translates of an arbitrary but fixed test function ex which
83
equals one on some open set containing supp(f).
On the other hand,
if we denote feet) by f(t), then from (4.3.9) we have
A
f(T na )
2W
n
A
=
fee n)
2W
=
f(2W) , and (4.3.4) reads
00
f(¢)
I
=
n=-oo
f(2~) (T nKw) (¢) , ¢€s,
2W
so that a tempered distribution f with compact spectrum can be reconA
= feet).
structed using the samples of the function f(t)
We now show that the sampling theorem for tempered distributions
with compact spectrum includes as special cases the sampling theorems
for conventionally bandlimited functions (Example 4.3.1) as well as
for bandlimited functions in L2(~k) (Example 4.3.2).
Example 4.3.1.
f€L 2 (]Rl)
E.
(Conventionally bandlimited functions).
be a continuous function such that
f has
Let
compact support
Then f determines a tempered distribution:
00
(4.3.14)
f(¢)
f
=
¢€S,
f(t)¢(t)dt,
-00
A
and its distributional Fourier transform (denoted also by f) is
defined by f(¢)
= f(¢), ¢€S, or equivalently by
00
f(¢)
f
=
f(u)¢(u)du, ¢€S
-00
A
f (as a tempered distribution) is supported by E.
Hence (4.3.4)
applies and, if W> 0 is defined as in Theorem 4.3.1, we have from
(4.3.14)
f(T
n
~) = fee
2W
)
n
2W
=
f
W
-W
f(u)e
• n
TIl
Wu
du
=
f(2~)
"
84
For v > 0, define the function
for I'!'\
v
>
1 ,
00
where Cv
=
J exP·f -~ 2}dt. For each
00
v > 0,
<P
1'" (v-)
-00
V
EV and
J <pv(t)dt
=
1,
-00
and for each continuous function g and every tEIR 1 ,
00
J g(u)<pv(t-u)du
From (4.3.4) we thus have that
get) as VtO.
+
-00
for each tdR 1 and v > 0
00
00
(4.3.15)
J f(uHv(t-u)dt
=
-00
,
e
00
f(2~) J Kw(u-2~)<Pv(t-u)du.
I
n=-oo-oo
Since f and KW are uniformly continuous, we have for each fixed tEJR l
and nElN
00
J f(u)<pv(t-u)du
+
f(t)
as vtO
-00
Now by Theorem 24 of Lighthill (1958, p. 64), if for any sequence
00
{bn } which is O(n) as n
Ln=-oobn an,v is absolutely convergent
00, \
+
and tends to a finite limit as v
00
(4.3.16)
lim
I
+
0, then
00
an v
\>+0 n=-oo'
But, for each fixed tcIR 1 ,
=
I
lim an v .
n=-oov+O
'
85
00
L
n=-oo
00
Ibn f(2~)
J Kw(u - 2~)¢V(t-u)dul
-00
since f is bounded,lbn I~ Bini, and for k
>
1,
It follows that the right hand side of (4.3.15) satisfies the conditions leading to (4.3.16), and hence by letting v+O, we obtain
00
(4.3.17)
f(t) =
\
n
n
1
f(2W)Kw(t - 2W) , tElR
n=-oo
L
which is the sampling theorem for a conventionally band1imitedfunction with compact spectrum.
Example 4.3.2.
k
~
(Band1imited functions in L2(~k)).
0 , be a continuous function.
trihution by (4.3.14).
Then f determines a tempered dis-
If its distributional Fourier transform '"f
has a compact support, then (4.3.4) applies and we have
'"
'"
n
f(T na ) = fee n) = f(2W)
2W
2W
(see Lee, 1979).
for constant Ck
>
Since f is a Coo-function and If(t)1 ~ C (l+lt])k,
k
0 (Lee, 1977), then (4.3.15) holds and following
86
the arguments used in Example 4.3.1, we obtain (4.3.17) which is
similar to (3.2.3) and is identical to (3.2.4).
It should be noted,
though, that (4.3.4) cannot be obtained from Campbell's result
•
(Theorem 3.2.4), since the convergence in (3.2.4) is not uniform
on compact sets.
4.4.
Sampling Expansions for Random Distributions.
In this section sampling expansions for stationary random
butions are considered.
distribution.
Let X ='
~(cjJ),
distri~
cjJES} be a second order random
X is said to be weakly stationary, if for every h > 0
and cjJ,l/JES,
If X is a weakly stationary random distribution (WSRD), then there
exists a unique tempered distribution PES' such that for every cjJ,l/JES,
(4.4.1)
where ~(t)
R(cjJ,l/J)
=
=
E(X(cjJ) • X(l/J))
=
p(cjJ*~) ,
l/J(-t) (Ito, 1954), and P has the spectral representation
00
(4.4.2)
p(cjJ) =
f
~(u)d]..l(u) , cjJES ,
-00
where]..l is a non-negative measure on IR 1 such that
00
J d]..l(~)k
-00
for some integer k.
<00
(1 +u )
In this case X is said to be of type k, and ]..I
is called the spectral measure of X.
Let B* be the set of all Borel sets with finite ]..I-measure.
An
2
L (Q)-valued function Z defined on B* is called a random measure with
respect to ]..I if
87
Hence E(ZZ(B)) = ~(B), and Z(B l ) i Z(B Z) if Bl and BZ are disjoint.
Since ~ is a-additive, then Z(B)
. = loo_1Z(B
nn ), whenever Bl,B Z""
00
are disjoint sets in B* with Un=lBn = B. It follows by (4.4.1)
and (4.4.Z) that there exists a random measure Z with respect to
~
such that
00
X(¢) =
J
¢(u)dZ(u) ,
¢ES.
-00
If H(X) is the linear subspace of LZ(n) generated by {X(¢),¢ES}, then
H(X) and LZ(~) are isometrically isomorphic under the correspondence
X(¢}I-> ¢,
~
A WSRD X is said to be WO-bandlimited, Wo > 0, if
¢ES.
c
{[-WO,WO] } =
o.
Theorem 4.4.1.
W> WO' aEV and
(a)
~EV[-W,W]
If X = {X(¢), ¢ES} is a WO-bandlimited WSRD,
with aCt) = 1 = ~(t) on [-WO,WO]' then for
every ¢ES,
00
(4.4.3)
X(¢) =
X(T n&) (T nGw) (~*¢)
n=-oo zw
ZW
I
in mean-square, where Gw(¢) = Joo
S~~W~1TWt
¢(t)dt.
-00
(b)
Let X = (X(¢), ¢ES} be a WSRD with spectral measure
~ supp(~)
supp(~).
and the translates' {E+ZnW} , n
Let a and
E, and aCt) = 1 =
which
Let the closed set E and W> 0 be such that
has compact support.
E
~
~
~(t)
r
0, are disjoint from
be any test functions such that
on
supp(~).
Then
~
has support
88
00
L
X(CP) =
(4.4.4)
X(T nex) (T n-1V
IL)
n=-oo
v
ZW
(CP) , cp€S ,
ZW
in mean-square, where Kw(t) = Z~
J ~(u)eZTIitudu.
E
""
To prove (a), first let cp€S be such that cpeV[-W,W].
oo
Then ~(u) = Ln-_ oo¢(u+ZnW) is a Coo-function which is periodic with
Proof.
period ZW and has the Fourier series
~(u) =
(4.4.5)
.n
n TIl Wu
1
ZW CP(ZW)e
,UEm,
1
00
L
n=-oo
which converges uniformly on m I .
(T
"" -W,W] ,
Since cp€V[
n~'T)(ep) = Joo sin TI(ZWt-n) CP(t)dt
ZW
TI(ZWt-n)
tv
-00
• n
W TIl W U"
1
zw J e
=
.
CP(u)du
-W
we have for the mean square error
There exists a constant
N
J
"
Icp(u) "
.
SInce, by (4.4.5),
M > 0
N
1
L
such that for all
n
1
Ln=-N zw
and U€m l ,
. n
ZWep(ZW)e
n=-N
N
TIl
Wu
I
~ M.
.n
n TIl W u
""
CP(ZW)e
converges to CP(u) on
[-Wo'Wo], by the dominated convergence theorem, e~(cp)
+
0 as N +
00.
89
Thus for every <pEi) [ -W,W], we have
00
(4.4.6)
X(<P)
=
L X(T na)(T nGw)(<P) .
n=-oo
ZW
Now for every <PES and
ZW
Was in part (a) of the statement of
the theorem, we have
W
o
X(<P) = J ¢(u)dZ(u) =
-Wo
W
o
(4.4. 7)
=
J
-Wo
V
A
1'\
where W*<P
=
A
(~*<p)I\(u)dZ(u)
=
X(~*<P) ,
A
(W<P) EV[-W,W], and (4.4.3) follows from (4.4.6) and (4.4.7).
The proof of part (b) is similar to that of (a) with the obvious modifi-
o
cation and hence is omitted.
It should be noted that, since a
= 1 on [-WO,WO],
o -TIl.
W
=
n
-u
J e W dZ(u) ,
ne:lN •
-Wo
If we define
, te: IR
1
,
l
then {x(t), tElR } is a weakly stationary WO-bandlimited stochastic
process, X(T
n
&)
=
n
x(ZW) , and (4.4.3) reads
ZW
00
(4.4.8)
90
i.e., the random distribution X is reconstructed using the samples
of the ordinary stochastic process x.
Hence there is a one-to-one
correspondence between WO-bandlimitedweakly stationary random distributions X and WO-bandlimited weakly stationary processes x deterW
W 2 .
mined by X(~) = f O$(u)dZ(u) and x(t) = f oe TIltudZ(U) and satisfying
-Wo
-W0
(4.4.8) .
We now show that the sampling theorem for bandlimited weakly
stationary random distributions includes as a particular case the
sampling theorem for bandlimited weakly stationary processes.
Example 4.4.1.
Let x = {x(t), tEIR 1} be a measurable, mean-square
continuous, weakly stationary process which is WO-bandlimited, i.e.,
Wo
.
x(t) = f e-2TIltudZ(u)
-WO
(4.4.9)
where Z is a random measure with respect to the spectral
x with
]J {[
measure~
-Wo,Wo]c} = O.
Then x determines a WO-bandlimited WSRD by
Wo
X(¢) = f ~(u)dZ(u) , ¢ES ,
-WO
which can also be written as
Wo
00
X(~) = J (f e-2TIitu~(t)dt)dZ(u)
-W -00
O
00
=
J x(t)¢ (t) dt
,
-00
where the latter integral exists both with probability one as well
as in quadratic mean. Then by (4.4. 4) we have for each tE IR 1 and
\) > 0,
of
91
00
(4.4.10)
00
J x(u)¢v(t-u)du
=
-00
in quadratic mean.
00
L x(2~) J KW(u - 2~)¢v(t-u)du
n=-oo-oo
As in Example 4.3.1,
00
J x(u)¢v(t-u)du
+
x(t)
as v{O
-00
00
in quadratic mean,
J KW(u - 2~Hv(t-u)du
+
KW(t - 2~) as v-l-O, and
-00
the right hand side of (4.4.10) converges in quadratic mean to
We thus obtain
00
x (t)
in quadratic mean.
=
\
L
X
n=-oo
n
n
(2W) KW(t - 2W)
tElR
1
BIBLIOGRAPHY
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Sampling Representations and Approximations for
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';UPPL[Mr:NTARY NOTES
19
KEY WORDS (ColltJnuft on rttvcr.¥(J
";I(J"
JI necftlu,arY4l1d Identlfv by block number)
sampling representations, communication and information theory, bounded
linear operators, bandlimited functions
20.
ABSTRACT (Continue on revorae aide /I neceBeary IUId Identify by block number)
Sampling representations and approximations of deterministic and random
signals are important in communication and information theory. Finite sampling
approximations are derived for certain functions and processes which are not
bandlimited, as well as bounds on the approximation errors. Also derived are
sampling representations for bounded linear operators acting on certain classes
of bandlimited functions and processes; these representations enable one to
reconstruct the image of a function (or a process) under a bounded linear
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20.
operator using the samples of the function (or the process) rather than
the samples of the image. Finally, sampling representations for distributions
and random distributions are obtained.
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