* This author's research was sponsored by the Air Force Office of Scientific
Research under Grant AFOSR-72-2386.
** This author's research was supported by the Office of Naval Research under
Contract N00014-69-A-0200-6037
ZAKAI'S CLASS OF BANDLIMITED FUNCTIONS AND
PROCESSES: ITS CHARACTERIZATION AND PROPERTIES
Stamatis Cambanis *
Department of Statistics
University of North Carolina at Chapel Hill
Elias Masry **
Department of Applied Physics &Information Science
University of California, San Diego
Institute of Statistics ~meo Series No. 915
March 1974
ZAKAI I S CLASS OF BA1WLIttlITED FUNCTIONS AND PROCESSES:
ITSCI1ARACTERIZATION AND PROPERTIES
by
Elias Masry **
Department of
Information
University of
La Jolla, CA
Stamatis Cambanis *
Department of Statistics
University of North Carolina
Chapel Hill, NC 27514
Applied Physics &
Sciences
California, San Diego
92037
ABSTRACT
This paper characterizes Zakai's class [3] of bandlimited
functions and processes in terms of conventionally band1imited
functions and processes.
This characterization was first con-
jectured by Zakai and is used here to derive sharper sampling
representations and to study further properties of functions
and processes bandlimited in the sense of Zakai.
*This author's research was sponsored by the Air Force Office of Scientific
Research under Grant AFOSR-72-2386.
** This author's research was supported by the Office of Naval Research under
Contract NOOOI4-69-A-0200-6037.
I.
Introduction
The concept of a band1imited function (and a band1imited process) has
been extended by Zakai [3] to a class of functions (and processes) which do
not have a Fourier integral representation.
In this paper Zakai's class of
band1imited functions is characterized by a representation in terms of conventionally band1imited functions (Theorem 1).
This characterization was
conj ectured by Zakai and resu1 ts in a sharper sampling theorem for the
entire class (Theorem 2).
It is shown that this class of band1imited functions
can be defined by using a host of reproducing kernels other than the one used
by Zakai (Theorem 4).
Also, the properties of the class of conventionally
band1imited functions as a subset of Zakai's class of band1imited functions
are studied (Theorem 5).
Similar results are obtained for Zakai's class of band1imited processes:
a sharper sampling theorem (Theorem 6), a characterization by a representation in terms of conventionally band1imited harmonizab1e processes
(Theorem 7), and the relationship between conventionally bandlimited stationary or harmonizab1e processes oold processes bandlimited in Zakai's sense
(Theorem 8).
II. Bandlimited Functions
The following notation is used throughout this paper.
measure on the real line,
real line defined by
~
[d~] (t)
dm
m is the Lebesgue
is the finite measure on the Borel set
= _1_, and
1+t2
of the
is the Hilbert space of
Borel measurable complex-valued functions on the real line satisfying
2
for I>.1
I
H(>')
(1)
= HP;W,o) =
~ for
I
0
o
!~J
<
::;; ':.1
I >.1
::;; w+o
for W+o < 1>.1
and denote its inverse Fourier transform by
(la)
h(t)
= h(t;W,o) = ~ sineW + !)t sin! t
1TO
Zakai [3] defines the class
B(W,o) of functions "bandlimited to (W, 0)"
as the set of all functions
f
in
satisfying for almost all
L2(~)
t
00
(2)
f(t)
=
f f(T)h(t-T)dT = (f*h)(t)
_00
B(W,o)
is then a subspace of
L2(~)
and every function in B(W,o) is
equal almost everywhere to a continuous flmction (the right hand side of
(2)).
As in [3] only these continuous modifications will be considered in
this paper.
Denote by
CB(W) the class of functions in
ally bandlimited to W", i.e.
where
CB(W)
L2 (m)
= {fd 2 (m):
F denotes the Fourier transform of
f.
F(>')
Then
which are "convention-
=0
for fAI > W}
CB(W)
is a subspace
of
L2 (m) and CB(W) c B(W,o) for all 0 > O. Thus Zakai's concept of
bandlimited functions generalizes the conventional one.
The following properties of functions in
LEMMA I [3].
(a)
If
fe:B(W,o)
then
B(W,o) were obtained by Zakai.
f(t) = f(O) + tg(t)
where
ge:CB(W+o).
(b)
o
> O.
If
ge:CB(A)
then
f(t)
=c
+ tg(t)e:B(W,o)
for all
W ~ A and
3
It was conjectured by Zakai that in Lemma la, g€CB(W).
This is proved
in the following theorem which thus provides the characterization of
THEOREM 1.
f€B(W,o)
B(W,o).
if and only if
f(t) = f(O) + tg(t)
(3)
where
g€CB(W).
Proof.
In view of (2) and the fact that
h
is a real-valued function,
it suffices to prove the theorem for real-valued
(a)
Let
f€B(W,o).
f.
By repeated convolutions of f
with
h, (2)
implies
n=I,2, •..
where
hn
is the (n-l)st fold convolution of h with itself, Le., hn
is the inverse Fourier transform of Hn(A) = Hn(A).
f(t) = f(O) + tg(t),
that
t
g€CB(W+o).
Substituting
in
* hn ,
f
and noting
(f(O) * hn Jet) = f(O), we have for all
!hn(t)dt = Hn(O) = 1 implies
and n = 1,2, ...
f
Now by Lemma la,
00
tg(t) =
f
(t-T)g(t-T)hn(T)dT
_00
It can be seen from (1) that
H~€L2(m)
H
and thus [1, Theorem 61]
theorem and
g€CB(W+o)
2~ -(W+o)
J
that
we have for all
W+O
where
is absolutely continuous with derivative
n
thn (t)€L2 (m).
t
Hence by Parseval's
and n=I,2, ...
\11+0
G(A)eiAtdA = 2;
I
-(W+o)
G is the Fourier transform of
g.
Since
Hn(A) = 1 for
A€[-W,W],
n=1,2, ... , it can be seen that the integrals over [-W,W] on the right and
left hand side are identical.and hence for all
t
and
n=I,2, ...
4
aCt)
(4)
~ J G(A)eiAt{t[Hn(A)
- 1] -
iH~(A)}dA = 0
E
where
E
a.e. on
= [-W-o,-W]
[W,W+o].
u
E and hence
We will show that (4) implies G(A)
g€CB(W)
=0
which proves the Ilonly if" part of the
theorem.
Let
set)
be any complex-valued function on the real line such that
s(t)€Ll(m)
and
ts(t)€Ll(m), and let
Then since
G[Hn-l]
implies that for all
I
and
GH~
be its Fourier transform.
LZ(m) n Ll(m),
Fubini's theorem
= 1,Z, ...
n
00
o=
are in
SeA)
s(t)a*(t)dt
=i
I
G*(A){S'(A) [Hn(A) - 1] +
S(A)H~(A)}dA
E
_00
which can be written in the form
I G*(A) d~
(5)
{S(A) [Hn(A) - l]}dA
=0
.
E
Now choose
ts(t)€Ll(m)
SeA)
set)
and SeA)
to be real, symmetric and such that
= 1 for A€E.
s(t)€L (m),
1
This is certainly possible and in fact
can be taken to be real, symmetric and infinitely differentiable
function with support [-(W+O+E), (W+O+E)], E > O.
For such a function
set),
G = G1 + iG Z into its real and imaginary
part, we have from (5) that for all n=1,2, ... and i=l,Z,
and using the decomposition of
I Gi(A)H~(A)dA
= 0 .
E
Note that for i = 1 the integrand is an odd function since
11'
n
G1 is even and
is odd and thus the integral is equal to zero automatically.
i = Z the integrand is an even function, since
for n = 1,2, ...
GZ and H'
n
For
are odd, and hence
w+O
5
G2(A)H~(A)dA
f
= 0 •
W
and
variable
H(A)
=1
A-W
- --0-- on [W.W+o]. the change of
u = H(A) gives that for n = 1.2 •...
1
f
Gz(w+o-ou)un-ldU
=0
.
o
Since
G2 (w+o-ou)EL 2 ([O.1].m)
[2. Theorem 11.2.1]
[0.1]
and the set of functions
in L 2 ([O.1].m). it follows that
Next choose
AEE.
set)
such that
=0
a.e. for
This can be done as follows:
E > O. such that
Now define
~(A) = 1 on [-
SeA)
Let
f. I];
and
= sgn
A
be a real. symmetric.
~(A)
[-
t-
E.t + E]. for some
such functions are known to exist.
o
= ~(A-W- 2)
-
~(A+W
0
+ 2) .
such that
SeA)
= sgn
A for
AEE.
It is then clear
s(t)EL1 (m). ts(t)EL 1 (m). and it follows from (5). since
E and
SeA)
is a real. odd. infinitely differentiable function with support
[-(W+O+E). (W+O+E)]
on
a.e. on
S by
SeA)
that
=0
AEE.
s(t)ELl(m). ts(t)ELl(m)
infinitely differentiable function with support
Then
G2 (W+o-ou)
is complete
and thus
G2 (A)
for
{uk}~=O
Gl • Hn are even. that for n
W+o
I G1 (A) [Hn(A)
or equivalently
= 1.2 •...
- l]'dA
W
w+o
f
w
Gl(A)H~(A)dA = 0
.
=0
GZ
=0
a.e.
6
As for
G2, it now follows that
Thus
G(A)
=0
a.e. on
E and hence
=0
G(A)
a.e. for
IAI > W, i.e.
g€CB(W).
(b)
Conversely, let
f€B(W,o)
(a)
for all
= f(O)
f(t)
0 > O.
tg(t)
+
where
Indeed, it is clear that
g€CB(W).
We show that
f€L2(~)
and as in part
we have
co
(f*h)(t)
= f(O)
co
f h(T)dT f (t-T)g(t-T)h(T)dT
+
_00
_CX)
W
= f(O)
+
J G(A)eitA{tH(A)
2;
-l\'
= f(O)
f
2;
+
W
G(A)eitAdA
- iH' (A) }dA
= f(O)
+
tg(t)
= f(t).
0
-W
B(W;o) is in fact independent
It follows from Theorem 1 that the class
of 0
>
0
and we shall therefore denote it by
B(W) "bandlimited to
B(W) and call functions in
W."
Zakai [3] defines the bandwidth WO(f) of a function
smallest
A.
A such that
Note that
WO(f)
[f(t) - f(O)]/t
depends on
bandwidth Wo of the class
f
B(W)
f€B(W) as the
is conventionally bandlimited to
and it is reasonable to define the
by
Wo = sup 1'1 0 (f)
f€B(W)
As a consequence of Theorem 1 we have
from [3] one can only conclude
Wo
S
WO(f)
W + o.
S
W and Wo
= W;
whereas
This determination of the
7
bandwidth of the class
B(W)
results in a sampling representation
(Theorem 2) with slower rates than those in [3].
We note that if fEB(W), then
f
can be extended to the complex plane
CXl
z
=t
+
ia via (~as in [3], i.e.
= !f(T)h(z-T)dT,
fez)
and
fez)
is then
_00
entire.
Theorem 1 offers an alternative method of extension, i.e.,
= f(O)
fez)
+
W
f G(A)eiAzdA
~
2n
-w
from which an exponential bound for
independent of
fez)
can be easily obtained which is
0 in contrast to the bound obtained in [3] via the convol-
ution integral.
A direct consequence of the characterization of
B(W)
provided in
Theorem 1 is the following sampling theorem whose proof is carried out as
in [3] and is thus omitted.
THEOREM 2.
For all
fEB(W)
and 0
<
T<
n
W
we
have
00
fez)
=
t f(nT) sin[(n/T)(z-nT)]
(niT) (z-nT)
n=-oo
L
and the convergence is uniform in any bounded region of the z-plane.
Another consequence of TIleorem 1 is that if
duced via a convolution integral
f
=f
*
~,
fEB(W)
then
f
is repro-
for a variety of kernels
whose Fourier transforms are essentially arbitrary outside [-W,W].
we have the following characterization of such kernels:
class of all complex-valued functions
~
In fact
Let;oK be the
on the real line such that
~
8
These are the weakest properties of
with
$ such that the convolution
f * $
fEB(W) be well defined.
THEOREM 3.
Let
Then
$EK.
the Fourier transform
of
~
(a)
Theorem 1,
Assume that
= f(O)
f(t)
+
=1
I
= f(O)
for
and
$EK
tg(t)
fEB(W) if and only if
AE[-W,W] .
=f
f
where
00
(f*$)(t)
* $ for all
$ satisfies
~(A)
Proof.
=f
f
* $ for all
gECB(W)
I
00
J g(t-t)$(t)dt
t
+
By
and thus
00
$(t)dT
fEB(W).
-
g(t-t)t$(t)dt .
_00
-00
All integrals are well defined; the first since
$ELl(m)~
the second is an
L2 (m) function as a convolution of gEL 2 (m) with ¢ELl(m), and the third
is either a function in L2(~) as a convolution of gEL2 (m) with
or
a convolution of
have
(f*$)(t)
gEL 2 (m)
and
t$(t)EL 2 (m).
W
= f(O)~(O)
+
2; I G(A)eiAt{t~*(A)
-
We then
i[~'(A)]*}dA
.
-w
Since
(f*$)(t)
= f(t) = f(O)
+
t
--2
'IT
f(O)
(6)
for all
t
[l-~(O)]
and all
G(A)eitAdA, we finally have
W
+
2;
IG(A)eitA{t[l-~*(A)] + i[~'(A)]*}dA = 0
-w
fEB(W), i.e., for all complex numbers
For
G = 0 we obtain
in (6) is equal to 0 for all
for all
JW
-w
t
and all
= 1.
~(O)
Then the second term
GEL ([-W,W],m).
2
f(O) and
It follows that
t
t[l-~ * (A)]
+ i[~'(A)] *
=0
a.e. on [-W,W]
9
for
=1
~(A)
and thus
a.e. on [-W,W].
~
Since
~(A)
is continuous,
=1
A€ [-W, W] .
(b)
The sufficiency follows from the fact that for
~€K
and
f€B(W),
f - f*4l
is given by the left hand side of (6), and the latter is equal to
o
~(A)
when
o
= 1 for A€[-W,W].
The method of the proof of Theorem 1 suggests that kernels other than
(1) can be used to define the class
property that
H must satisfy
behavior of H outside [-W,W]
is
B(W).
H(A)
=1
In the following
Denote by H the class of functions
defined as the inverse Fourier transform of real, symmetric, twice con-
tinuous1y differentiable functions
a
on [-W,W], and that the
is essentially arbitrary.
theorem we give such a class of kernels.
h
It appears that the basic
> 0,
and such that
=1
H(A)
H with support [-W;O, W+o], for some
for
A€[-W,W].
Then
W+o
h(z)
= -l
I
21T
H(A)eiAZdA
z = t+ia,
-(W+d)
is entire and for some finite constant
C,
(W+o)lal
(7)
Ih(z)1
:'5:
C _e_---::::--_
2
1+t
This is seen as follows.
Integrating by parts we have
(-iz)2h (z)
= 2;
W+o
I
H(2)(A)e
-(W+O)
and hence
iAt
dA
10
W+o
2
(1+lzI )lh(z)! s e(w+o)lcr l {2;
J
(IH(X)I + IH(2)(x)l)dX}.
-(W+O)
For
h€H
define the class
Then the class
B(Wjh)
by
B(Wjh) has properties similar to those of the class
B(W) and under an additional assumption on
THEOREM 4.
then
B(W;h)
Proof.
If h€H
h
we have the following
and H(X) is strictly decreasing on [W,W+o],
= B(W).
(a)
We first show that
B(Wjh)
c
B(W).
Let
[3, Lemma 2 and Theorem 1], it follows from (7) that
where
h
g€CB(W+o).
In view of the definition of
is real, it suffices to consider real
f
= f(O)
f(t)
B(W;h)
= 1,2
+ tg(t)
and the fact that
and prove that
f€B(W).
Now as in part (a) of the proof of Theorem 1 we obtain that for n
and i
As in
f€B(W;h).
= 1,2, ...
W+o
J
Gi(X) [Hn(A)]'dA
=0
W
G = G + iG Z is the Fourier transform of G. Since H(A) is
l
strictly monotone decreasing on [W,w+o] with H(W) = 1 and H(W+o) =.0,
where
the change of variable
u = H(A)
gives that for i = 1,2, and
n = 1,2, ...
1
J Gi[H-l(u)]un-ldu = 0
Note that
o
flG~[H-1(u)]du = - 1+oG~(X)H'(X)dX
O 1
W 1
ous and hence bounded on [W,W+o], and
< w
.
since
H'
Gi €L ([-W-o, W+o],m).
2
is continuHence
11
G [H- 1 (U)]EL2 ([0,1],m), and since the set {uk}~=o is complete in
i
-1
L2 ([O,1],m), it follows that Gi[H (u)] = a.e. on [0,1] and thus
G.(A)
=
1
° a.e.
on [W,W+o].
gECH(W)anl,i hence
(b)
Thus
G=
°
°
a.e. on [-W-o,-W] u [W,W+o],
fEH(W).
The inclusion
B(W) c 8(W;h)
is proved as in part (b) of the proof
0
of Theorem 1.
The class
h
H of functions used in Theorem 4 does not include the function
given by (la) which was originally used in defining the class
8(W).
However, the essential property needed for the proof of Theorem 4 is the
H satisfy.
inequality (7) which both (la) and functions in
Therefore
it should be pointed out that Theorem 4 is valid with the class
by the class
H'
of functions
h
form of a real symmetric function
H(A) = 1 for
AE[-W,W]
defined as the inverse Fourier transH having bounded derivative, with
and such that the inequality (7) is satisfied.
It was noted earlier that if
fEL 2 (m)
and
fEE:;B(W)
The question arises whether the only functions in
are those in CB(W).
of B(W)
sentation of Theorem 1.
Also
CB(W)
(ii)
Let
(i)
tg(t), where gECH(W)
if for some
fEB(W).
that are in
CH(W)
L2 (m)
as a subset
in terms of the repre-
These questions are answered in the following
If
fEL 2 (m),
is dense in
fEH(W)
fECH(W)
then
B(W)
It is also of interest to study
and to characterize functions
THEOREM 5.
H replaced
H(W)
then
f€H(W)
if and only if
with respect to the metric of
fECH(W).
L2(~)'
have the representation of Theorem 1, f(t) = f(O) +
with Fourier transform G.
~EL2([-W,W],m)
Then
fECH(W)
if and only
lZ
f
>.
ep(u)du
a.e. on (-W,O)
= -w
W
f 4>(u)du
G(>.)
a.e. on (O,W)
A
and
frO) =
2~i J
O(u)du.
-VI
Proof.
(i)
Let
F(A)H(>') a.e.
(1).
Hence
for all
where
f€B(W)
0 > 0
It follows that
fELZ(m).
Then
f
= f*h
is equivalent to
F is the Fourier transform of
is equivalent to
and thus to
F(A)
=0
FCA)
and
f~CB(W).
f€B(W)
Then
and H is given by
a.e. outside [-(W+o),W+o]
= 0 a.e. outside [-W,W], i.e., fECB(W).
Next we will show that the closure of CB(W)
f€B(W)
=
= CB(W).
B(W) n LZ(m)
It suffices to show that
f
F(A)
f~q
and
f~CB(W)
for all
in
LZ(~)
implies
f
is equal to B(W).
= O.
Assume that
q€CB(W), i.e.,
00
o = J f(t)q*(t)
_00
Since
for all
q(t)
=-! /WQ(A)eitAdA,
21T -W
QELZ([-W,W],m)
it follows from Fubini' s theorem that
we have
W
o = -!
21T
dt z
l+t
00
J Q* (A) ( Jr
-w
f(t)e -itA~) dA .
1+t2J
_00
f(~) € L1(m) n LZ(m); hence the function inside the
l+t
parentheses is continuous and in L2 (m) and it follows that
implies
co
=0
f f(t)e-itA~
1+t 2
-co
for all
A€[-W,W].
13
Since
= f(O)
f€B(W), then by Theorem 1, f(t)
with Fourier transform
G.
Since
t
1+t
2
J ~
_00
1+t 2
g(t)e-itAdt
-i~
g€CB(W)
A€[-W,W]
l+t
with Fourier transform
---- €
where
00
1_ e-itAdt +
_00
tg(t)
It follows that for all
00
f(O)I
+
= °.
sgn A e- 1A1 , it follows
by Parseval's theorem that
w
I e-1A-u1sgn(A-u)G(u)du = -i2~f(0)e-IAI,
Ad-W,W].
-w
This integral equation can be written as
A
e- A
f
W
eUG(u)du - e A
-W
where we have set
I e-UG(u)du = ce- 1A1
,
A
c
= -2i~f(0).
Since all functions are absolutely con-
tinuous we obtain by differentiation
A
_e-
W
A
IeuG(U)dU - e
-W
A
J e -uG(u)du +
2G(A)
A
a.e. A€{-W,W].
It follows that on each of the intervals (-W,D) and (O,W),
G is a.e.
equal to an absolutely continuous function, and hence we can consider a
modification of G which is absolutely continuous on (-W,O) and (O,W),
and thus a.e. differentiable.
I
By adding the last two equations we have
W
_2e A
A
or equivalently
e-UG(u)du
+
2G(A)
= ce-1A1(1-sgn
A)
a.e. A€[-W,W]
14
W
I e-UG(u)du + e-AG(A)
={ ~ ,
A > 0
a.e.
A< 0
A€[-W,W].
A
By differentiation we have
Thus
G'
=0
a.e. on [-W,W], and since it is absolutely continuous on (-W,O)
and (O,W), it follows that
G(A)
= { ba
Now substituting G in the original integral equation we find (after
some calculations)
[a
+
(e-W_l)b_c]e A - (ae-W)e- A = 0
(be-W)e A + [a(l-e-W)-b-c]e- A = 0
Hence
a
= 0,
and therefore
(ii)
tg(t)
= 0,
b
f
c
= O.
- f(O)
for all
G = 0, hence
t
= it
2~
lAeitUdu
we have that for all t ~ 0
W
G(A)eitAdA
-w
g
= 0,
and f(O)
F.
Then
can be written as
I G(A)eitAdA = -! I F(A) (eitA_l)dt
J
A€(O,W) .
W
-w
Using eitA_l
for
f€CB(W) with Fourier transform
W
-!
2~
A€(-W,O)
= 0, which completes the proof.
First assume that
= f(t)
It follows that
for
= it
.
-w
!:eitUX(O,A) (u)du
and Fubini's theorem
w w
= i I ( J F(A)X(O,A) (U)dA)eitudu
-w -w
=0
IS
Since both integrals are continuous functions in
all
t
and thus
t, equality holds for
W
G(A)
= i f F(u) X(O,u) (A)du
a.e. on [-W,W]
-W
from which the desired expression for
have
w
G follows with
~
= iF. We also
W
I
J F(u)du = f(O).
~
~(u)du = -l
2~
2~1
-w
-w
Conversely, assume that fEB(N)
with
f(O) and g
(i.e.G)
satisfying
the expressions in part (ii) of the theorem. Then, as before,
W
G(A) = f ~(u)X(O,u)(A)du a.e. on [-W,W] and
-W
W
W
tg(t)
= -!
f
2~
G(A)eitAdA
=~
f ~(u)(eiut_l)du
2~1
-W
-W
W
and finally
f(t)
= f(O)
+ tg(t) =2;i!
~(u)e
iut
du,
-w
and thus
0
f€CB(W).
It should be remarked that
in
L2(~)'
~
SeW) , the orthogonal complement of B(W)
can be characterized as follows:
complement of CB(W)
in
Let
CB(W)~ be the orthogonal
L2 (m); then it can be seen from the fact that
CB(W) is dense in B(W) (Theorem 5 i)
that f€B(W)~ if and only if
f(t)
~
f(t)
~~2~ € CB(W) , i.e., if and only if the
L2-Fourier transform of ----2
l+t
l+t
vanishes on [-W,W].
16
III.
Let
X = {X(t,w),
Bandlimited Stochastic Processes
-~ <
t
<~}
be a second order, mean square continuous
(~,F,P)
stochastic process on a probability space
R(t,s).
Without loss of generality the process
X may be assumed to be
Accordingi,to Zakai I s definition [3J and the remark following
measurable.
TIleorem 1,
with correlation function
X is called "bandlimited to W" if
(8)
f
-~
R(t,t) dt
1+t 2
<
~
and with probability one its sample functions are bandlimited to
(X(o,w)€B(W) a.s.).
W
It is shown in [3J that wide sense stationary and
harmonizah1e processes that are "conventionally bandlimited to W," Le.,
and [-W,W]x[-W,W]
whose spectral measure is concentrated on [-W,W]
respectively, are band1imited to
W.
The characterization of processes bandlimited to
W [3, Theorem 5] and
their sampling representation [3, p. 154] can be stated as follows in view
of Theorems 1 and 2.
THEOREM 6.
(i)
A second order process
only if its correlation function
R(t,o)
(ii)
is bandlimited to
t.
Wand
o
<
T
<
1T
IV'
then with probability
one
~
X(t,w)
for all
t.
= I
W if and
R is continuous, satisfies (8) and
W for all
If X is bandlimited to
X is band1imited to
X(nT,w) sin[(1T/T)(t-nT)]
(1T/T) (t-nT)
n=-~
17
Part (i) of Theorem 6 characterizes the correlation function of bandlimited processes.
A characterization in terms of a representation of the
process itself, similar to that of Theorem I, can be obtained as follows.
Zakai proves in [3, Theorem 6] that bandlimited processes have properties
similar to those stated in Lemma 1 for bandlimited functions.
Using
Theorem 1, these properties can be strengthened to the following
THEOREM 7.
X is bandlimited to
X(t,w)
for all
t
and almost all
ventionally bandlimited to
W if and only if
= X(O,w)
+
w , where
tY(t,w)
Y is a harmonizable process con-
W with correlation function
Ry such that
00
J Ry(t,t)dt
<
00.
_00
Finally it is shown that the concepts of "conventionally bandlimited"
and "bandlimited" process coincide for the class of wide sense stationary
and harmonizable processes; also the conventionally bandlimited processes
are characterized in terms of the representation of Theorem 7.
THEOREM 8.
(i)
Let
X be a harmonizable or a mean square continuous,
wide sense stationary process.
Then
X is bandlimited to
if it is conventionally bandlimited to
(ii)
X(t)
Let
= X(O)
+
W if and only
W.
X be bandlimited to
Wand have the representation
tY(t)
Then
of Theorem 7.
ventionally bandlimited to
X is wide sense stationary con-
W if and only if for some finite measure
on the Borel subsets of [-W,W],
~
18
W
E[Y(t)Y* (s)] =
I-w
W
E[Y (t) X*(0)] =
E[I X(O)
and then
lJ
1
2
I-w
e itA - 1 e
t
-isA
-1 dlJ(A)
s
e itA - 1
dlJ(A)
t
] = lJ{[ -W, w]}
is the spectral measure of X.
cient conditions can be expressed for
bandlimited to
Similar necessary and suffi-
X to be harrnonizable conventionally
W.
Note that the first two expressions in (ii) of Theorem 8 can be written
in an equivalent form exhibiting the harmonizable character of Y, i.e.,
w
II
-w
=I
E[Y(t)Y * (s)] =
E[Y(t)X * (0)]
ei(tT-SO)¢(T,o)dTdo
W•
e
1tL
1jJ(T)dT
-w
where
~
and
Ware given by
HT,O)
w("r)
Proof.
(when
(i)
=
=
{
lJ{ [max(T, 0), W]}
lJ{[-W,min(T,o))}
0
for 0
T,
:S
0
:S
W
for -w ~ T, 0 < 0
for -w ~ T < 0 ~ 0
and -W :s 0 < 0 ~ T
lJ{[T, W]}
lJ{ [-W,T)}
U
We will give the proof for
for
for
o~
-w
T
~
W
W
W
~
T
~
~
<
0
X a harmonizable process
X is stationary the proof is even simpler).
Then
19
00
R(t,s)
= If ei(tu-SV)d~(u,v)
_00
where
~
is the two-dimensional spectral measure of X
(~
is a finite,
complex measure, nonnegative definite on Borel measurable rectangles).
Assume first that
X is bandlimited to W.
that
for all
R(t,o)€B(W)
R(o,t)€B(W).
t, and since
Hence for all
it follows
From Theorem 6 i
R(t,s)
= R* (s,t) then
t,s,
II
00
R(t,s)
=
R(u,v)h(t-u)h(s-v)dudv
_00
where
h is given by (la).
Replacing
R on the right hand side by its
spectral representation and using Fubini's theorem, we obtain
II ei(tT-SO)H(T)H(o)d~(T,o)
00
R(t,s)
=
_00
= H(T;W,O)
where
li(T)
o>
and limo-+O H(T;W,O) = X[_W,W](T), it follows from the bounded converg-
0
is given by
(1).
Since this is true for all
ence theorem that
II
WW
R(t,s)
and thus
=
ei(tT-SO)d~(T,o)
-w -w
X is conventionally bandlimited to
conventionally bandlimited to
{R(t,o) * h(o)}(s)
hence
W.
Conversely, if X is
W then it is easily checked that
R(t,s)
and (8) follows from IR(t,t)1 ~ ~{[-W,W] x [-W,W]} ;
X is bandlimited.
The proof of (ii) is straightforward and is thus omitted.
0
=
20
References
[1]
S. Bochner and K. Chandrasekharen, Fourier Transforms, Princeton
University Press, Princeton, N.J., 1949.
[2]
P.J. Davis, Interpolation and Approximation, Blaisdell Publishing
Co., New York, 1963.
[3]
M. Zakai, Band-limited functions and the sampling theorem, Information
and Control, 8 (1965), pp. 143-158.