A Note on the Campbell Sampling Theorem
Alan J. Lee
University of Auckland
and
University of North Carolina at Chapel Hill
.
ABSTRACT.
Campbell's 1968 sampling theorem is examined and a more explicit
formula for the truncation error is given.
The result is shown to apply to
random processes bandlimited in a general sense.
§1.
Many versions of the Shannon sampling series have been
INTRODUCTION.
proposed; see e.g. [IJ for a recent review.
«
In 1968 Campbell [2J proposed
the sampling series
00
f(t)
(1)
L f(nh)
=
Sin'rrh- 1 Ct-nh)
n=-oo
valid for
a
COO
-1
TIh
(t-nh)
h -1 > 2w , S < h -1 /2 - w where
function
t
supported by
[-I,IJ
_OO<t<OO
t(S(t-nh))
is the Fourier transform of
with
ftdx
= 1
and
f
is any
function whose fourier transform (in the distriblltional sense) is g1rpported
by
f-w,wJ.
The series (1) is also valid for stationary random processes.
Campbell
also gives an expression for the truncation error committed in using
terms of (1) to evaluate f.
2N
+
In this note we give a more explicit form of
the truncation error and show that the sampling theorem is valid for a
general type of bandlimited random process, giving an expression for the
truncation error in this case also.
The work of this author was partially supported by the Air Force Office of
Scientific Research under Contract AFOSR-75-2796.
1
2
§2.
BANDLIMITED FUNCTIONS.
2
= Jlf(t)1 (1+t 2 )-kdt < 00
I/fl/~
form of
f
is supported by
Call a function
for some
[-w,w].
~ a
k
f
bandlimited to w
if
and i f the Fourier trans-
The following form of the Paley-Wiener
theorem will be useful.
Theorem.
I/fl/
<
k
For all real
C (w) ::: 4 (W + 1) K for constants
k
k
Proof.
Let
Then if
F
~
be any
00
C
w satisfying
Now the distribution
on the space
F
E of all
defined below.
f,
!.:
2
A
2
}
2 k }~
JIf6(u)1 2 (l+u)
du
A
{
can be extended to a continuous linear functional
00
C
functions topologized by the usual family of
semi norms (see e.g. [3] p 88) and if
then for all
K
k
: : {J Ifeu) I2 (l+u)2 -kdu.! I16 (u) I2 (l+u) kdu
= /If/l k
~(x)
bandlimited to
function with compact support, i.e. a testfunction.
is the Fourier transform of
IF(~)I = I!f(u)¢(u)dul
[- w, w]
f
satisfies
00
where
t, a function
~ E E
X is a testfunction equal to 1 on
F (~) = F (~X), so in particular, setting
= e 21Titx we obtain
Now let
y(x)
= {K e:p(l/ex2-11)
Ixl ::: 1
Ixl > 1
where
and
-1
K
y(x)
=
exp(1/(x 2-1))dx
!
Ixl:::1
= 2.2523
is a testfunction supported by
then
[-1,1].
!y(x)dx
=1
Also let for
6
>
0
3
lex) ={01
w+o
Then choose for
X
the function
Ixl
:::
w + 0 ,
Ixl
>
W
X(x) =
i-
f'
-00
then
X(x)
is
1
on
[-w,w]
+ 0 .
I (v ) y(X
\'1+0
and is supported by
6V) dv
(-w - 0,
W
+ 0),
and
00
is
A
C .
A(u) =
Then denoting the Fourier transform of a function
fOO
_00
e
-2~iux
A(x)dx
A
by
we have
X(u) = y(uo) Sin 2~ (w+6)u
and so
~u
I f(t) I
•
~
Ilfll k{[ 1,,(110) 12
[Sin 2~~(:~UU-t)r(l+u2)kdU}"
=
IIfll k{[ !,,(uo)!2 {Sin;~ (w+O)u} (1+ (nH) 2)kdU}"
::: Ilfll (1+ I t
k
Then (2) is true
Il{I I y(uo) 12 Sin22~ (~+o)u
(~u)
2
Ck(w)
with
2 . 2
.
II"
= lnf
. y(uo) 1 Sln
0>0
For an upper bound on
2
C (w),
k
2
(1+ lu 1 ) k dU}!z
consider setting
2
2~(w+o)u
2 (1+ I u 1 ) k du .
(TIu)
0
= 1,
then since
I Si~ x I - 1+l2x l
<
C~(W) ::: 16(w+1)2Ily(u)12(1+2~(w+1)lulr2(1+luI2l
::: 16(w+1)2Ily(u)1 2 (1+u 2 )k-l du
= 16(W+1)2K~
(3)
say, and hence
4
The constant
K
k
can be calculated from
2
k-1
(k-1)
I
yU) (x)
= I . f
. dx.
1
j=O
The derivatives of
J
(27T)J
can be generated by a simple recursive scheme
y
described in [2J, a numerical integration then allows the calculation
of the
K , given below in Table 1 for
k
Table 1.
k
= 1,2,3,4,5 .
Values of K to 4 significant figures.
k
K
k
k
8.217
2
1. 003 x 10°
3
1. 649 x 10°
4
8.446
5
1. 252 x 10
We note in passing that for
k
=
= I f e27Titx~(X)dXI ~
-w
w
{
=
2
-w
a with Co(w) =
!;z
f 1dXf
(2w)
Il(x) 12 dX}
_00
k
k
10°
x
00
= (2w) 211 fll
so (2) is valid for
x
0, the inequality takes the simple form
w
If(tll
10
-1
1
°
k
2
•
For convenience, and foliowing [2J we propose to take for
series (1) the function
for
y
y
defined above.
~
in the
A version of (2) appropriate
Vy
is obtained simply by integrating the f.t. of y(V)(= d ]
by parts, obtaining
l
ax v
5
and so for
UFO
we obtain
ly(u}1 :s
(4)
c lui-v, say, for any integer
v
=
Again the constants
c
o.
may be calculated simply by numerical quadrature.
v
An upper bound for the
v ~
c
may be obtained by the methods of
v
Table 2 below gives the first few values of c :
v
Table 2.
Values of
[2].
c .
v
v
c
a
1.000
v
1
.2637
2
.1822
3
.3237
4
1.5550
5
13.6874
By means of (3) and (4) we can obtain a precise inequality on the
truncation error of (1) in terms of the norm
method of
[2]:
Theorem.
Let
Ilfllk <
co.
f
y
be a function bandlimited to
Then if
theorem 1 and if
IIfllk
y
of
f, using the
wand satisfying
is the function defined in the proof of
PN(t)
denotes the truncation error using the function
= f(t)
-
in (1) i.e. if
.
-1
(t-nh) Y(S(t-nh))
I f(nh) SJ.n7Th-7Th1 (t-nh)
Inl ~N
6
k
-1
then for
v
>
2 C (w) C)fll Sin 1fh t (l+Nh)
k
k
:::------'-~-------V
SV1fINh _ Itll
k
8 (w+l) KkC)fll
k
ISin 1fh-It I (l+nh/
~--------------V
SV1fINh _ Itll
Similar to §4 of [2].
Proof.
§3.
BANDLIMITED PROCESSES.
x(t) , and let R(t,s)
Consider a zero mean second order process
= E(x(t)
xes))
be the covariance function of x(t).
For the properties of such processes see [4].
A key property is the following.
consisting of all functions
operator
Consider the Hilbert space
!
If 1 2
(l+t 2 )
f
satisfying
=
!R(t,s)f(t) (l+t 2 ) -k dt
-kdt .
H
k
Then the
R defined by
R f(s)
H . Let A., f., j = l, 2 , 3, •••
k
J
J
be the eigenvalues and eigenvectors of this operator. Then the following
is a trace-class operator from
H
k
to
results are true (details may be found in [4] and [5])
1.
There exist random variables
e.
J
satisfying
such that
00
(5)
x(t)
= L f.(t)e.
j =1 J
J
1
the convergence of (5) being in mean square;
E(e
e.) = 0 ..
J
JJ
A.
J
7
2.
Each function
f.
satisfies
J
Ilf.ll
= 1
J k
and is bandlimited to
.
w;
00
3.
For each
t, s
R(t,s) =
I
A.f.(t)f.(s),
j=1 J J
the series converging
J
absolutely.
Using these we may prove the following
Theorem 3.
Let
x(t)
be a zero mean process whose covariance function
fR(t, t) (l+t 2 ) -kdt <
R satisfies
Then
x(t)
00
and that is bandlimi ted to
w.
satisfies the sampling expansion (1) and the root mean square
truncation error
{
E x(t) -
1
I x (nh) Sin 1Th- (t-nh) y(p(t-nh))
1
Inl $N
1Th- (t-nh)
2}~
is bound by
for
k < v,
Proof.
S< h
-1
1
It I
/2 - w, h - /2 > w) and
< Nh .
Using (5) above,
E x(t) _
I x(nh)
In I ~ij. N
00
:::
I
E
f.(t) J
j=1
-1
2
Sin 1Th (t-nh)
1Th - 1 (t - nh )
-1
I
f(nh) Sin1Th (t-nh)
Inl~.N
1Th- 1 (t-nh)
-1
00
=
I
A. f. (t) J
j=1
I
J
I f . (nh)Sin 1Th (t-nh)
1
Inl:;;n J
1Th- (t-nh)
1
00
:;;
Y(S(t-nh))
J
y(s (t-nh))
8(w+1)K C 1If.ll lSin 1Th- tl (1+Nh)k 2
k v J k
A. _
j=l J
2
Y(S(t-nh))e.
2
8
00
Now consider
R(t,t)
=
L A.lf.(t) 12
j=1 J
respect to
(1+t 2 )-k dt
Integrating term by term with
J
we obtain
00
L A.
2
flf.(t)1 (1+t 2 )-kdt
j=1 J
J
00
.
2
= L Lllf.ll k
j =1 J
=L
J
A
j j
For
k < v, we see that the truncation error converges to zero as
N -+
00
,
thus proving the theorem.
The series (1) converges in mean square for a bandlimited process
e
x(t)
~
since the sample paths of
probability
1,
x(t)
are bandlimited to
(see [4]); it follows that
x(t)
w with
satisfies the series
(1) with probability 1 .
REFERENCES
[1]
A. J. Jerri, "The Shannon sampling theorem - its various extensions
and applications: a tutorial review", Proc. IEEE vol 65, pp 1565-1596,
1977.
[2]
L. L. Campbell, "Sampling theorem for the Fourier transform of a
distribution of bounded support", SIAM J. Appl. Math. Vol 16
pp 626-636, 1968.
[3]
W. F. Donoghue, "Distributions and Fourier Transforms" New York,
Academic Press, 1969.
[4]
A. J. Lee, "Characterization of bandlimited functions and processes"
Inform. Contr. vol 31 pp 258-271, 1976.
[5]
S. Cambanis and E. Masry "On the representation of weakly continuous
stochastic processes", Information Sciences Vol 3 pp 277-290, 1971.
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A Note on the Campbell Sampling Theorem
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Mimeo Series No. 1256
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Alan J. Lee
Contract AFOSR-75-2796
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sampling theorem, distributions, band1imited processes
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Campbell's 1968 sampling theorem is examined and a more explicit formula
for the truncation error is given. The result is shown to apply to random
processes bandlimi ted in a general sense.
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