Kramer’s generalized sampling of stochastic
processes
Sinuk Kang
Division of Mathematics and Informational Statistics
College of Natural Sciences, Wonkwang University, South Korea
Email: skang@wku.ac.kr
Abstract—Several sampling theorems for deterministic signals
are known to have their counterparts for stochastic signals.
Motivated by Kramer’s result on generalized sampling of bandlimited deterministic signals, we obtain generalized sampling
expansions of a certain class of stochastic processes which are
not necessarily wide sense stationary.
I. I NTRODUCTION
Sampling and reconstruction procedure of deterministic
and/or stochastic signals are fundamental and important issues
in signal analysis. As is well known, for deterministic signals
the Whittaker-Shannon-Kotel’nikov (WSK) sampling theorem
[9] has played a role as the backbone of the subject. Since the
WSK sampling theorem, much development has been achieved
in theoretic and practical viewpoint. We refer to the survey
article [11] by Jerri and references therein for details.
The WSK sampling theorem states that for any signal f with
bounded spectrum (or band) [−πω, πω](ω > 0) is uniquely
determined by its equidistant samples taken at the rate ω̃ no
less than the Nyquist rate ω, and can be reconstructed via
ωX n
n
f (t) =
f ( )sincω(t − )
ω̃
ω̃
ω̃
n∈Z
sin πt
πt .
where sinct :=
The Fourier transform (FT) maps the
ˆ
signal f (t) into f (λ) in L2 [−πω, πω], the space of square
integrable functions on [−πω, πω]. A key observation is that
point sample f ( ω̃n ) is the n-th coefficient of some orthogonal
basis expansion of fˆ in L2 [−π ω̃, π ω̃]: the orthogonal basis is
given by the Fourier kernel eitλ and hence the orthogonal basis
expansion corresponds to the Fourier series of fˆ on [−π ω̃, π ω̃].
To extend the WSK sampling theorem Kramer [13] used a
itλ
general kernel K(t, λ) instead of the Fourier
R kernel e . He
showed that for any f (t) satisfying f (t) = I g(λ)K(t, λ)dλ,
t ∈ I, for some interval I and for g ∈ L2 (I) if there exist
{tn : n ∈ Z} such that {K(tn , λ) : n ∈ Z} forms a complete
orthogonal set on L2 (I) then
X
f (t) =
f (tn )sn (t)
n∈Z
where
R
sn (t) =
I
K(t, λ)K(tn , λ)dλ
R
.
|K(tn , λ)|2 dλ
I
c
978-1-4673-7353-1/15/$31.00 2015
IEEE
Compared to sampling expansions of deterministic signals,
those of stochastic signals seem to have been less investigated.
Balakrishnan [1] first addressed a stochastic version of the
WSK sampling theorem for wide sense stationary (WSS)
stochastic processes for which the spectral representation is
well established [4]. The WSK sampling theorem for WSS
stochastic processes is further developed, e.g., by Beutler [2],
Boche and Mönich [3], Houdré [10] and Lloyd [17]. Lee
[14] and Zakai [26] extended the earlier sampling theorem
by introducing a notion of bounded spectrum in wider sense.
Gardner [8] obtained the WSK sampling theorem for nonstationary stochastic processes. Taking into account a general
integral representation [24] of stochastic processes, Lee [15]
and Piranashvili [23] provided sampling expansions for nonstationary stochastic processes with certain integral representations.
In this paper we establish Kramer’s sampling for stochastic
processes.
II. N OTATIONS AND DEFINITIONS
WeR take the FT as F[f ](λ)
=
fˆ(λ)
:=
∞
−itλ
1
2
f
(t)e
dt
for
f
(t)
∈
L
(R)
∩
L
(R)
with
−∞
R∞
the inverse FT F −1 [fˆ](t) = f (t) = √1
fˆ(λ)eitλ dλ.
√1
2π
2π
−∞
As a generalization of the Fourier transform(FT) F, the
fractional Fourier transform(FrFT) is introduced in 1980s [19],
[20]. For any θ ∈ R, we let
Z ∞
ˆ
Fθ [f ](λ) = fθ (λ) :=
f (t)Uθ (t, λ)dt
−∞
2
be the FrFT of f ∈ L (R) with respect to θ, where

if θ = 2πn, n ∈ Z
 δ(t − λ)
δ(t + λ)
if θ + π = 2πn, n ∈ Z
Uθ (t, λ) =
2
2

c(θ)eia(θ)(t +λ )−ib(θ)tλ
otherwise,
q
cot θ
, a(θ) =
is the transformation kernel with c(θ) = 1−i2π
cot θ
,
and
b(θ)
=
csc
θ.
The
inverse
FrFT
with
respect
to θ is
2
defined by the FrFT with respect to −θ, that is,
Z ∞
f (t) =
fˆθ (λ)U−θ (t, λ)dλ.
−∞
Note that the FrFT with respect to
i.e., F π2 = F.
π
2
corresponds to the FT,
A sequence {φn : n ∈ Z} of vectors in a separable Hilbert
space H is
• a frame of H with bounds (A, B) if there are constants
B ≥ A > 0 such that
X
Akf k2H ≤
|hf, φn iH |2 ≤ Bkf k2H , f ∈ H;
where
sn (t) = hK(t, ·), K̃n (·)iL2 (I)
(4)
where {K̃n (λ) : n ∈ Z} is a dual frame of {K(tn , λ) : n ∈
Z}. Especially, if {K(tn , λ) : n ∈ Z} is an orthogonal basis
of L2 (I) then
n∈Z
K̃n (λ) =
K(tn , λ)
kK(tn , ·)k2L2 (I)
a Riesz (or stable) basis of H with bounds (A, B) if
{φn : n ∈ Z} is complete in H and there are constants
which result in the same synthesis function as in [13].
B ≥ A > 0 such that
X
2
Theorem III.1. Let x(t), −∞ < t < ∞, be a stochastic
Akck2 ≤ c(n)φn ≤ Bkck2 , c := {c(n)}n ∈ `2 (Z)process which can be represented as (1). If Φ is weakly
H
n∈Z
absolutely continuous, then for any t ∈ R
P
X
2
2
where kck = n∈Z |c(n)| .
x(t) =
x(tn )sn (t)
We denote by E(·) the expectation of a given probability
n∈Z
space. A stochastic process x(t), −∞ < t < ∞, is WSS if
which converges in mean square and with probability 1, where
E(x(t)) = 0 and E|x(t)|2 < ∞ for t ∈ R and E(x(t)x(t0 ))
tn satisfies (2) and sn (t) is given by (4).
0
depends only on the difference t − t . It is well known [4]
that any WSS stochastic process x(t), −∞ < t < ∞, has the
When K(t, λ) = eitλ and I = [−π, π], the frame expansion
spectral representation
(3) is reduced to the conventional Fourier or the nonharmonic
Z ∞
Fourier series expansion of eitλ on [−π, π], depending on tn ,
itλ
x(t) =
e Φ(dλ)
n ∈ Z. Since eitλ is infinitely many differentiable on [−π, π],
−∞
(3) converges uniformly on (−π, π). Thus sufficient conditions
where the process Φ has orthogonal increments and of Example III.3 and Example III.4 below can be relaxed
E|Φ(dλ)|2 = F (dλ). Here F is the spectral measure (or into those as in Theorem 1 of [2] and Theorem 3.3 of [10],
spectral distribution function) of x.
respectively.
n
Since { √ 1
U−θ ( ω csc
θ , λ) : n ∈ Z} is an orthonormal
ω| csc θ|
III. K RAMER ’ S GENERALIZED SAMPLING
2
basis of L [−πω, πω] for ω > 0 [22], we have a stochastic
Let I be a bounded and closed interval in R. Φ is said to
analogue of the sampling theorem in FrFT domain [25]:
have support I if it vanishes on all Lebesgue measurable sets
outside of I. Consider a complex-valued stochastic process Corollary III.2. For any θ ∈ R and ω > 0 let x(t), −∞ <
t < ∞, be a stochastic process which can be represented in
x(t) which can be represented in the form
Z ∞
Z
the form
Z πω
x(t) =
K(t, λ)Φ(dλ) = K(t, λ)Φ(dλ), t ∈ R (1)
x(t) =
Uθ (t, λ)Φ(dλ).
•
−∞
I
−πω
where Φ is of bounded variation and has support I, and
K(t, λ) is a complex-valued measurable function on R × I
such that
2
• K(t, ·) ∈ L (I) for any t ∈ R;
• there is a countable set E for which
{K(tn , ·) : tn ∈ E} forms a frame of L2 (I).
(2)
The standard example of such K(t, λ) is the Fourier kernel
eitλ . Precisely, {K( 2n
|I| , λ) : n ∈ Z} is an orthogonal basis
2
of L (I), where |I| denotes the length of the interval I, and
{K(tn , λ) : n ∈ Z} is a frame of L2 [−π, π] provided that
supn∈Z |tn − n| < 14 [16]. x(t) of the form (1) with K(t, λ)
being the Fourier kernel is said to be harmonisable. In this
case we shall say that x(t) has spectrum I if Φ has support I.
The transform kernel of FrFT gives a nonstandard example.
See Corollary III.2 below.
As a frame expansion of K(t, λ) in L2 (I), we have for any
t∈R
X
K(t, λ) =
sn (t)K(tn , λ)
(3)
n∈Z
If Φ is weakly absolutely continuous then for any t ∈ R
x(t) =
∞
X
x(
n=1
n
)sn (t)
ω csc θ
which converges in mean square and with probability 1, where
sn (t) := e−i
2
2
cot θ
n
2 (t −( ω csc θ ) )
sinc(ωt csc θ − n).
(5)
Example III.3. Let x(t), −∞ < t < ∞, be a harmonisable
process with bounded spectrum [πω1 , πω2 ](ω1 < ω2 ), i.e.,
Z πω2
x(t) =
eitλ Φ(dλ).
πω1
If Φ is weakly absolutely continuous then for any t ∈ R
X
2n
2n
x(t) =
x(
)s(t −
)
(6)
ω2 − ω1
ω2 − ω1
n∈R
which converges in mean square and with probability 1, where
s(t) :=
eiπω2 t − eiπω1 t
.
iπ(ω2 − ω1 )t
(7)
Especially, when ω2 = −ω1 = ω we have s(t) = sinc(ωt).
When θ = π2 , Corollary III.2 reduces to Example III.3 with
ω2 = −ω1 = ω.
If the process x(t) is real-valued then its spectrum is
symmetric interval, i.e., ω1 = −ω2 . The converse is also true.
Sampling expansion (6) can also be derived from the stochastic
ω2 +ω1
version of the WSK sampling theorem since e−itπ 2 x(t)
1
1
, π ω2 −ω
].
is of spectrum [−π ω2 −ω
2
2
It is well known [16] that if supn∈Z |t − tn | < 41 then
{eitn λ : n ∈ Z} is a frame of L2 [−π, π] and its dual frame
{hn (λ) : n ∈ Z} is given by
Z π
1
hn (λ)eitλ dλ = Hn (t)
2π −π
where
Hn (t) =
with
G(t) := (t − t0 )
G0 (t
∞
Y
G(t)
n )(t − tn )
(1 −
n=1
t
t
)(1 −
).
tn
t−n
Moreover, the ordinary and the nonharmonic Fourier series are
uniformly equiconvergent over (−π, π).
Example III.4. Let the notation and the assumption be the
same as Example III.3 with ω2 = −ω1 = 1. If Φ is weakly
absolutely continuous and supn∈Z |t − tn | < 41 then for any
t∈R
X
x(t) = 2π
x(tn )Hn (t)
n∈R
which converges in mean square and with probability 1.
IV. C ONCLUSION
We present Kramer’s generalised sampling expansion for
stochastic processes. Motivated by Kramer’s generalisation of
WSK sampling theorem, we obtain a similar sampling expansion for stochastic processes which have a certain integral
representation.
ACKNOWLEDGMENT
This work is partially supported by Wonkwang University
Research Grant 2015. The author thanks the referees for many
valuable comments, which improve the paper.
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